diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjorc" "b/data_all_eng_slimpj/shuffled/split2/finalzzjorc" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjorc" @@ -0,0 +1,5 @@ +{"text":"\n\\section{Demystifying The Declared SDK Versions and Their Two Side Effects}\n\\label{sec:backg}\n\n\nIn this section, we first demystify the declared platform SDK versions in Android apps, and then explain their two side effects if inappropriate \\texttt{DSDK}\\xspace versions are being used.\n\n\n\n\\vspace{-1.5ex}\n\\subsection{Declared SDK Versions in Android Apps}\n\\label{sec:declared}\n\n\\vspace{-2ex}\n\\lstinputlisting[\nlanguage=XML,\nbasicstyle=\\ttfamily\\small,\nlabel={lst:sdkversion},\ncaption={\\small The syntax for declaring the platform SDK versions in Android apps.}\n]{sdkversion.xml}\n\nListing \\ref{lst:sdkversion} illustrates how to declare the supported platform SDK versions in Android apps by defining the \\texttt{} element in apps' manifest files (i.e., \\texttt{AndroidManifest.xml}).\nThese \\texttt{DSDK}\\xspace versions are for the runtime Android system to check apps' compatibility, which is different from the compiling-time SDK for compiling source codes.\nThe value of each \\texttt{DSDK}\\xspace version is an integer, which represents the API level of the corresponding SDK.\nFor example, if a developer wants to declare the SDK version 5.0, he\/she sets its value as 21 (the API level of Android 5.0 is 21).\nSince each API level has a precise mapping of the corresponding SDK version~\\cite{AndroidVersion}, we do not use another term, \\textit{declared API level}, to represent the same meaning of \\texttt{DSDK}\\xspace throughout this paper.\n\n\nWe explain the three \\texttt{DSDK}\\xspace attributes as follows: \n\\begin{itemize}\n \\item The \\texttt{minSdkVersion}\\xspace integer specifies the minimum platform API level required for the app to run. The Android system refuses to install an app if its \\texttt{minSdkVersion}\\xspace value is greater than the system's API level. Note that if an app does not declare this attribute, the system by default assigns the value of ``1'', which means that the app can be installed in all versions of Android.\n\n \\item The \\texttt{targetSdkVersion}\\xspace integer designates the platform API level that the app targets at. An important \\textit{implication} of this attribute is that Android adopts the back-compatible API behaviors of the declared target SDK version, even when an app is running on a higher version of the Android platform. Android makes such compromised design because it aims to guarantee the same app behaviors as developers expect, even when apps run on newer platforms. It is worth noting that if this attribute is not set, the default value equals to the value of \\texttt{minSdkVersion}\\xspace.\n \n\n \\item The \\texttt{maxSdkVersion}\\xspace integer specifies the maximum platform API level on which an app can run. However, this attribute is \\textit{not} recommended and already \\textit{deprecated} since Android 2.1 (API level 7). That said, modern Android no longer checks or enforces this attribute during the app installation or re-validation. The only effect is that Google Play continues to use this attribute as a filter when it presents users a list of applications available for download. Not that if this attribute is not set, it implies no any restriction on the maximum platform API level.\n\\end{itemize}\n\n\n\n\n\n\\subsection{Two Side Effects of Inappropriate DSDK Versions}\n\\label{sec:sideeffect}\n\n\nFig.\\xspace \\ref{fig:demystify} illustrates the two side effects of inappropriate \\texttt{DSDK}\\xspace versions.\nWe first explain the symbols used in this figure, and then describe the two side effects in the subsequent paragraphs. \nAs shown in Fig.\\xspace~\\ref{fig:demystify}, we can obtain $minSDK$, $targetSDK$, and $maxSDK$ from an app manifest file.\nBased on the API calls of an app, we can calculate the minimum and maximum API levels it requires, i.e., $minLevel$ and $maxLevel$. \nEventually, the app will be deployed to a range of Android platforms between $minSDK$ and $maxSDK$.\n\n\n\\begin{figure}[t!]\n\\vspace{-4ex}\n\\begin{adjustbox}{center}\n\\includegraphics[width=0.6\\textwidth]{demystify}\n\\end{adjustbox}\n\\vspace{-4ex}\n\\caption{Illustrating the two side effects of inappropriate \\texttt{DSDK}\\xspace versions.}\n\\vspace{-4ex}\n\\label{fig:demystify}\n\\end{figure}\n\n\n\n\n\\noindent\n\\textbf{Side Effect I: Causing Runtime Crash Bugs}\nThe blue part of Fig.\\xspace \\ref{fig:demystify} shows two scenarios in which inappropriate \\texttt{DSDK}\\xspace versions can cause app crash.\nThe first scenario is $minLevel > minSDK$, which means a new API is introduced after the $minSDK$.\nConsequently, when an app runs on the Android platforms between $minSDK$ and $minLevel$ (marked as the block 1 in Fig.\\xspace \\ref{fig:demystify}), it will crash.\nWe verified this case by using the \\texttt{VpnService.Builder.addDisallowedApplication()} API, which was introduced at Android 5.0 at the API level 21.\nWe called this API at the MopEye app \\cite{MopEyePoster15} and ran MopEye on an Android 4.4 device.\nWhen the app executed the \\texttt{addDisallowedApplication()} API, it crashed with the \\texttt{java.lang.NoSuchMethodError} exception.\n\n\nThe second crash scenario is $maxSDK > maxLevel$, which means an old API is removed at the $maxLevel$.\nSimilar to the first scenario, the app will crash when it runs on the Android platforms between $maxLevel$ and $maxSDK$.\n\n\n\n\n\\noindent\n\\textbf{Side Effect II: Making Apps Less Secure}\nThe red part of Fig.\\xspace \\ref{fig:demystify} shows the scenario in which inappropriate \\texttt{DSDK}\\xspace versions cause apps fail to be patched that they originally should be able to.\nSuppose an app calls an API (e.g., \\texttt{addJavascriptInterface()} \\cite{addJavascriptInterfaceSaga}) that is vulnerable before the $targetSDK$.\nHowever, if the \\texttt{targetSdkVersion} of the app is lower than the patched API level, Android will still take the compatibility behaviors, i.e., the non-patched API behavior in this case, even when the app runs on the patched platforms (between $targetSDK$ and $maxLevel$).\nSome such vulnerable app examples are available in \\url{https:\/\/sites.google.com\/site\/androidrce\/}.\n\n\\section{Conclusion and Future Work}\n\\label{sec:conclude}\n\\vspace{-1.5ex}\n\n\nIn this paper, we made a first effort to systematically study the declared SDK versions in Android apps, a modern software mechanism that has received little attention. \nWe measured the current practice of the declared SDK versions or \\texttt{DSDK}\\xspace versions in a large dataset of apps, and the consistency between the \\texttt{DSDK}\\xspace versions and their app API calls.\nTo facilitate the analysis, we proposed a three-dimensional analysis method that operates at both Google Play, Android document, and Android app levels.\nWe have obtained some interesting and novel findings, including (i) around 17\\% apps do not claim the targeted \\texttt{DSDK}\\xspace versions or declare them wrongly, (ii) around 1.8K apps under-set the minimum \\texttt{DSDK}\\xspace versions, causing them would crash when running on lower Android versions, and (iii) over 400 apps under-claim the targeted \\texttt{DSDK}\\xspace versions, making them potentially exploitable by remote code execution.\nIn the future, we plan to contact the authors of the apps to inform them about the detected issues and collect their feedback, release a publicly available tool to let app developers detect and fix issues, and improve our approach to further mitigate the threats to validity (e.g., by designing and incorporating a suitable control-flow analysis technique).\n\n\n\\vspace{-1.5ex}\n\n\\section{Threats To Validity}\n\\label{sec:discuss}\n\n\nIn this section, we discuss a couple of threats to the validity of our study.\n\n\nFirst, we have not performed the control-flow analysis to determine whether an API call will be invoked only when running on certain Android versions.\nDuring the experiments, we noticed that many library codes take \\texttt{if-else} blocks to call higher-version APIs on when the app is running on the corresponding versions.\nTo mitigate its impact to our analysis, we currently exclude the library codes for consistency analysis (Section~\\ref{sec:AppAnalysis}), and use a threshold value to minimize the potential version-related \\texttt{if-else} blocks in app codes (Section~\\ref{sec:rq3}).\n\n\nApps may employ Java reflection to call private Android APIs \\cite{privateAPI} that are not included in the SDK but contained in Android framework. \nSimilarly, developers may use native codes to access Android APIs.\nCurrently we have not handled these two cases and leave them as our future work.\n\n\nOur assumption in Section~\\ref{sec:overview} that multiple-apk apps do not have compatibility issues may not be always true.\nIn particular, developers may provide only one apk for several Android platforms to share. \nIn this case, those shared apks are similar to single-apk apps.\n\n\\section{Evaluation}\n\\label{sec:evaluate}\n\\vspace{-2ex}\n\n\nOur evaluation aims to answer the following three research questions:\n\\begin{description}\n \\item[\\textbf{RQ1}] What are the \\textit{characteristics} of the \\texttt{DSDK}\\xspace versions in real-world apps?\n\n \\item[\\textbf{RQ2}] What are the \\textit{characteristics} of the API calls in real-world apps?\n\n \\item[\\textbf{RQ3}] Could we identify the \\textit{inconsistency} between \\texttt{DSDK}\\xspace versions and API calls in real apps?\n In particular, could we discover crash bugs and potential security vulnerabilities?\n\\end{description}\n\n\n\n\\vspace{-3ex}\n\\subsection{RQ1: Characteristics of the Declared SDK Versions}\n\\label{sec:rq1}\n\n\nIn this section, we report a total of four findings regarding the RQ1.\n\n\n\\textbf{Finding 1: Not all apps define the \\texttt{minSdkVersion}\\xspace and \\texttt{targetSdkVersion}\\xspace attributes, and 16.5\\% apps do not claim the \\texttt{targetSdkVersion}\\xspace attributes.}\nFrom Table \\ref{tab:nondefined}, we can see that rare apps (about 0.22\\%) do not define the \\texttt{minSdkVersion}\\xspace, while a noticeable portion of apps (over 15\\%) do not define the \\texttt{targetSdkVersion}\\xspace.\nOut of these apps, 48 apps declare neither the \\texttt{minSdkVersion}\\xspace, nor the \\texttt{targetSdkVersion}\\xspace.\nConsequently, the values of both \\texttt{minSdkVersion}\\xspace and \\texttt{targetSdkVersion}\\xspace will be assigned to ``1'' by the system.\nWe also notice that almost all apps (over 99\\%) do not define the \\texttt{maxSdkVersion}\\xspace.\nThis result is reasonable because, as we described in Section \\ref{sec:declared}, the \\texttt{maxSdkVersion}\\xspace attribute is strongly suggested \\textit{not} to define.\n\n\n\\begin{table}[t!]\n\\vspace{-2ex}\n\\caption{\\small The number and percentage of non-defined \\texttt{DSDK}\\xspace attributes in our dataset.}\n\\begin{adjustbox}{center}\n\\begin{tabular}{ c| c | c}\n\n\\hline\n & \\# Non-defined & \\% Non-defined \\tabularnewline\n\\hline\n\\hline\n\n\\texttt{minSdkVersion}\\xspace & 51 & 0.22\\% \\tabularnewline\n\\hline\n\\texttt{targetSdkVersion}\\xspace & 3,826 & 16.54\\% \\tabularnewline\n\\hline\n\\texttt{maxSdkVersion}\\xspace & 23,109 & 99.93\\% \\tabularnewline\n\\hline\n\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:nondefined}\n\\vspace{-4ex}\n\\end{table}\n\n\n\\textbf{Finding 2: There are 53 outlier \\texttt{targetSdkVersion}\\xspace values.}\nWe also find out some declared \\texttt{targetSdkVersion}\\xspace are outlier values.\nOne app defines its \\texttt{targetSdkVersion}\\xspace as 0, which is lower than the \\texttt{minSdkVersion}\\xspace.\nOthers' \\texttt{targetSdkVersion}\\xspace are larger than the newest SDK version (API level 23 at that time).\nSome apps declare \\texttt{targetSdkVersion}\\xspace as 24, 25, 26 or larger, however, these SDK versions have not been released yet in year 2015.\nEven more surprisingly, one app sets the \\texttt{targetSdkVersion}\\xspace value to ``10000''.\nIn general, \\texttt{targetSdkVersion}\\xspace should be always greater than or equal to the \\texttt{minSdkVersion}\\xspace, but 34 apps have negative \\texttt{targetSdkVersion}\\xspace - \\texttt{minSdkVersion}\\xspace value.\n\n\n\\textbf{Finding 3: The minimal platform versions most apps support are Android 2.3 and 2.2, whereas the most targeted platform versions are Android 4.4 and 5.0.}\nIn Fig.\\xspace \\ref{fig:minSDKdestribute} and Fig.\\xspace \\ref{fig:tarSDKdestribute}, we plot the distribution of \\texttt{minSdkVersion}\\xspace and \\texttt{targetSdkVersion}\\xspace, respectively.\nWe can see that most apps (around 85\\%) have \\texttt{minSdkVersion}\\xspace lower than or equal to level 11 (i.e., Android 3.0), which means that they can run on the majority of Android devices in the market~\\cite{dashboards}.\nMoreover, the minimal platform versions most apps support are Android 2.3 and 2.2.\nFig.\\xspace \\ref{fig:tarSDKdestribute} shows that more than 89\\% apps test their apps on platform versions larger than Android 4.0, and the most targeted platform versions are Android 4.4 and 5.0.\n\n\n\\begin{figure}[t!]\n\\vspace{-2ex}\n\\begin{adjustbox}{center}\n \\begin{minipage}{0.53\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{minSDKdestribute}\n \\vspace{-4ex}\n \\caption{Distribution of \\texttt{minSdkVersion}\\xspace.}\n \\label{fig:minSDKdestribute}\n \\end{minipage}\n \n \\begin{minipage}{0.53\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{tarSDKdestribute}\n \\vspace{-4ex}\n \\caption{Distribution of \\texttt{targetSdkVersion}\\xspace.}\n \\label{fig:tarSDKdestribute}\n \\end{minipage}\n\\end{adjustbox}\n\\vspace{-2ex}\n\\end{figure}\n\n\n\\textbf{Finding 4: The mean version difference between \\texttt{targetSdkVersion}\\xspace and \\texttt{minSdkVersion}\\xspace is 8.}\nWe define a new metric called \\texttt{lagSdkVersion}\\xspace to measure the version difference between \\texttt{targetSdkVersion}\\xspace and \\texttt{minSdkVersion}\\xspace, as shown in Equation \\ref{equ:lagversion}.\n\\begin{equation}\n \\texttt{lagSdkVersion}\\xspace = \\texttt{targetSdkVersion}\\xspace - \\texttt{minSdkVersion}\\xspace\n\\label{equ:lagversion}\n\\end{equation}\nAfter removing negative \\texttt{targetSdkVersion}\\xspace values and outliers, we draw the CDF (Cumulative Distribution Function) plot of \\texttt{lagSdkVersion}\\xspace in Fig.\\xspace \\ref{fig:lagSDK}.\nIt shows that more than 20\\% apps have equal \\texttt{targetSdkVersion}\\xspace and \\texttt{minSdkVersion}\\xspace.\nFurthermore, the majority of apps (more than 95\\% apps) have a \\texttt{lagSdkVersion}\\xspace less than 12.\n\n\n\\begin{figure}[t!]\n\\begin{adjustbox}{center}\n \\begin{minipage}{0.36\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{lagSDK}\n \\caption{\\small CDF plot of \\texttt{lagSdkVersion}\\xspace.}\n \\label{fig:lagSDK}\n \\end{minipage}\n \n \\begin{minipage}{0.46\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{APInum}\n \\caption{\\small CDF plot of the number of each app's API calls.}\n \\label{fig:APInum}\n \\end{minipage}\n \n \\begin{minipage}{0.36\\textwidth}\n \\centering\n \n \\includegraphics[width=1\\textwidth]{minOverNum}\n \\caption{\\small CDF plot of each app's number of API calls that have higher API level than \\texttt{minSdkVersion}\\xspace.}\n \\label{fig:minOverNum}\n \\end{minipage}\n\\end{adjustbox}\n\\vspace{-2ex}\n\\end{figure}\n\n\n\n\\subsection{RQ2: Characteristics of the API Calls}\n\\label{sec:rq2}\n\n\nIn this section, we briefly present two more findings related to the RQ2.\nIt is worth noting that here we consider all API calls that include the API calls in libraries.\n\n\n\\textbf{Finding 5: Around 500 apps call less than 50 APIs, making them lightweight apps. On the other hand, half of apps call over 1.8K APIs.}\nWe find that 446 apps call less than 50 APIs.\nThe majority of them are about user interface improvement, such as system theme and wallpaper apps.\nThese apps are regarded as lightweight ones that have less dependency on the SDK versions.\nAdditionally, many other apps contain several thousand API calls.\nWe plot the distribution of apps by API call numbers in Fig.\\xspace~\\ref{fig:APInum}.\n\n\n\\begin{figure}[t!]\n\\vspace{-2ex}\n\\begin{adjustbox}{center}\n \\subfigure[All API calls with library code.] {\n\t\\label{fig:allMinimumAPIlevel}\n \\includegraphics[height = 33ex]{allMinimumAPIlevel}\n }\n \\subfigure[App's own API calls without library code.] {\n\t\\label{fig:ownMinimumAPIlevel}\n \\includegraphics[height = 33ex]{ownMinimumAPIlevel}\n }\n\\end{adjustbox}\n\\vspace{-3ex}\n\\caption{\\small The distribution of \\texttt{minLevel}\\xspace that is calculated from API calls w\/o library.}\n\\label{fig:MinimumAPIlevel}\n\\end{figure}\n\n\n\\textbf{Finding 6: Library codes contribute more higher-version API calls than apps' own codes.}\nLibraries such as Android support library provide backward-compatible versions of Android framework APIs, as well as the features that are only available through the library APIs.\nEach support library is backward-compatible to a specific API level, which allows an app that contains higher-version APIs run correctly on a lower version of Android system.\nFig.\\xspace~\\ref{fig:allMinimumAPIlevel} shows that distribution of the \\texttt{minLevel}\\xspace of API calls with the library code, whereas Fig.\\xspace~\\ref{fig:ownMinimumAPIlevel} presents the distribution of the \\texttt{minLevel}\\xspace of API calls without the library code. \nBy analyzing and de-compiling the support library, we found that they can redirect the APIs calls in a higher-version SDK to some similar APIs which are already in a lower SDK or to an empty function.\n\n\n\n\\subsection{RQ3: Inconsistency Results}\n\\label{sec:rq3}\n\nIn this section, we report two important findings regarding the RQ3.\n\n\n\\textbf{Finding 7: Around 1.8K apps under-set the \\texttt{minSdkVersion}\\xspace value, causing them would crash when they run on lower Android versions.}\nWe find that 1,750 apps have over five API calls, the levels of which are larger than the declared \\texttt{minSdkVersion}\\xspace.\nIn 692 apps, more than ten API calls have higher API level than \\texttt{minSdkVersion}\\xspace.\nIn Fig.\\xspace~\\ref{fig:minOverNum}, we draw the CDF plot of the number of API calls that have higher API level than \\texttt{minSdkVersion}\\xspace.\nBased on this figure, we find that several apps have more than 50 API calls whose API level is higher than \\texttt{minSdkVersion}\\xspace.\n\n\n\n\n\n\n\n\n\\textbf{Finding 8: Around 400 apps fail to update their \\texttt{targetSdkVersion}\\xspace values, making them potentially exploitable by remote code execution.}\nThe \\texttt{addJavascriptInterface()} API \\cite{addJavascriptInterfaceSaga} has a serious security issue.\nBy exploiting this API, attackers are able to inject malicious codes, which may obtain any information from SD card.\nGoogle later fixed this bug on Android 4.2 and afterward.\nHowever, as mentioned in the side effect II, if an app has the \\texttt{targetSdkVersion}\\xspace lower than 17 and calls this API, the system will still call the vulnerable API even when running in Android 4.2 and afterward.\nIn our dataset, we find that 909 apps call the \\texttt{addJavascriptInterface()} API.\nAmong these apps, 413 apps are vulnerable, which may cause privacy information leakage.\nIn particular, out of these 413 apps, 238 apps do not define the \\texttt{targetSdkVersion}\\xspace attribute (i.e., \\texttt{targetSdkVersion}\\xspace is null).\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\\vspace{-1ex}\n\n\nRecent years have witnessed the extraordinary success of Android, a smartphone operating system owned by Google.\nAt the end of 2013, Android became the most sold phone and tablet OS.\nAs of 2015, Android evolved into the largest installed base of all operating systems.\nAlong with the fast-evolving Android, its fragmentation problem becomes more and more serious.\nAlthough new devices ship with the recent Android versions, there are still huge amounts of existing devices running old Android versions~\\cite{dashboards}.\n\n\nTo better manage the application's compatibility with multiple platform versions, Android allows apps to declare the supported platform SDK versions in their manifest files.\nWe term these declared SDK versions as \\texttt{DSDK}\\xspace versions.\nThe \\texttt{DSDK}\\xspace mechanism is a modern software mechanism that to the best of our knowledge, few systems are equipped with such mechanism until Android.\nNevertheless, so far the \\texttt{DSDK}\\xspace receives little attention and few understandings are known about the effectiveness of the \\texttt{DSDK}\\xspace mechanism.\n\n\nIn this paper, we make a first attempt to systematically study the \\texttt{DSDK}\\xspace mechanism.\nIn particular, our objective is to measure the current practice of \\texttt{DSDK}\\xspace versions in real apps, and the consistency between \\texttt{DSDK}\\xspace versions and their apps' API calls.\nTo this end, we perform a three-dimensional analysis that analyzes Google Play, Android documents, and each individual app.\nWe use a large dataset that contains over 24K apps crawled from Google Play in July 2015.\nOur study sheds light on the current \\texttt{DSDK}\\xspace practice and quantitatively measures the two side effects of inappropriate \\texttt{DSDK}\\xspace versions. \n\n\nWe summarize the contributions of this paper as follows:\n\\begin{compactitem}\n\\item (\\textit{New problem}) We study a modern software mechanism, i.e., allowing apps to declare the supported platform SDK versions. In particular, we are the first to measure the declared SDK versions and their consistency with API calls in Android apps.\n \n\\item (\\textit{New understanding}) We give the first demystification of the \\texttt{DSDK}\\xspace mechanism and its two side effects of inappropriate \\texttt{DSDK}\\xspace versions. \n\n\\item (\\textit{Hybrid approach}) We propose a three-dimensional analysis method that operates at both Google Play, Android document, and Android app levels.\n \n\\item (\\textit{Insightful results}) We have three major findings, including (i) around 17\\% apps do not claim the targeted \\texttt{DSDK}\\xspace versions or declare them wrongly, (ii) around 1.8K apps under-set the minimum \\texttt{DSDK}\\xspace versions, causing them crash when running on lower Android versions, and (iii) over 400 apps under-claim the targeted \\texttt{DSDK}\\xspace versions, making them potentially exploitable by remote code execution.\n\n\\end{compactitem}\n\n\n\n\n\\section{Methodology}\n\\label{sec:method}\n\n\nIn this section, we present an overview of our methodology and its three major components.\n\n\n\\subsection{Overview}\n\\label{sec:overview}\n\n\nFig.\\xspace \\ref{fig:overview} illustrates the overall design of our method.\nIt performs the analysis at three levels.\nFirst, we crawl and analyze each app's Google Play page to filter \\textit{multiple-apk} apps that provide different app binaries (i.e., \\textit{apks}) for different Android platforms. \nSince each apk of these apps is tailored for a particular Android version, its declared platform SDK version is no longer important.\nWe therefore exclude these multiple-apk apps for further analysis.\nSecond, we parse Android API documents to build a complete mapping between each API and their corresponding platform versions.\nWe call this mapping the \\textit{API-SDK mapping}.\n\n\n\\begin{figure}[t!]\n\\begin{adjustbox}{center}\n \\begin{minipage}{0.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{ourmethod}\n \\caption{The overview of our methodology.}\n \\label{fig:overview}\n \\end{minipage}\n \\hspace{0ex}\n \\begin{minipage}{0.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{facebookPage}\n \n \\caption{The Facebook app's Google Play page (with irrelevant contents removed).}\n \\label{fig:playPage}\n \\vspace{5ex}\n \\centering\n \\captionof{table}{The dataset of our study.}\n \\label{tab:dataset}\n\\begin{tabular}{ c | c | c }\n\\hline\n & \\# & Note \\tabularnewline\n\\hline\n\\hline\nAll crawled apps & 24,426 & The initial dataset \\tabularnewline\n\\hline\nMultiple-apk apps & 1,301 & Filtered apps\\tabularnewline\n\\hline\nSingle-apk apps & 23,125 & The final dataset \\tabularnewline\n\\hline\n\\end{tabular}\n\\label{tab:dataset}\n \\end{minipage}\n\\end{adjustbox}\n\\end{figure}\n\n\nIn the final app analysis phase, we first extract apps' declared SDK versions and API calls, then leverage the existing API-SDK mapping to infer the range of SDK versions from API calls, and finally compare these two SDK versions (i.e., the declared SDK versions and the SDK versions inferred from API calls). \nThe output is the (in)consistency results between declared SDK versions and API calls, which can be further leveraged to detect bugs and vulnerabilities.\n\n\n\n\n\\subsection{Google Play Analysis}\n\\label{sec:PlayAnalysis}\n\n\n\\noindent\n\\textbf{Design and Implementation}\nThe main objective of running Google Play analysis is to filter multiple-apk apps.\nWe explain this step using a representative Google Play page, the Facebook app's page as shown in Fig.\\xspace~\\ref{fig:playPage}.\nWe can notice that three attributes (``Size'', ``Current Version'', and ``Requires Android'') all have the same value of ``Varies with device''. \nThis indicates that Facebook employs the multiple-apk approach to handle the app compatibility over different versions of Android platforms.\nThe apps that do not have the value of ``Varies with device'' are thus the single-apk apps.\n\n\nTo implement the Google Play analysis, we write Python scripts based on our previous codes~\\cite{FileCross14,MoST15} and Selenium, a web browser automation tool.\nWe use Selenium's Firefox driver to load each app's Google Play page, and extract the attribute values we are interested by parsing the page's HTML source.\n\n\n\n\n\n\\noindent\n\\textbf{Dataset}\nTable \\ref{tab:dataset} lists the dataset used in this paper.\nWe have crawled 24,426 apps from Google Play in July 2015.\nWe run Google Play analysis for all these apps, among which we identify and filter 1,301 multiple-apk apps.\nTherefore, the remaining 23,125 single-apk apps assemble our final dataset, which will be further analyzed in Section \\ref{sec:AppAnalysis}.\nUnless stated otherwise, we refer to our dataset as these 23,125 apps in this paper.\n\n\n\n\n\n\n\n\n\n\\subsection{Android Document Analysis}\n\\label{sec:DocAnalysis}\n\n\n\\noindent\n\\textbf{Method}\nTo build the API-SDK mapping, we analyze Android SDK documents based on a previous work~\\cite{ICSM13}.\nSpecifically, we first build a list of all Android APIs and the corresponding platform versions they were introduced to by parsing a SDK document called \\texttt{api-versions.xml}.\nThis file covers both initial APIs (those introduced in the first Android version) and other newly added APIs in subsequent Android versions.\nWe further count the API change (e.g., deprecated and removed APIs) by analyzing the HTML files in the \\texttt{api\\_diff} directory.\n\n\nAfter running the document analysis for 23 Android versions (from 1.0 to 6.0), we recorded a total of 30,083 APIs, out of which 794 APIs were afterwards deprecated and 190 APIs were finally removed.\nHowever, we found that the lists of deprecated and removed APIs are not fully accurate, probably due to the mistakes made by Google developers when they wrote SDK documents.\nFor example, the \\texttt{removeAccount(Account, Callback, Handler)} API in the \\texttt{AccountManager} class was recorded as ``removed in SDK version 22'' in the documents, but actually it is still available in the SDK version 23.\nThis result implies that such a document-based analysis employed by the previous work~\\cite{ICSM13} requires further improvement.\nAs a future work, we will explore to retrieve the API-SDK mapping directly from each SDK \\texttt{jar} file.\nIn this paper, since the list of added APIs is accurate, we use only this part of results for the subsequent \\texttt{DSDK}\\xspace analysis in Section~\\ref{sec:evaluate}.\n\n\n\\begin{figure}[t!]\n\\begin{adjustbox}{center}\n \\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{APIadd}\n \\caption{\\footnotesize The comparison between initial and added APIs.}\n \\label{fig:APIadd}\n \\end{minipage}\n \n \\begin{minipage}{0.65\\textwidth}\n \\centering\n \\includegraphics[width=1\\textwidth]{APIlevel}\n \\caption{The distribution of added Android APIs.}\n \\label{fig:APIlevel}\n \\end{minipage}\n\\end{adjustbox}\n\\end{figure}\n\n\n\\noindent\n\\textbf{Results}\nWe now present the results of document analysis.\nFig.\\xspace~\\ref{fig:APIadd} shows the comparison between the initial Android APIs and those subsequently added APIs.\nWe can see that almost half of all APIs were added afterwards.\nThis indicates that Android evolves dramatically along the whole process. \nIn Fig.\\xspace~\\ref{fig:APIlevel}, we further plot the distribution of those subsequently added APIs since API level~2.\nAndroid 5.0 (API level 21) changed most, with 2,581 new API introduced.\nThe following two most changed versions are Android 3.0 (API level 11) and Android 6.0 (API level 23), with 1,760 and 1,657 new APIs, respectively.\n\n\n\n\\subsection{Android App Analysis}\n\\label{sec:AppAnalysis}\n\n\n\n\\noindent\n\\textbf{Retrieving Declared SDK Versions}\nWe leverage \\texttt{aapt}\\xspace (Android Asset Packaging Tool) to retrieve \\texttt{DSDK}\\xspace versions \\textit{directly} from each app without extracting the manifest file.\nThis method is more robust than the traditional \\texttt{apktool}\\xspace-based manifest extraction employed in many other works.\nIndeed, our \\texttt{aapt}\\xspace-based approach can successfully analyze all 23,125 apps, whereas a recent work \\cite{ECVDetector14} shows that \\texttt{apktool}\\xspace fails six times in the analysis of top 1K apps.\n\n\n\n\nIn the course of implementation, we observed and handled two kinds of special cases.\nFirst, some apps define \\texttt{minSdkVersion}\\xspace multiple times, for which we only extract the first value.\nSecond, we apply the by-default rules (see Section \\ref{sec:declared}) for the non-defined \\texttt{minSdkVersion}\\xspace and \\texttt{targetSdkVersion}\\xspace.\nMore specifically, we set the value of \\texttt{minSdkVersion}\\xspace to 1 if it is not defined, and set the value of \\texttt{targetSdkVersion}\\xspace (if it is not defined) using the \\texttt{minSdkVersion}\\xspace value.\n\n\n\n\n\n\\noindent\n\\textbf{Extracting API Calls and Their SDK Versions}\nTo extract API calls from apps' bytecodes, we first translate the compressed bytecodes into readable texts by using the \\texttt{dexdump}\\xspace tool.\nWe then use a set of Linux bash commands to extract each app's method calls from their \\texttt{dexdump}\\xspace outputs.\n\n\nWith the extracted API calls, we use the API-SDK mapping to compute their corresponding SDK versions (i.e., \\texttt{minLevel}\\xspace and \\texttt{maxLevel}\\xspace, as explained in Fig.\\xspace~\\ref{fig:demystify}).\nTo compute the \\texttt{minLevel}\\xspace, we calculate a maximum value of all API calls' added SDK versions.\nSimilarly, to compute the \\texttt{maxLevel}\\xspace, we calculate a minimum value of all API calls' removed SDK versions.\nIf an API is never removed, we set its removed SDK version to a large flag value (e.g., 100,000).\n\n\nDuring the experiments, we find that it is necessary to exclude library codes' API calls from host apps' own API calls.\nLibraries such as Android Support Library provide the stub implementation of higher-version APIs on lower-version platforms to ensure the backward-compatibility of higher-version APIs.\nIf an app is running on a higher-version platform, the library directly calls the corresponding API.\nOtherwise, the library calls the stub implementation, which actually does nothing but would not crash the app.\nSince we currently do not differentiate such control-flow information, we exclude library codes for the consistency analysis.\n\n\n\n\n\\noindent\n\\textbf{Comparing Consistency}\nWith the \\texttt{DSDK}\\xspace and API level information, it is easy to compare their consistency.\nWe compute the following three kinds of inconsistency (as previously mentioned in Section~\\ref{sec:sideeffect}):\n\\begin{itemize}\n\n\\item $\\texttt{minSdkVersion}\\xspace < \\texttt{minLevel}\\xspace$: the \\texttt{minSdkVersion}\\xspace is set too low and the app would crash when it runs on platform versions between \\texttt{minSdkVersion}\\xspace and \\texttt{minLevel}\\xspace.\n\n\\item $\\texttt{targetSdkVersion}\\xspace < \\texttt{maxLevel}\\xspace$: the \\texttt{targetSdkVersion}\\xspace is set too low and the app could be updated to the version of \\texttt{maxLevel}\\xspace. If the \\texttt{maxLevel}\\xspace is infinite, the \\texttt{targetSdkVersion}\\xspace could be adjusted to the latest Android version.\n\n\\item $\\texttt{maxSdkVersion}\\xspace > \\texttt{maxLevel}\\xspace$: the \\texttt{maxSdkVersion}\\xspace is set too large and the app would crash when it runs on platform versions between \\texttt{maxLevel}\\xspace and \\texttt{maxSdkVersion}\\xspace.\n\n\\end{itemize}\n\n\\section{Optimization}\n\\label{sec:optimize}\n\n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\n\nOur paper is mainly related to prior works that also study Android APIs or SDKs.\nThe work performed by McDonnell et al.~\\cite{ICSM13} is the closest to our paper.\nThey studied the Android API evolution and how client apps follow Android API changes, which is different from our focus on the consistency between apps' \\texttt{DSDK}\\xspace and API calls.\nIn the methodology part, we followed their document analysis method for extracting the API-SDK mapping.\nBut in the future we plan to directly analyze Android SDKs instead of documents for more accurate mapping extraction.\nOther related works have studied the coefficient between apps' API change and their success~\\cite{AppSuccess13}, the deprecated API usage in Java-based systems~\\cite{DeprecatedAPI16}, and the inaccessible APIs in Android framework and their usage in third-party apps~\\cite{Inaccessible16}.\nTwo recent works~\\cite{TargetFragment16,TameFragment16} also focused on the fragmentation issues in Android.\nCompared to all these works, our study is the first systematic work on \\texttt{DSDK}\\xspace versions and their consistency with API calls.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Introduction%\n \\label{introduction}%\n}\n\n\nIn the early 21st century, astronomy is truly a multi-wavelength enterprise. Ground and space-based instruments across the electromagnetic spectrum, from radio waves to gamma rays, provide the most complete picture of the various physical processes governing the evolution of astrophysical sources. In particular, X-ray astronomy probes high-energy processes in astrophysics, including high-temperature thermal plasmas (e.g., the solar wind, the intracluster medium) and relativistic cosmic rays (e.g., from active galactic nuclei). X-ray astronomy has a long and successful pedigree, with a number of observatories. These include \\href{http:\/\/heasarc.gsfc.nasa.gov\/docs\/einstein\/heao2.html}{\\emph{Einstein}}, \\href{http:\/\/science.nasa.gov\/missions\/rosat\/}{\\emph{ROSAT}}, \\href{http:\/\/chandra.harvard.edu}{\\emph{Chandra}}, \\href{http:\/\/xmm.esac.esa.int\/}{\\emph{XMM-Newton}}, \\href{http:\/\/www.isas.jaxa.jp\/e\/enterp\/missions\/suzaku\/}{\\emph{Suzaku}}, and \\href{http:\/\/www.nustar.caltech.edu\/}{\\emph{NuSTAR}}, as well as upcoming missions such as \\href{http:\/\/astro-h.isas.jaxa.jp\/en\/}{\\emph{Astro-H}} and \\href{http:\/\/www.the-athena-x-ray-observatory.eu\/}{\\emph{Athena}}.\n\nAn important distinguishing feature of X-ray astronomy from that of studies at longer wavelengths is that it is inherently \\DUroletitlereference{discrete}, e.g., the numbers of photons per second that reach the detectors are small enough that the continuum approximation, valid for longer-wavelength photons such as those in the visible light, infrared, microwave, and radio bands, is invalid. Instead of images, the fundamental data products of X-ray astronomy are tables of individual photon positions, energies, and arrival times.\n\nDue to modeling uncertainties, projection effects, and contaminating backgrounds, combining the insights from observations and numerical simulations is not necessarily straightforward. In contrast to simulations, where all of the physical quantities in 3 dimensions are completely known to the precision of the simulation algorithm, astronomical observations are by definition 2-D projections of 3-D sources along a given sight line, and the observed spectrum of emission is a complicated function of the fundamental physical properties (e.g., density, temperature, composition) of the source.\n\nSuch difficulties in bridging these two worlds have given rise to efforts to close the gap in the direction of the creation of synthetic observations from simulated data (see, e.g., \\cite{Gardini04}, \\cite{Nagai06}, \\cite{ZuHone09}, and \\cite{Heinz11} for recent examples). This involves the determination of the spectrum of the emission from the properties of the source, the projection of this emission along the chosen line of sight, and, in the case of X-ray (and $\\gamma$-ray) astronomy, the generation of synthetic photon samples. These photons are then convolved with the instrumental responses and (if necessary) background effects are added. One implementation of such a procedure, \\href{http:\/\/www.mpa-garching.mpg.de\/~kdolag\/Phox\/}{\\texttt{PHOX}}, was described in \\cite{Biffi12} and \\cite{Biffi13}, and used for the analysis of simulated galaxy clusters from smoothed-particle hydrodynamics (SPH) cosmological simulations. \\texttt{PHOX} was originally implemented in C using outputs from the \\texttt{Gadget} SPH code. \\texttt{PHOX} takes the inputs of density, temperature, velocity, and metallicity from a 3D \\texttt{Gadget} simulation, using them as inputs to create synthetic spectra (using spectral models from the X-ray spectral fitting package \\href{http:\/\/heasarc.gsfc.nasa.gov\/xanadu\/xspec}{\\texttt{XSPEC}}). Finally, \\texttt{PHOX} uses these synthetic spectra convolved with instrument response functions to simulate samples of observed photons.\n\nIn this work, we describe an extension of this algorithm to outputs from other simulation codes. We developed the module \\texttt{photon\\_simulator}, an implementation of \\texttt{PHOX} within the Python-based \\href{http:\/\/yt-project.org}{\\texttt{yt}} simulation analysis package. We outline the design of the \\texttt{PHOX} algorithm, the specific advantages to implementing it in Python and \\texttt{yt}, and provide a workable example of the generation of a synthetic X-ray observation from a simulation dataset.\n\n\\subsection{Model%\n \\label{model}%\n}\n\n\nThe overall model that underlies the \\texttt{PHOX} algorithm may be split up into roughly three steps: first, constructing an original large sample of simulated photons for a given source, second, choosing a subset of these photons corresponding to parameters appropriate to an actual observation and projecting them onto the sky plane, and finally, applying instrumental responses for a given detector. We briefly describe each of these in turn.\\begin{figure*}[]\\noindent\\makebox[\\textwidth][c]{\\includegraphics[scale=0.25]{schematic.png}}\n\\caption{Schematic representation of a roughly spherical X-ray emitting object, such as a\ngalaxy cluster. The volume element $\\Delta{V}_i$ at position ${\\bf r}_i$\nin the coordinate system ${\\cal O}$ of the source has a velocity\n${\\bf v}_i$. Photons emitted along the direction given by $\\hat{\\bf n}$\nwill be received in the observer's frame in the coordinate system ${\\cal O}'$,\nand will be Doppler-shifted by the line-of-sight velocity component $v_{i,z'}$.\n${\\rm Chandra}$ telescope image credit: NASA\/CXC. \\DUrole{label}{schematic}}\n\\end{figure*}\n\n\\subsubsection{Step 1: Generating the Original Photon Sample%\n \\label{step-1-generating-the-original-photon-sample}%\n}\n\n\nIn the first step of the \\texttt{PHOX} algorithm, we generate a large sample of photons in three dimensions, with energies in the rest frame of the source. These photons will serve as a \\textquotedbl{}Monte-Carlo\\textquotedbl{} sample from which we may draw subsets to construct realistic observations.\n\nFirst, to determine the energies of the photons, a spectral model for the photon emissivity must be specified. In general, the normalization of the photon emissivity for a given volume element will be set by the number density of emitting particles, and the shape of the spectrum will be set by the energetics of the same particles.\n\nAs a specific and highly relevant example, one of the most common sources of X-ray emission is that from a low-density, high-temperature, thermal plasma, such as that found in the solar corona, supernova remnants, \\textquotedbl{}early-type\\textquotedbl{} galaxies, galaxy groups, and galaxy clusters. The specific photon count emissivity associated with a given density, temperature $T$, and metallicity $Z$ of such a plasma is given by\\begin{equation}\n\\label{emissivity}\n\\epsilon_E^\\gamma = n_en_H\\Lambda_E(T,Z)~{\\rm photons~s^{-1}~cm^{-3}~keV^{-1}}\n\\end{equation}where the superscript $\\gamma$ refers to the fact that this is a photon count emissivity, $E$ is the photon energy in keV, $n_e$ and $n_H$ are the electron and proton number densities in ${\\rm cm^{-3}}$, and $\\Lambda_E(T,Z)$ is the spectral model in units of ${\\rm photons~s^{-1}~cm^{3}~keV^{-1}}$. The dominant contributions to $\\Lambda_E$ for an optically-thin, fully-ionized plasma are bremmstrahlung (\\textquotedbl{}free-free\\textquotedbl{}) emission and collisional line excitation. A number of models for the emissivity of such a plasma have been developed, including Raymond-Smith \\cite{Raymond77}, MeKaL \\cite{Mewe95}, and APEC \\cite{Smith01}. These models (and others) are all built into the \\texttt{XSPEC} package, which includes a Python interface, \\href{http:\/\/heasarc.gsfc.nasa.gov\/xanadu\/xspec\/python\/html\/}{\\texttt{PyXspec}}, which is a package we will use to supply the input spectral models to generate the photon energies.\n\nThe original \\texttt{PHOX} algorithm only allowed for emission from variants of the APEC model for a thermal plasma. However, astrophysical X-ray emission arises from a variety of physical processes and sources, and in some cases multiple sources may be emitting from within the same volume. For example, cosmic-ray electrons in galaxy clusters produce a power-law spectrum of X-ray emission at high energies via inverse-Compton scattering of the cosmic microwave background. Recently, the detection of previously unidentified line emission, potentially from decaying sterile neutrinos, was made in stacked spectra of galaxy clusters \\cite{Bulbul14}. The flexibility of our approach allows us to implement one or several models for the X-ray emission arising from a variety of physical processes as the situation requires.\n\nGiven a spectral model, for a given volume element $i$ with volume $\\Delta{V}_i$ (which may be grid cells or Lagrangian particles), a spectrum of photons may be generated. The \\emph{total number} of photons that are generated in our initial sample per volume element $i$ is determined by other factors. We determine the number of photons for each volume element by artificially inflating the parameters that determine the number of photons received by an observer to values that are large compared to more realistic values. The inflated Monte-Carlo sample should be large enough that realistic sized subsets from it are statistically representative. In the description that follows, parameters with subscript \\textquotedbl{}0\\textquotedbl{} indicate those with \\textquotedbl{}inflated\\textquotedbl{} values, whereas we will drop the subscripts in the second step when choosing more realistic values.\n\nTo begin with, the bolometric flux of photons received by the observer from the volume element $i$ is\\begin{equation}\n\\label{flux}\nF^{\\gamma}_i = \\frac{n_{e,i}n_{H,i}\\Lambda(T_i,Z_i)\\Delta{V}_i}{4\\pi{D_{A,0}^2}(1+z_0)^2}~{\\rm photons~s^{-1}~cm^{-2}}\n\\end{equation}where $z_0$ is the cosmological redshift and $D_{A,0}$ is the angular diameter distance to the source (if the source is nearby, $z_0 \\approx 0$ and $D_{A,0}$ is simply the distance to the source). The physical quantities of interest are constant across the volume element. The total number of photons associated with this flux for an instrument with a collecting area $A_{\\rm det,0}$ and an observation with exposure time $t_{\\rm exp,0}$ is given by\\begin{equation}\n\\label{n_phot}\nN_{\\rm phot} = t_{\\rm exp,0}A_{\\rm det,0}\\displaystyle\\sum_i{F^{\\gamma}_i}\n\\end{equation}By setting $t_{\\rm exp,0}$ and $A_{\\rm det,0}$ to values that are much larger than those associated with typical exposure times and actual detector areas, and setting $z_0$ to a value that corresponds to a nearby source (thus ensuring $D_{A,0}$ is similarly small), we ensure that we create suitably large Monte-Carlo sample to draw subsets of photons for more realistic observational parameters. Figure \\DUrole{ref}{schematic} shows a schematic representation of this model for a roughly spherical source of X-ray photons, such as a galaxy cluster.\n\n\\subsubsection{Step 2: Projecting Photons to Create Specific Observations%\n \\label{step-2-projecting-photons-to-create-specific-observations}%\n}\n\n\nThe second step in the \\texttt{PHOX} algorithm involves using this large 3-D sample of photons to create 2-D projections of simulated events, where a subsample of photons from the original Monte-Carlo sample is selected.\n\nFirst, we choose a line-of-sight vector $\\hat{\\bf n}$ to define the primed coordinate system from which the photon sky positions $(x',y')$ in the observer's coordinate system ${\\cal O}'$ are determined (c.f. Figure \\DUrole{ref}{schematic}). The total emission from any extended object as a function of position on the sky is a projection of the total emission along the line of sight, minus the emission that has been either absorbed or scattered out of the sight-line along the way. In the current state of our implementation, we assume that the source is optically thin to the photons, so they pass essentially unimpeded from the source to the observer (with the caveat that some photons are absorbed by Galactic foreground gas). This is appropriate for most X-ray sources of interest.\n\nNext, we must take into account processes that affect on the photon energies. The first, occurring at the source itself, is Doppler shifting and broadening of spectral lines, which arises from bulk motion of the gas and turbulence. Each volume element has a velocity ${\\bf v}_i$ in ${\\cal O}$, and the component $v_{i,z'}$ of this velocity along the line of sight results in a Doppler shift of each photon's energy of\\begin{equation}\n\\label{doppler}\nE_1 = E_0\\sqrt{\\frac{c+v_{z'}}{c-v_{z'}}}\n\\end{equation}where $E_1$ and $E_0$ are the Doppler-shifted and rest-frame energies of the photon, respectively, and $c$ is the speed of light in vacuum. Second, since many X-ray sources are at cosmological distances, each photon is cosmologically redshifted, reducing its energy further by a factor of $1\/(1+z)$ before being received in the observer's frame.\n\nSince we are now simulating an actual observation, we choose more realistic values for the exposure time $t_{\\rm exp}$ and detector area $A_{\\rm det}$ than we did in the first step to determine the number of photons to use from the original Monte-Carlo sample. Similarly, we may also specify alternative values for the angular diameter distance $D_A$ and the cosmological redshift $z$, if desired. The fraction $f$ of the photons that will be used in the actual observation is then given by\\begin{equation}\n\\label{fraction}\nf = \\frac{t_{\\rm exp}}{t_{\\rm exp,0}}\\frac{A_{\\rm det}}{A_{\\rm det,0}}\\frac{D_{A,0}^2}{D_A^2}\\frac{(1+z_0)^3}{(1+z)^3}\n\\end{equation}where $f \\leq 1$.\n\nBefore being received by the observer, a number of the photons, particularly on the softer end of the spectrum, are absorbed by foregrounds of hydrogen gas in the Milky Way Galaxy. The last operation that is applied in our implementation of the \\texttt{PHOX} algorithm is to use a tabulated model for the absorption cross-section as a function of energy (examples include \\texttt{wabs} \\cite{Morrison83}, \\texttt{phabs} \\cite{Balucinska-Church92}, \\texttt{tbabs} \\cite{Wilms00}, all included in \\texttt{XSPEC}) as an acceptance-rejection criterion for which photons will be retained in the final sample, e.g., which of them are actually received by the observer.\n\nThe advantage of the \\texttt{PHOX} algorithm is that the two steps of generating the photons in the source frame and projecting them along a given line of sight are separated, so that the first step, which is the most computationally expensive, need only be done once for a given source, whereas the typically cheaper second step may be repeated many times for many different lines of sight, different instruments, and different exposure times.\n\n\\subsubsection{Step 3: Modeling Instrumental Effects%\n \\label{step-3-modeling-instrumental-effects}%\n}\n\n\nUnfortunately, the data products of X-ray observations do not simply consist of the original sky positions and energies of the received photons. Spatially, the positions of the received photons on the detector are affected by a number of instrumental factors. These include vignetting, the layout of the CCD chips, and a typically spatially dependent point-spread function. Similarly, the photon energies are binned up by the detectors into a set of discrete energy channels, and there is typically not a simple one-to-one mapping between which channel a given photon ends up in and its original energy, but is instead represented by a non-diagonal response matrix. Finally, the \\textquotedbl{}effective\\textquotedbl{} collecting area of the telescope is also energy-dependent, and also varies with position on the detector. When performing analysis of X-ray data, the mapping between the detector channel and the photon energy is generally encapsulated in a \\href{http:\/\/cxc.harvard.edu\/ciao\/dictionary\/rmf.html}{redistribution matrix file (RMF)}, and the effective area curve as a function of energy is encapsulated in an \\href{http:\/\/cxc.harvard.edu\/ciao\/dictionary\/arf.html}{ancillary response file} (ARF).\n\nIn our framework, we provide two ways of convolving the detected photons with instrumental responses, depending on the level of sophistication required. The first is a \\textquotedbl{}bare-bones\\textquotedbl{} approach, where the photon positions are convolved with a user-specified point-spread function, and the photon energies are convolved with a user-input energy response functions. This will result in photon distributions that are similar enough to the final data products of real observations to be sufficient for most purposes.\n\nHowever, some users may require a full simulation of a given telescope or may wish to compare observations of the same simulated system by multiple instruments. Several software packages exist for this purpose. The venerable \\href{http:\/\/space.mit.edu\/ASC\/MARX\/}{\\texttt{MARX}} software performs detailed ray-trace simulations of how \\DUroletitlereference{Chandra} responds to a variety of astrophysical sources, and produces standard event data files in the same FITS formats as standard \\DUroletitlereference{Chandra} data products. \\href{http:\/\/hea-www.harvard.edu\/simx\/}{\\texttt{SIMX}} and \\href{http:\/\/www.sternwarte.uni-erlangen.de\/research\/sixte\/}{\\texttt{Sixte}} are similar packages that simulate most of the effects of the instrumental responses for a variety of current and planned X-ray missions. We provide convenient output formats for the synthetic photons in order that they may be easily imported into these packages.\\begin{figure*}[]\\noindent\\makebox[\\textwidth][c]{\\includegraphics[]{sloshing.png}}\n\\caption{Slices of density (left) and temperature (right) of an \\texttt{Athena} dataset of a\ngalaxy cluster core. \\DUrole{label}{sloshing}}\n\\end{figure*}\n\n\\subsection{Implementation%\n \\label{implementation}%\n}\n\n\nThe model described here has been implemented as the analysis module \\texttt{photon\\_simulator} in \\texttt{yt} \\cite{Turk11}, a Python-based visualization and analysis toolkit for volumetric data. \\texttt{yt} has a number of strengths that make it an ideal package for implementing our algorithm.\n\nThe first is that \\texttt{yt} has support for analyzing data from a large number of astrophysical simulation codes (e.g., \\href{http:\/\/flash.uchicago.edu}{\\texttt{FLASH}}, \\href{http:\/\/www.enzo-project.org}{\\texttt{Enzo}}, \\href{http:\/\/www.mpa-garching.mpg.de\/gadget\/}{\\texttt{Gadget}}, \\href{http:\/\/www.astro.princeton.edu\/~jstone\/athena.html}{\\texttt{Athena}}), which simulate the formation and evolution of astrophysical systems using models for the relevant physics, including magnetohydrodynamics, gravity, dark matter, plasmas, etc. The simulation-specific code is contained within various \\textquotedbl{}frontend\\textquotedbl{} implementations, and the user-facing API to perform the analysis on the data is the same regardless of the type of simulation being analyzed. This enables the same function calls to easily generate photons from models produced by any of these simulation codes making it possible to use the \\texttt{PHOX} algorithm beyond the original application to \\texttt{Gadget} simulations only. In fact, most previous approaches to simulating X-ray observations were limited to datasets from particular simulation codes.\n\nThe second strength is related, in that by largely abstracting out the simulation-specific concepts of \\textquotedbl{}cells\\textquotedbl{}, \\textquotedbl{}grids\\textquotedbl{}, \\textquotedbl{}particles\\textquotedbl{}, \\textquotedbl{}smoothing lengths\\textquotedbl{}, etc., \\texttt{yt} provides a window on to the data defined primarily in terms of physically motivated volumetric region objects. These include spheres, disks, rectangular regions, regions defined on particular cuts on fields, etc. Arbitrary combinations of these region types are also possible. These volumetric region objects serve as natural starting points for generating X-ray photons from not only physically relevant regions within a complex hydrodynamical simulation, but also from simple \\textquotedbl{}toy\\textquotedbl{} models which have been constructed from scratch, when complex, expensive simulations are not necessary.\n\nThe third major strength is that implementing our model in \\texttt{yt} makes it possible to easily make use of the wide variety of useful libraries available within the scientific Python ecosystem. Our implementation uses \\href{http:\/\/www.scipy.org}{\\texttt{SciPy}} for integration, \\href{http:\/\/www.astropy.org}{\\texttt{AstroPy}} for handling celestial coordinate systems and FITS I\/O, and \\texttt{PyXspec} for generating X-ray spectral models. Tools for analyzing astrophysical X-ray data are also implemented in Python (e.g., \\href{http:\/\/cxc.harvard.edu\/ciao\/}{\\texttt{CIAO}}'s \\href{http:\/\/cxc.harvard.edu\/sherpa\/}{\\texttt{Sherpa}} package, \\cite{Refsdal09}), enabling an easy comparison between models and observations.\n\n\\subsection{Example%\n \\label{example}%\n}\n\n\nHere we present a workable example of creating simulated X-ray events using \\texttt{yt}'s \\texttt{photon\\_simulator} analysis module. We implemented the module in \\texttt{yt} v. 3.0 as \\texttt{yt.analysis\\_modules.photon\\_simulator}. \\texttt{yt} v. 3.0 can be downloaded from \\url{http:\/\/yt-project.org}. The example code here is available \\href{http:\/\/nbviewer.ipython.org\/url\/www.jzuhone.com\/files\/photon_simulator_example.ipynb}{as an IPython notebook}. This is not meant to be an exhaustive explanation of all of the \\texttt{photon\\_simulator}'s features and options-{}-for these the reader is encouraged to visit the \\href{http:\/\/yt-project.org\/doc\/}{yt documentation}.\n\nAs our input dataset, we will use an \\texttt{Athena} simulation of a galaxy cluster core, which can be downloaded from the \\texttt{yt} website at \\url{http:\/\/yt-project.org\/data\/MHDSloshing.tar.gz}.\nYou will also need to download a version of \\texttt{APEC} from \\url{http:\/\/www.atomdb.org}. Finally, the absorption cross section table used here and the \\emph{Chandra} response files may be downloaded from \\url{http:\/\/yt-project.org\/data\/xray_data.tar.gz}.\n\nFirst, we must import the necessary modules:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{k+kn}{import} \\PY{n+nn}{yt}\n\\PY{k+kn}{from} \\PY{n+nn}{yt.analysis\\char`\\_{}modules.photon\\char`\\_{}simulator.api} \\char`\\\\{}\n \\PY{k+kn}{import} \\PY{n}{TableApecModel}\\PY{p}{,} \\PY{n}{ThermalPhotonModel}\\PY{p}{,} \\char`\\\\{}\n \\PY{n}{PhotonList}\\PY{p}{,} \\PY{n}{TableAbsorbModel}\n\\PY{k+kn}{from} \\PY{n+nn}{yt.utilities.cosmology} \\PY{k+kn}{import} \\PY{n}{Cosmology}\n\\end{Verbatim}\n\\vspace{1mm}\nNext, we load the dataset \\texttt{ds}, which comes from a set of simulations presented in \\cite{ZuHone14}. \\texttt{Athena} datasets require a \\texttt{parameters} dictionary to be supplied to provide unit conversions to Gaussian units; for most datasets generated by other simulation codes that can be read by \\texttt{yt}, this is not necessary.\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{parameters}\\PY{o}{=}\\PY{p}{\\char`\\{{}}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{time\\char`\\_{}unit}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{:}\\PY{p}{(}\\PY{l+m+mf}{1.0}\\PY{p}{,}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{Myr}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{)}\\PY{p}{,}\n \\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{length\\char`\\_{}unit}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{:}\\PY{p}{(}\\PY{l+m+mf}{1.0}\\PY{p}{,}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{Mpc}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{)}\\PY{p}{,}\n \\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{mass\\char`\\_{}unit}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{:}\\PY{p}{(}\\PY{l+m+mf}{1.0e14}\\PY{p}{,}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{Msun}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{)}\\PY{p}{\\char`\\}{}}\n\n\\PY{n}{ds} \\PY{o}{=} \\PY{n}{yt}\\PY{o}{.}\\PY{n}{load}\\PY{p}{(}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{MHDSloshing\/virgo\\char`\\_{}low\\char`\\_{}res.0054.vtk}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{,}\n \\PY{n}{parameters}\\PY{o}{=}\\PY{n}{parameters}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nSlices through the density and temperature of the simulation dataset are shown in Figure \\DUrole{ref}{sloshing}. The luminosity and temperature of our model galaxy cluster roughly match that of Virgo. The photons will be created from a spherical region centered on the domain center, with a radius of 250 kpc:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{sp} \\PY{o}{=} \\PY{n}{ds}\\PY{o}{.}\\PY{n}{sphere}\\PY{p}{(}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{c}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{,} \\PY{p}{(}\\PY{l+m+mf}{250.}\\PY{p}{,} \\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{kpc}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{)}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nThis will serve as our \\texttt{data\\_source} that we will use later. Now, we are ready to use the \\texttt{photon\\_simulator} analysis module to create synthetic X-ray photons from this dataset.\n\n\\subsubsection{Step 1: Generating the Original Photon Sample%\n \\label{id17}%\n}\n\n\nFirst, we need to create the \\texttt{SpectralModel} instance that will determine how\nthe data in the grid cells will generate photons. A number of options are available, but we will use the \\texttt{TableApecModel}, which allows one to use the \\texttt{APEC} data tables:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{atomdb\\char`\\_{}path} \\PY{o}{=} \\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{\/Users\/jzuhone\/Data\/atomdb}\\PY{l+s}{\\char`\\\"{}}\n\n\\PY{n}{apec\\char`\\_{}model} \\PY{o}{=} \\PY{n}{TableApecModel}\\PY{p}{(}\\PY{n}{atomdb\\char`\\_{}path}\\PY{p}{,}\n \\PY{l+m+mf}{0.01}\\PY{p}{,} \\PY{l+m+mf}{10.0}\\PY{p}{,} \\PY{l+m+mi}{2000}\\PY{p}{,}\n \\PY{n}{apec\\char`\\_{}vers}\\PY{o}{=}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{2.0.2}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{,}\n \\PY{n}{thermal\\char`\\_{}broad}\\PY{o}{=}\\PY{n+nb+bp}{False}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nwhere the first argument specifies the path to the \\texttt{APEC} files, the next three specify the bounds in keV of the energy spectrum and the number of bins in the table, and the remaining arguments specify the \\texttt{APEC} version to use and whether or not to apply thermal broadening to the spectral lines.\n\nNow that we have our \\texttt{SpectralModel}, we need to connect this model to a \\texttt{PhotonModel} that will connect the field data in the \\texttt{data\\_source} to the spectral model to and generate the photons which will serve as the sample distribution for observations. For thermal spectra, we have a special \\texttt{PhotonModel} called \\texttt{ThermalPhotonModel}:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{thermal\\char`\\_{}model} \\PY{o}{=} \\PY{n}{ThermalPhotonModel}\\PY{p}{(}\\PY{n}{apec\\char`\\_{}model}\\PY{p}{,}\n \\PY{n}{X\\char`\\_{}H}\\PY{o}{=}\\PY{l+m+mf}{0.75}\\PY{p}{,}\n \\PY{n}{Zmet}\\PY{o}{=}\\PY{l+m+mf}{0.3}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nWhere we pass in the \\texttt{SpectralModel}, and can optionally set values for\nthe hydrogen mass fraction \\texttt{X\\_H} and metallicity \\texttt{Z\\_met}, the latter of which may be a single floating-point value or the name of the \\texttt{yt} field representing the spatially-dependent metallicity.\n\nNext, we need to specify \\textquotedbl{}fiducial\\textquotedbl{} values for the telescope collecting area in ${\\rm cm}^2$, exposure time in seconds, and cosmological redshift, choosing generous values so that there will be a large number of photons in the Monte-Carlo sample. We also construct a \\texttt{Cosmology} object, which will be used to determine the source distance from its redshift.\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{A} \\PY{o}{=} \\PY{l+m+mf}{6000.} \\PY{c}{\\char`\\#{} must be in cm**2!}\n\\PY{n}{exp\\char`\\_{}time} \\PY{o}{=} \\PY{l+m+mf}{4.0e5} \\PY{c}{\\char`\\#{} must be in seconds!}\n\\PY{n}{redshift} \\PY{o}{=} \\PY{l+m+mf}{0.05}\n\\PY{n}{cosmo} \\PY{o}{=} \\PY{n}{Cosmology}\\PY{p}{(}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nBy default the \\texttt{Cosmology} object uses the WMAP7 cosmological parameters from \\cite{Komatsu11}, but others may be supplied, such as the \\cite{Planck13} parameters:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{cosmo} \\PY{o}{=} \\PY{n}{Cosmology}\\PY{p}{(}\\PY{n}{hubble\\char`\\_{}constant} \\PY{o}{=} \\PY{l+m+mf}{0.67}\\PY{p}{,}\n \\PY{n}{omega\\char`\\_{}matter} \\PY{o}{=} \\PY{l+m+mf}{0.32}\\PY{p}{,}\n \\PY{n}{omega\\char`\\_{}lambda} \\PY{o}{=} \\PY{l+m+mf}{0.68}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nNow, we finally combine everything together and create a \\texttt{PhotonList}\ninstance, which contains the photon samples:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{photons} \\PY{o}{=} \\PY{n}{PhotonList}\\PY{o}{.}\\PY{n}{from\\char`\\_{}scratch}\\PY{p}{(}\\PY{n}{sp}\\PY{p}{,} \\PY{n}{redshift}\\PY{p}{,} \\PY{n}{A}\\PY{p}{,}\n \\PY{n}{exp\\char`\\_{}time}\\PY{p}{,}\n \\PY{n}{thermal\\char`\\_{}model}\\PY{p}{,}\n \\PY{n}{center}\\PY{o}{=}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{c}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{,}\n \\PY{n}{cosmology}\\PY{o}{=}\\PY{n}{cosmo}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nwhere we have used all of the parameters defined above, and \\texttt{center} defines the reference coordinate which will become the origin of the photon coordinates, which in this case is \\texttt{\\textquotedbl{}c\\textquotedbl{}}, the center of the simulation domain. This object contains the positions and velocities of the originating volume elements of the photons, as well as their rest-frame energies.\n\nGenerating this Monte-Carlo sample is the most computationally intensive part of the PHOX algorithm. Once a sample has been generated it can be saved to disk and loaded as needed rather than needing to be regenerated for different observational scenarios (instruments, redshifts, etc). The photons object can be saved to disk in the \\href{http:\/\/www.hdfgroup.org}{HDF5} format with the following method:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{photons}\\PY{o}{.}\\PY{n}{write\\char`\\_{}h5\\char`\\_{}file}\\PY{p}{(}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{my\\char`\\_{}photons.h5}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nTo load these photons at a later time, we use the \\texttt{from\\_file} method:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{photons} \\PY{o}{=} \\PY{n}{PhotonList}\\PY{o}{.}\\PY{n}{from\\char`\\_{}file}\\PY{p}{(}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{my\\char`\\_{}photons.h5}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\n\n\n\\subsubsection{Step 2: Projecting Photons to Create Specific Observations%\n \\label{id20}%\n}\nAt this point the photons can be projected along a line of sight to create a specific synthetic observation. First, it is necessary to set up a spectral model for the Galactic absorption cross-section, similar to the spectral model for the emitted photons set up previously. Here again, there are multiple options, but for the current example we use \\texttt{TableAbsorbModel}, which allows one to use an absorption cross section vs. energy table written in HDF5 format (available in the \\href{http:\/\/yt-project.org\/data\/xray_data.tar.gz}{xray\\_data.tar.gz} file mentioned previously). This method also takes the column density \\texttt{N\\_H} in units of $10^{22}~{\\rm cm}^{-2}$ as an additional argument.\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{N\\char`\\_{}H} \\PY{o}{=} \\PY{l+m+mf}{0.1}\n\\PY{n}{a\\char`\\_{}mod} \\PY{o}{=} \\PY{n}{TableAbsorbModel}\\PY{p}{(}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{tbabs\\char`\\_{}table.h5}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{,} \\PY{n}{N\\char`\\_{}H}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nWe next set a line-of-sight vector \\texttt{L}:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{L} \\PY{o}{=} \\PY{p}{[}\\PY{l+m+mf}{0.0}\\PY{p}{,} \\PY{l+m+mf}{0.0}\\PY{p}{,} \\PY{l+m+mf}{1.0}\\PY{p}{]}\n\\end{Verbatim}\n\\vspace{1mm}\nwhich corresponds to the direction within the simulation domain along which the photons will be projected. The exposure time, telescope area, and source redshift may also be optionally set to more appropriate values for a particular observation:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{texp} \\PY{o}{=} \\PY{l+m+mf}{1.0e5}\n\\PY{n}{z} \\PY{o}{=} \\PY{l+m+mf}{0.07}\n\\end{Verbatim}\n\\vspace{1mm}\nIf any of them are not set, those parameters will be set to the original values used when creating the \\texttt{photons} object.\n\nFinally, an \\texttt{events} object is created using the line-of-sight vector, modified observation parameters, and the absorption model:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{events} \\PY{o}{=} \\PY{n}{photons}\\PY{o}{.}\\PY{n}{project\\char`\\_{}photons}\\PY{p}{(}\\PY{n}{L}\\PY{p}{,}\n \\PY{n}{exp\\char`\\_{}time\\char`\\_{}new}\\PY{o}{=}\\PY{n}{texp}\\PY{p}{,}\n \\PY{n}{redshift\\char`\\_{}new}\\PY{o}{=}\\PY{n}{z}\\PY{p}{,}\n \\PY{n}{absorb\\char`\\_{}model}\\PY{o}{=}\\PY{n}{a\\char`\\_{}mod}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\n\\texttt{project\\_photons} draws events uniformly from the \\texttt{photons} sample, orients their positions in the coordinate frame defined by \\texttt{L}, and applies the Doppler and cosmological energy shifts, and removes a number of events corresponding to the supplied Galactic absorption model.\\begin{figure}[]\\noindent\\makebox[\\columnwidth][c]{\\includegraphics[scale=0.33]{aplpy_figure.png}}\n\\caption{100 ks exposure of our simulated galaxy cluster, from a FITS image plotted with\n\\texttt{APLpy}. \\DUrole{label}{image}}\n\\end{figure}\n\n\\subsubsection{Step 3: Modeling Instrumental Effects%\n \\label{id21}%\n}\n\n\nIf desired, instrumental response functions may be supplied to convolve the photons with a particular instrumental model. The files containing these functions are defined and put in a single list \\texttt{resp}:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{ARF} \\PY{o}{=} \\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{chandra\\char`\\_{}ACIS\\char`\\-{}S3\\char`\\_{}onaxis\\char`\\_{}arf.fits}\\PY{l+s}{\\char`\\\"{}}\n\\PY{n}{RMF} \\PY{o}{=} \\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{chandra\\char`\\_{}ACIS\\char`\\-{}S3\\char`\\_{}onaxis\\char`\\_{}rmf.fits}\\PY{l+s}{\\char`\\\"{}}\n\\PY{n}{resp} \\PY{o}{=} \\PY{p}{[}\\PY{n}{ARF}\\PY{p}{,}\\PY{n}{RMF}\\PY{p}{]}\n\\end{Verbatim}\n\\vspace{1mm}\nIn this case, we would replace our previous call to \\texttt{project\\_photons} with one that supplies \\texttt{resp} as the \\texttt{responses} argument:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{events} \\PY{o}{=} \\PY{n}{photons}\\PY{o}{.}\\PY{n}{project\\char`\\_{}photons}\\PY{p}{(}\\PY{n}{L}\\PY{p}{,}\n \\PY{n}{exp\\char`\\_{}time\\char`\\_{}new}\\PY{o}{=}\\PY{n}{texp}\\PY{p}{,}\n \\PY{n}{redshift\\char`\\_{}new}\\PY{o}{=}\\PY{n}{z}\\PY{p}{,}\n \\PY{n}{absorb\\char`\\_{}model}\\PY{o}{=}\\PY{n}{a\\char`\\_{}mod}\\PY{p}{,}\n \\PY{n}{responses}\\PY{o}{=}\\PY{n}{resp}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nSupplying instrumental responses is optional. If they are provided, \\texttt{project\\_photons} performs 2 additional calculations. If an ARF is provided, the maximum value of the effective area curve will serve as the \\texttt{area\\_new} parameter, and after the absorption step a number of events are further removed using the effective area curve as the acceptance\/rejection criterion. If an RMF is provided, it will be convolved with the event energies to produce a new array with the resulting spectral channels.\\begin{figure}[]\\noindent\\makebox[\\columnwidth][c]{\\includegraphics[scale=0.33]{spectrum.png}}\n\\caption{Spectral energy distribution of our simulated observation. \\DUrole{label}{spectrum}}\n\\end{figure}\\begin{figure*}[]\\noindent\\makebox[\\textwidth][c]{\\includegraphics[scale=0.50]{comparison.png}}\n\\caption{100 ks exposures of our simulated galaxy cluster, observed with several\ndifferent existing and planned X-ray detectors. The \\DUroletitlereference{Chandra} image\nwas made with \\texttt{MARX}, while the others were made with \\texttt{SIMX}. All images have the\nsame angular scale. \\DUrole{label}{comparison}}\n\\end{figure*}\n\nHowever, if a more accurate simulation of a particular X-ray instrument is needed, or if one wishes to simulate multiple instruments, there are a couple of options for outputting our simulated events to be used by other software that performs such simulations. Since these external packages apply instrument response functions to the events list, the original \\texttt{events} object generated from the \\texttt{project\\_photons} method must not be convolved with instrument responses (e.g., the ARF and RMF) in that step. For input to \\texttt{MARX}, we provide an implementation of a \\texttt{MARX} \\textquotedbl{}user source\\textquotedbl{} at \\url{http:\/\/bitbucket.org\/jzuhone\/yt_marx_source}, which takes as input an HDF5 file. The events list can be written in the HDF5 file format with the following method:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{events}\\PY{o}{.}\\PY{n}{write\\char`\\_{}h5\\char`\\_{}file}\\PY{p}{(}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{my\\char`\\_{}events.h5}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nInput to \\texttt{SIMX} and \\texttt{Sixte} is handled via \\href{http:\/\/hea-www.harvard.edu\/heasarc\/formats\/simput-1.0.0.pdf}{\\texttt{SIMPUT}}, a file format designed specifically for the output of simulated X-ray data. The events list can be written in SIMPUT file format with the following method:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{events}\\PY{o}{.}\\PY{n}{write\\char`\\_{}simput\\char`\\_{}file}\\PY{p}{(}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{my\\char`\\_{}events}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{,}\n \\PY{n}{clobber}\\PY{o}{=}\\PY{n+nb+bp}{True}\\PY{p}{,}\n \\PY{n}{emin}\\PY{o}{=}\\PY{l+m+mf}{0.1}\\PY{p}{,} \\PY{n}{emax}\\PY{o}{=}\\PY{l+m+mf}{10.0}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nwhere \\texttt{emin} and \\texttt{emax} are the energy range in keV of the outputted events. Figure \\DUrole{ref}{comparison} shows several examples of the generated photons passed through various instrument simulations. \\texttt{SIMX} and \\texttt{MARX} produce FITS event files that are the same format as the data products of the individual telescope pipelines, so they can be analyzed by the same tools as real observations (e.g., \\texttt{XSPEC}, \\texttt{CIAO}).\n\n\\subsubsection{Examining the Data%\n \\label{examining-the-data}%\n}\n\n\nThe \\texttt{events} may be binned into an image and written to a FITS file:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{events}\\PY{o}{.}\\PY{n}{write\\char`\\_{}fits\\char`\\_{}image}\\PY{p}{(}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{my\\char`\\_{}image.fits}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{,}\n \\PY{n}{clobber}\\PY{o}{=}\\PY{n+nb+bp}{True}\\PY{p}{,}\n \\PY{n}{emin}\\PY{o}{=}\\PY{l+m+mf}{0.5}\\PY{p}{,} \\PY{n}{emax}\\PY{o}{=}\\PY{l+m+mf}{7.0}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nwhere \\texttt{emin} and \\texttt{emax} specify the energy range for the image. Figure \\DUrole{ref}{image} shows the resulting FITS image plotted using \\href{http:\/\/aplpy.github.io\/}{\\texttt{APLpy}}.\n\nWe can also create a spectral energy distribution (SED) by binning the spectrum into energy bins. The resulting SED can be saved as a FITS binary table using the \\texttt{write\\_spectrum} method. In this example we bin up the spectrum according to the original photon energy, before it was convolved with the instrumental responses:\\vspace{1mm}\n\\begin{Verbatim}[commandchars=\\\\\\{\\},fontsize=\\footnotesize]\n\\PY{n}{events}\\PY{o}{.}\\PY{n}{write\\char`\\_{}spectrum}\\PY{p}{(}\\PY{l+s}{\\char`\\\"{}}\\PY{l+s}{my\\char`\\_{}spec.fits}\\PY{l+s}{\\char`\\\"{}}\\PY{p}{,}\n \\PY{n}{energy\\char`\\_{}bins}\\PY{o}{=}\\PY{n+nb+bp}{True}\\PY{p}{,}\n \\PY{n}{emin}\\PY{o}{=}\\PY{l+m+mf}{0.1}\\PY{p}{,} \\PY{n}{emax}\\PY{o}{=}\\PY{l+m+mf}{10.0}\\PY{p}{,}\n \\PY{n}{nchan}\\PY{o}{=}\\PY{l+m+mi}{2000}\\PY{p}{,} \\PY{n}{clobber}\\PY{o}{=}\\PY{n+nb+bp}{True}\\PY{p}{)}\n\\end{Verbatim}\n\\vspace{1mm}\nhere \\texttt{energy\\_bins} specifies whether we want to bin the events in unconvolved photon energy or convolved photon channel. Figure \\DUrole{ref}{spectrum} shows the resulting spectrum.\n\n\\subsection{Summary%\n \\label{summary}%\n}\n\n\nWe have developed an analysis module within the Python-based volumetric data analysis toolkit \\texttt{yt} to construct synthetic X-ray observations of astrophysical sources from simulation datasets, based on the \\texttt{PHOX} algorithm. This algorithm generates a large sample of X-ray photons in the rest frame of the source from the physical quantities of the simulation dataset, and uses these as a sample from which a smaller number of photons are drawn and projected onto the sky plane, to simulate observations with a real detector. The utility of this algorithm lies in the fact that the most expensive step, namely that of generating the photons from the source, need only be done once, and these may be used as a Monte Carlo sample from which to generate as many simulated observations along as many projections and with as many instrument models as desired.\n\nWe implement PHOX in Python, using \\texttt{yt} as an interface to the underlying simulation dataset. Our implementation takes advantage of the full range of capabilities of \\texttt{yt}, especially its focus on physically motivated representations of simulation data and its support for a wide variety of simulation codes as well as generic \\texttt{NumPy} array data generated on-the-fly. We also benefit from the object-oriented capabilities of Python as well as the ability to interface with existing astronomical and scientific Python packages.\n\nOur module provides a crucial link between observations of astronomical sources and the simulations designed to represent the objects that are detected via their electromagnetic radiation, enabling some of the most direct testing of these simulations. Also, it is useful as a proposer's tool, allowing observers to generate simulated observations of astrophysical systems, to precisely quantify and motivate the needs of a proposal for observing time on a particular instrument. Our software also serves as a model for how similar modules in other wavebands may be designed, particularly in its use of several important Python packages for astronomy.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nOne of the key feature of the hierarchical picture of galaxy \nformation and growth involves continuous interactions and mergers\nwith other galaxies. Various galaxy properties such as morphology, luminosity, structure parameters, star-formation\nrate (SFR) and dust properties are strongly affected by these interactions \\citep{2006asup.book..115S,2007AJ....134..527W,2009ARA&A..47..159B}. The location of galaxies in particular environment is known to significantly affect their physical properties\\citep{1936rene.book.....H,1974ApJ...194....1O,1980ApJ...236..351D,1984ApJ...281...95P,2004MNRAS.353..713K,2005ApJ...631..208B,2006MNRAS.366....2W,2007ApJ...664..791B,2012A&A...539A..45L}. With the advent of large spectroscopic and photometric surveys, the study of the role of the environment in galaxy formation has got a new impetus \\citep{2004MNRAS.348.1355B,2007ApJ...658..898P,2006MNRAS.370..198C, 2006A&A...458...39C,2012MNRAS.423.1485S}. With the availability of large galaxy samples, such as Sloan Digital\nSky Survey Data Release 7 \\citep[SDSS-DR7][]{2009ApJS..182..543A}, we are in a position to explore many dimensions of galaxy properties simultaneously and homogeneously. This kind of analysis helps us put galaxy scaling relationships in context with respect to one another as well as probe the underlying physics affecting galaxy formation and evolution.\n\n\nStarting from the suggestions of \\citet{1977egsp.conf..401T} that elliptical galaxies can be formed by the merger between spiral galaxies, there is growing evidence for a change in galaxy morphology (e.g. \\citealt{2008ApJ...674..784P,2013pss6.book....1B}) and galaxy structure as a result of merger (e.g. \\citealt{2004MNRAS.355..874N,2005AJ....130.2043P,2005AJ....129..682H,2009ApJ...691.1828P}). On the other hand, at fixed color, the residual dependence of galaxy morphology on environmental density is reported to be rather weak \\citep{2008MNRAS.383..907B,2008MNRAS.391..607B}. With an earlier release of SDSS data (DR4) \\citet{2008ApJ...674..784P} showed that when a galaxy is located within the virial radius of its nearest neighbor, its morphology tends to be the same as that of the neighbor. This points to an important role of hydrodynamic interactions with neighbors within the virial radius. Further studies comprising of galaxies in galaxy clusters \\citep{2009ApJ...699.1595P,2011MNRAS.414..587C}, and involving hosts and their satellite galaxies \\citep{2008MNRAS.389...86A,2010ApJ...718..762W} seem to favor this proposition.\n\nIn regard to star formation activity, following Zwicky's suggestion that collisions would be frequent within dense galaxy clusters, \\citet{1951ApJ...113..413S} argued that strong shock waves could push the interstellar gas out of these galaxies resulting in scarcity of late-type spiral galaxies with substantial ongoing star formation in clusters. \\citet{1978ApJ...219...46L} identified interacting galaxies from Atlas of Peculiar Galaxy, and determined that many of these galaxies had recently undergone a burst of star formation \\citep[see also][]{2010ApJ...721..297M}. Several groups (e.g. \\citealt{1982ApJ...252..102C,1985AJ.....90..708K,1987AJ.....93.1011K}) have observed and found bursts of star formation associated with tidal interactions. \\citet{2007ApJ...671.1538B} constructed an isolated ``field'' galaxy mock sample using cosmological N-Body simulations and compared it with ``field'' galaxy sample from 2dF data to establish that a large fraction of the galaxies that experience close passes respond with triggered star formation.\\citet{2012MNRAS.425.2313T} find the tendency of close pairs in high-density environments to have fewer low specific SFR galaxies than non-pairs, whereas pairs in low-density environments have the opposite trend. Star formation related properties, such as color and emission-line flux, are found to be directly correlated with environmental density \\citep{2004MNRAS.353..713K,2005ApJ...631..208B,2005ApJ...621..201C,2011MNRAS.412..591P,2012ApJ...747...42I}. In addition, SFR and the fraction of star-forming galaxies is reported to be decreasing with increasing galaxy density \\citep{2001ApJ...559..606C,2002MNRAS.334..673L,2003ApJ...584..210G,2004MNRAS.349.1251M,2010MNRAS.404.1745M,2011MNRAS.416.2182E}. \n\nRegarding studies of structure parameters of galaxies, \\citet{2013A&A...553A..67B} studied the normalized rates and radial distributions of star formation in galaxies within low-redshift clusters to report that morphological type classifications of cluster galaxies correlate only weakly with their concentration indices, whereas this correlation is strong for non-cluster populations of disk galaxies.\n\\citet{2009ApJ...691.1828P} studied the variation of velocity dispersion and found almost no change for early type galaxies while for late type galaxies the velocity dispersion becomes larger as they approach their neighbors. \n\\citet{2013MNRAS.428.1715H} report that at intermediate redshifts, the size-mass relation for passive early-type galaxies is independent of environments ranging from field to groups which seem to contradict the findings by \\citet{2012MNRAS.419.3018C} at similar redshifts. \\citet{2013MNRAS.435..207L} find that the massive quiescent galaxies at $z > 1$ are typically 50\\% larger in the highest density environments compared to those in the lowest density environments whereas this relationship between size and environment is much weaker for star-forming galaxies\n\nIn addition to small scale dependence of galaxy properties discussed above, there are different studies which determine the dependence of various galaxy properties on large-scale structure \\citep{1982A&A...107..338B,2010MNRAS.408.1818W}. \\citet{2007A&A...464..815E} reported higher fraction of early type, passive, red galaxies in rich superclusters compared to poor superclusters thereby indicating the role of global environment in influencing galaxy morphology and their star formation activity \\citep[see also][]{2011ApJ...735...53P,2012A&A...545A.104L}. \\citet{2007ApJ...658..898P}, however, concluded that different galaxy properties are independent of large-scale density once morphology and luminosity are fixed \\citep[see also][]{2009ApJ...691.1828P,2013ApJ...768...51K}. \n\nThe extent to which the galaxy properties are driven by internal-physical-processes ('\\textit{nature}') as opposed to external-physical-processes ('\\textit{nurture}') is still a matter of debate \\citep{2011MNRAS.413.1036H}. A sample of galaxies whose physical properties are largely the result of internal-physical-processes can be used to address this issue in greater details. By comparing the physical properties of these galaxies in different environments we can have a better idea of the role played by nurture-induced processes. Keeping this in mind, for this work, we use a subsample of galaxies from SDSS-DR7 which are neither in the vicinity of a member of Abell Cluster of galaxies nor a part of large group of galaxies. The three-dimensional environment parameter space, used in this study, enables us better understand the effects of galaxy interactions.\n\nA brief outline of the paper follows. In section \\ref{data} we describe the data from Sloan Digital Sky Survey. Different subsections describe associated morphology classification. small and large scale density environments as well as definition of nearest neighbor galaxy as used in our analysis. The results of environment dependence of different galaxy properties are presented in detail in section \\ref{result}. We conclude this paper with discussions and conclusions in section \\ref{discuss} \n\n\\section{Data} \\label{data}\n\\subsection{SDSS Galaxy Sample}\nWe use galaxies from the SDSS-DR7. \nThis data release covers about 8,032 square degrees in the northern sky and consists of a series of three \ninterlocking imaging and spectroscopic surveys. A $2.5$m telescope located at Apache Point Observatory in Southern New Mexico is being used to carry out observations in five photometric bands ranging in wavelength from $3000$ to $11000$ $\\AA$ \\citep{1996AJ....111.1748F,\n1998AJ....116.3040G}. After careful photometric image reduction, calibration and classification of individual galaxies, a subsample of galaxies with Petrosian $r$ band magnitude $r < 17.77$ has been chosen for follow up spectroscopic observations \\citep{1999AJ....118.1406L,2001ASPC..238..269L,2002AJ....124.1810S}. The spectroscopic sky survey is being performed using two multi-object fiber spectrographs on the same telescope. Each spectroscopic fiber plug plate having a circular field-of-view, with a radius of 1.49 degrees, can accommodate a total of 640 fibers. Because of the finite size of the fiber plugs, the minimum separation of fiber centers is $55''$. If two galaxies are within $55''$ of each other, both of them can be observed only if they lie in the overlap between two adjacent fiber plug plates \\citep{2003AJ....125.2276B}.\n\nFor the present study we use the Korea Institute for Advanced Study Value-Added Galaxy Catalog (KIAS-VAGC) \\citep{2010JKAS...43..191C}. This catalog has been extracted from the New York University Value-Added Galaxy Catalog (NYU-VAGC) Large Scale Structure Sample (brvoid0) \\citep{2005AJ....129.2562B} that includes 583,946 galaxies within a $r$ band apparent magnitude range of $10 < r <17.6$. Our sample is supplemented by brighter galaxies that are not part of SDSS and whose redshifts are obtained from various literatures \\citep{2007ApJ...658..884C}. This leads to a total of $707,817$ galaxies in the flux limited sample. Such a large catalog of galaxies offers the advantage of an extended magnitude range with high completeness.\n\nThe volume limited sample used for the present study is a galaxy sample with $r-$band absolute magnitude $M_r < -19.0 +5~log~h$ and redshifts $ 0.020 <$ z $< 0.074$ or a comoving distance of $59.7$ $h^{-1}$Mpc $<$ d $< 218.9$ $h^{-1}$Mpc. This sample includes $114272$ galaxies. In order to select, manly passive galaxies from low density environments, we post process the volume limited sample by removing galaxies within a velocity separation of $\\pm$ $3000$~kms$^{-1}$ and a projected separation of $3$ Mpc from galaxies located inside Abell Clusters \\citep{1989ApJS...70....1A} of galaxies. Furthermore we apply finger of god corrections \\citep{1986ApJ...311...15B} to galaxies belonging to SDSS group catalog of \\citet{2007ApJ...671..153Y} and remove galaxies which are member of massive groups. This is done to minimize the effects of large scale density environments on various galaxy properties. In principle, the property of a galaxy may be affected by more than one neighbor galaxy. For the present study a target galaxy that is within virial radius of its second nearest neighbor (refer to \\ref{nn}) is also removed from sample. The removal enables us to better understand the effect of ``the nearest neighbor'' on different galaxy properties. The subsample of galaxies is plotted in figure \\ref{fig1}. \n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.0in,height=3in]{fig1.png}\n\\end{center}\n\\caption{The rectangular box in the upper panel encloses the target galaxies which are part of a total of $114,272$ galaxies contained in our volume limited subsample. The faint limit of these target galaxies is $0.5$ mag brighter than the full sample to achieve complete neighbor selection. The bottom panel shows the galaxies in the $color-magnitude$ diagram. Red points are early$-$morphological$-$type galaxies and blue points are late types.}\\label{fig1}\n\\end{figure}\n\n\n\n\\subsection{Morphology Classification}\n\n\\citet{2009ApJ...691.1828P} stressed the importance of accurate morphology classification since the effects of interaction strongly depend on morphology of interacting galaxies. For this work we use \\citet{2005ApJ...635L..29P} prescription to classify morphological types of galaxies in our sample. In this scheme the the location of galaxies, in the color-color gradient space, combined with their i-band concentration enables us in dividing the sample into early (ellipticals and lenticulars) and late (spirals and irregulars) morphological types with completeness and reliability of sample around 90\\%. An additional visual check of color images is undertaken to account for possible incorrect morphological classification of about 10,000 galaxies having close neighbors. This is necessary because \\citet{2005ApJ...635L..29P} prescription performs poorly when an early-type galaxy starts to overlap with other galaxies. Visual analysis guides us in identifying blue but elliptical-shaped, and dusty edge-on spiral galaxies. It is also useful in correcting central positions of merging galaxies. These procedures result in a volume limited subsample of 114,272 galaxies with $M_r$ $<$ $-\u2212$19.0. Removing the galaxies that are, part of massive SDSS groups \\citep{2007ApJ...671..153Y}, in the vicinity of Abell Clusters or are near the survey boundaries result in a population of \n$86,604$ galaxies out of which $30,298$ are early-type, and the rest are late-type galaxies. \n\nDifferent properties of galaxy population are studied in the absolute magnitude range of $-19.5 > M_r > -\u221220.5$. In such a subsample volume we have $14,244$ early types and $25,771$ late types. We identify these galaxies as those lying within a rectangular box in\nFigure \\ref{fig1}. As argued in literature (see e.g. \\citet{2007ApJ...658..884C,2009ApJ...691.1828P}), in order to minimize the effect of internal extinction on different galaxy properties, we restrict our late-type galaxy population to those having an isophotal axis ratio greater than $0.6$. This results in a further reduction of late-type galaxies to $14,581$ within the absolute magnitude limit of our subsample. \nDifferent physical properties of galaxies are studied by subdividing the sample of galaxies into four subsamples: $7106$ early types having early type nearest neighbor (the E$-$e galaxies), $7138$ early types having late-type nearest neighbor (E$-$l), $6894$ late types having early-type nearest neighbor (L$-$e), and $7687$ late types having late-type nearest neighbor (L$-$l). \n\\subsection{Environment}\nThere is a variety of methods to measure galaxy environment that may or may not correlate with each other \\citep{2012MNRAS.419.2670M}. We have followed \\citet{2009ApJ...691.1828P} in defining three different environment measures namely large-scale background density, small-scale mass density and morphology of the closest neighbor galaxy. In this scheme large scale background density at given location of an object in our sample is given by \n\\begin{equation}\n\\rho_{20}({\\bf x})\/\\bar\\rho = \\sum_{j=1}^{20}\\gamma_j L_j W_j(|{\\bf x_j - x}|) \/ \\bar\\rho\n\\end{equation}\nwhere $\\gamma$ and $L$ are mass to light ratio and luminosity of closest $20$ galaxies around an object in our volume limited sample. The weighting, $W$, is defined in terms of a spline kernel \\citep{1985A&A...149..135M,2007ApJ...658..898P} because it is adaptive, centrally weighted and has a finite tail, making it superior to tophat, cylindrical and Gaussian kernels. The smoothing scale determined by $20$ nearest neighbor in our volume limited sample is larger than typical cluster virial radius (about $1$-$2$h$^{-1}$Mpc) resulting into an estimation of $\\rho_{20}$ that never exceeds the virialization density.\n\nThe small scale density experienced by a target galaxy due to local mass density given by its nearest neighbor is defined as \n\\begin{equation}\n\\rho_n\/\\bar\\rho = 3\\gamma_n L_n\/4\\pi r_p^3 \\bar\\rho\n\\end{equation}\nwith $r_p$ being the projected separation of the nearest neighbor from target galaxy. In order to understand the interactions among various galaxies, we have studied the dependence of various galaxy properties on the projected separation from its nearest neighbor normalized by neighbor's virial radius (see Figure \\ref{fig3},\\ref{fig5},\\ref{fig7}). Following \\citet{2008ApJ...674..784P}, we have defined the virial radius of a galaxy as the projected radius where the mean mass density within the sphere of radius $r_p$ is $200$ times the critical density of the universe. According to our estimates, the virial radii of galaxies with $M_r = -19.5, -20.0$ and $-20.5$ are $264.74, 308.66$ and $359.87h^{-1}$kpc for early types and $210.12,~244.98$ and $285.63h^{-1}$kpc for late types, respectively. Clearly, the virial radius thus defined includes the galaxy as well as the surrounding dark matter halo region. \n\n\\subsection{The Nearest Neighbor} \\label{nn}\nThe definition of nearest neighbor galaxy in our study is similar to the one in \\citet{2009ApJ...691.1828P}. It requires the neighbor to be situated closest to the galaxy on the sky satisfying absolute magnitude and radial velocity conditions as well. The absolute magnitude condition requires the difference between neighbor and target galaxy's absolute magnitude to be smaller than $\\Delta M_r$ and larger than $\\Delta M_r - 1 $. For most of the work we chose $\\Delta M_r$ to be 0.5. We also discuss, in section \\ref{discuss}, the dependence of galaxy properties on a range of values of $\\Delta M_r$. The radial velocity condition demands that the neighbor galaxy must lie within $\\pm 600 (800) $kms$^{-1}$ if the target is late(early) type. The velocity difference between target and neighbor galaxy should also depend on their luminosity \\citep{1976ApJ...204..668F,1977A&A....54..661T} and projected separation. Accordingly velocity dispersion between early and late-type galaxies in our sample turns out to be between $1.3$ to $1.4$ across different separations(see Figure 2 of \\citet{2009ApJ...691.1828P} for details).\n\\section{Results}\\label{result}\nWe study the variation of galaxy properties on distance to the nearest neighbor, morphology of the nearest neighbor and background density contributed by $20$ neighbor galaxies. We leave out galaxies from very high density environments like Abell clusters and massive SDSS groups. For the galaxies in our subsample, we study different properties such as morphology, absolute magnitude, star formation activity (u-\u2212r color, g-\u2212i color gradient, equivalent width of the H$\\alpha$ line) and structure parameters (central velocity dispersion, i-band Petrosian radius, concentration index).\n\nFor most portions of our study, we fix the r-band absolute magnitude of target galaxies in a narrow range between $-19.5$ and $-\u221220.5$. This enables restricting effects due to the coupling of a parameter with luminosity. As pointed out in \\citet{2007ApJ...658..884C}, late-type target galaxies with the i-band isophotal axis ratio less than $0.6$ suffer from internal extinction and corresponding dispersion in luminosity. To avoid false trends in galaxy properties due to this issue, we have discarded such galaxies from our analysis.\n\nSmooth distributions of various physical parameters in figure \\ref{fig4}, \\ref{fig6} and \\ref{fig8} are found by the following method. At each point of parameter space, we first sort the values of physical parameter of galaxies contained within a certain spline radius from the point. The median value of a physical parameter is then given by the galaxy whose cumulative sum of spline kernel weights is $\\sum w_i\/2$, where $w_i$ is the individual spline weight of a galaxy around that point. The median is preferred over mean because of possible skewness in the distribution of various physical properties. \n \n\\subsection{\\it Morphology}\nWe build our study on previous similar analysis \\citep[e.g. ][]{2009ApJ...691.1828P,2009ApJ...699.1595P} of morphology dependence on local environment. With a much larger SDSS DR7 data sets we are in a position to study this dependence in greater details for galaxies lying in comparatively low density regimes. Figure \\ref{fig2} shows the variation of fraction $f_E$ of early type galaxies as function of large scale background density ($\\rho_{20}$) and projected separation $r_p$ from the galaxy's nearest neighbor. Early (red dot) and late (blue dot) type galaxies, having early type (upper panel) and late type (lower panel) nearest neighbor, are distributed in a triangular shaped region in this figure. This is due to the statistical correlation between $r_p$ and $\\rho_{20}$.\nAt each point of parameter space we obtain the value of $f_E$ from the ratio of weighted number of early type galaxies to the weighted number of total galaxies within a smoothing spline kernel of fixed size. The contours thus obtained are restricted to regions with statistical significance above 1$\\sigma$.\n\nIn case of target galaxies having early type neighbors (upper panel), we detect a gradual increase in $f_E$ as the galaxy approaches its neighbor. For galaxies in our sample we find no (or very little) dependence of $f_E$ on background density for galaxies that are situated outside (or inside) the virial radius of their neighbors, thus indicating the absence of any morphology density relation \\citep{1980ApJ...236..351D}. The behavior of $f_E$ for galaxies with late type nearest neighbor (lower panel) is similar at separations larger than neighbor's virial radius. However, when the galaxy is inside the virial radius of its neighbor, $f_E$ starts decreasing after attaining a maximum value around $r_p$ $\\sim$ 0.3 $r_{vir,nei}$. Significant hydrodynamic effects from neighbor galaxies appear to be important at these separations. Sensitivity of $f_E$ to $\\rho_{20}$ exists mainly within the virialized region at very high background density. This verifies earlier studies \\citep[e.g.][]{2008ApJ...674..784P} confirming that the effects of the nearest neighbors are critically important to galaxy morphology.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=4.0in]{fig2.png}\n\\end{center}\n\\caption{(Upper) Environment dependence of Morphology, when the nearest neighbor is an early-type galaxy. Red points are early-type target galaxies and blue points are late-type target galaxies. Absolute magnitude of galaxies is\nfixed to a narrow range of $\u2212-19.5 >$ $M_r$ $> -\u221220.0$. Contours show constant\nearly-type galaxy fraction $f_E$. Contours are limited to regions with \nstatistical significance above 1$\\sigma$. (Lower) Same, but for the late$-$type\n nearest neighbor case.}\\label{fig2}\n\\end{figure}\n\n\nIn the region of low merger and interaction ($r_p > r_{vir,nei}$), we report an early type fraction that asymptotically approaches about 0.2. This could be the inborn morphology fraction. In such an environment, early type galaxies couldn't have formed by processes such as strangulation because of lack of nearby large galaxies. According to recent observational \\citep{2008MNRAS.386.2285C} and theoretical \\citep{2013MNRAS.433.1479H} studies, such early type galaxies could either have formed due to AGN heating or other internal processes of galaxy formation. \n\\subsection{\\it Luminosity}\nFigure \\ref{fig3} shows the r-band absolute magnitudes of all\ngalaxies in a volume-limited sample with z $<$ 0.0741 and\n$M_r$ $<$ -\u221219.0. The lines with error bars are the median values as a function of projected separation from nearest neighbor. We find that the absolute magnitude of galaxies with early type neighbors is much brighter than those with late type neighbors. The neighbor separation does not seem to affect this behavior except for L$-$l, in which case the magnitude seem to rise as the galaxies come closer to each other.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=4.0in]{fig3.png}\n\\end{center}\n\\caption{Absolute magnitude - Neighbor separation relation. The upper panel is for the early-type target galaxies, and\nthe lower panel is for late types. All galaxies brighter than $M_r$ = $-\u221219.5$ \nare plotted. The median $M_r$ relations are drawn for early-type neighbor \n(red dots, magenta line) and late-type neighbor (blue dots, green line) cases.}\n\\label{fig3}\n\\end{figure}\n\nFigure \\ref{fig4} examines the environmental dependence of $M_r$ in the $r_p - \\rho_{20}$ plane. \nWe find that on smaller separation from the neighbor, i.e. at $r_p\/r_{vir,nei} <1 $, the galaxies irrespective of their morphology show no variation in $M_r$ as a function of separation and almost none to very little ($\\sim 0.2$) variation with respect to the underlying density field. At much smaller separation, we also observe a tendency for a galaxy to have neighbor of same morphology, which is indicated from a lesser population of $E-l$ galaxies compared to $E-e$ galaxies. \n\\citet{2008ApJ...674..784P} \\& \\citet{2009ApJ...691.1828P} have earlier reported an early to late type morphology transformation which we can explain \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=3.5in]{fig4.png}\n\\end{center}\n\\caption{Three dimensional (morphology, $r_p$ and $\\rho_{20}$) environment dependence of Median absolute magnitude. Points are all target galaxies brighter than $M_r$ = -\u221219.5. Four cases are given, the early-type target galaxies having an early-type neighbor (E-e), early-type targets having a late-type neighbor (E-l), late-type targets having an early-type neighbor (L-e), late-type targets having a late-type neighbor (L-l).}\\label{fig4}\n\\end{figure}\nby larger number of $L-l$ galaxies in comparison to $E-l$ galaxies at the scale of hydrodynamic interactions. Such a transformation can be explained by transfer of cold gas from late type neighbors to their early type target galaxies through interactions, thereby changing the morphology of early type galaxies to late type.\n\nHowever when the galaxies are well separated from their neighbor, we find significant dependence on both separation as well as density field. In such a situation, we find that galaxies that are farther from their neighbors tend to be the brighter one. The luminosity decreases as the galaxies come closer at this separation and this behavior is more apparent in high density environments. In the low density environments the variation with respect to $r_p\/r_{vir,nei}$ is slow possibly because of fewer number of neighbors \\citep{2009ApJ...691.1828P}. Another observation from figure \\ref{fig4} helps us to infer that higher luminosity contours have negative slopes, indicating that the change in $M_r$ due to $\\rho_{20}$ is much faster when the galaxies are at comparatively larger separation.\n\n\\subsection{\\it Star Formation Activity Parameters}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=5.50in]{fig5.png}\n\\end{center}\n\\caption{Star formation activity parameters of our target galaxies with $-\u221219.5>M_r > -\u221220.5$ and their dependence on the distance to the nearest neighbor normalized to the virial radius of the neighbor. Left panels are for early-type target galaxies and right panels are for late types. Median curves are drawn for the cases of early-type neighbor (magenta line, red dots) and of late-type neighbor (green line, blue dots). The $r_p$-space is uniformly binned in the logarithmic scale, and in each bin, the median value of physical parameter of the galaxies belonging to the bin is used for the median curve.}\n\\label{fig5}\n\\end{figure*}\nThe u - r color of a galaxy is a measure of the star formation activity of galaxies in the recent past. Equivalent width of H$\\alpha$ line is a well calibrated standard indicator of star formation rate \\citep{1998ApJ...498..541K}. This width can be defined as the ratio of the H$\\alpha$ luminosity to the underlying stellar continuum thus representing a measure of the the current to past average star formation. It is therefore a model independent, directly observed proxy for specific star formation rate of a galaxy. The color gradient of a galaxy enables us to study the star formation history along the disk. The {\\it u-}band surface photometry of galaxies in our sample is noisy. For this reason we use gradient in g - i color which has been corrected for the inclination and seeing effects. It is defined as the difference in g - i color between the central and annulus region of the galaxy in our sample. Observational studies indicate that the higher surface brightness galaxies have shallower color gradients. Figure \\ref{fig5} shows the distribution of u -\u2212 r color, equivalent width of H$\\alpha$ line and g--i color gradient of galaxies divided according to their neighbor morphology as a function of $r_p\/r_{vir,nei}$. The left(right) panels are for early(late) type target galaxies. Dots in Figure \\ref{fig5} and \\ref{fig6} are the galaxies with $-19.5\\ge M_r>-20.5$ spread in the two and three dimensional environmental parameter space respectively. \n\nFigure \\ref{fig5} shows the radial dependence of the star formation activity parameters. We find that when the target galaxy is \nearly type its median u - r color is constant, irrespective of the neighbor's morphology or separation. For late type target galaxies the u - r color is slightly larger for those with early type neighbors. On smaller separations the late type galaxies having late type neighbors show an increase in their u -r color, however, the sample size becomes so small that the increase is not statistically significant. In line with the constancy of u - r color, the equivalent width of H$\\alpha$ line for early type galaxies is constant and indistinguishable of neighbor's morphology at separations larger than 0.1 $r_{vir,nei}$ as shown in middle row of figure \\ref{fig5}. The situation is completely different when the neighbor morphology is late type. When any galaxy comes within the virial radius of its late type neighbor it shows a dramatic increase in its SFA. \\citet{2009ApJ...691.1828P} had noticed a sharp drop in SFA of a late type galaxy when it came in close contact with early type galaxy. We, however, don't see such a dramatic drop possibly because our galaxies are mainly taken from low density regions compared to the sample used in \\citet{2009ApJ...691.1828P}. The increase in width of H$\\alpha$ line can be attributed to the interactions, however small, that are going on in these galaxies even though they reside in comparatively low density environments. \nThe gradient in g - i color, as shown in the lowest panel, slightly increases as the galaxy approaches it neighbor. This indicates that central region of galaxies become slightly bluer compared to the outside region. Different neighbor morphology do not seem to affect this behavior significantly unlike the case of W(H$\\alpha$) and u - r color.\n\nIn order to better understand the dependence of these physical parameter on different environment parameters, in Figure \\ref{fig6}, we study their behavior in three dimensional space indicated by large scale background density, projected separation from the neighbor and neighbor's morphology. The u - r color for early type target galaxies changes little as we traverse the full $\\rho_{20}$ - $r_p$ parameter space in the top left panels of Figure \\ref{fig6}. For early type neighbor (E - e ) and late type neighbors (E - l) the color changes by 0.02 remaining constant when the target galaxy is outside virial radius of its nearest neighbor. As the galaxy enters the virial radius of the neighbor, the color decreases slightly and shows a weak dependence on $\\rho_{20}$ for E - l case. The decrease in color on small scales for E - l galaxies is consistent with our findings of decrease in early type fraction on smaller scales in lower panel of figure \\ref{fig2}. For late type target galaxies with early type neighbors, the u - r color increases as the galaxies come closer to each other within the virial radius. Outside the virial radius the color does not change with $r_p$ but shows a weak dependence on $\\rho_{20}$. For the case of L-l, we notice a peculiar increase in color as the galaxies come closer in the comparatively denser regions. In lower density regions, however, the color remains constant as a function of separation between nearest neighbors. We can conclude that color of early type galaxies remain constant as a function of small and large scale environment but for late type galaxies, we observe a weak but non zero dependence on underlying density field.\nThree dimensional contours of $W(H\\alpha)$ show an even clearer dependence of neighbor separation and morphology. The middle column in figure \\ref{fig6} shows that SFA of early type galaxies with early type neighbors in intermediate density field does not depend on either density or the neighbor separation. In the comparatively lower as well as higher density regions there is a slight decrease in star formation activity as the galaxies come closer. The situation is similar for L - e case where we notice a slight decrease in $W(H\\alpha)$ in high density regions.\nFor the E - l case, we notice a complete independence from local density in all density environments. At separations larger than virial radius, $W(H\\alpha)$ remains constant but as the galaxy enters the virial radius of the late type neighbor $W(H\\alpha)$ shows a gradual increase. \nThis situation is similar to L - l case, except the increase in equivalent width $W(H\\alpha)$ is almost double at close separation to what it was at separations beyond the virial radius. In the right column of Figure \\ref{fig6} contours represent the distribution of the median $\\Delta (g-i)$ at each location of the $r_p$-$\\rho_{20}$ space. The color gradient $\\Delta (g-i)$ of all galaxies irrespective of neighbor morphology is independent of density as can be seen from the horizontal contours. In case of E - e, E - l and L - e galaxies we notice a change in $\\Delta (g-i)$ of about 0.01 for well separated and closeby neighbor galaxies. When a late-type galaxy approaches another late type neighbor inside its virial radius, its color gradient increases (center becomes relatively bluer) significantly compared to three other cases. \\citep[see Figs. 6 and 7 of][]{2009ApJ...691.1828P}. \n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=6.0in]{fig6.png}\n\\end{center}\n\\caption{\nVariation of $u-\u2212r$ color, equivalent width of the the $H\\alpha$ line, $g-\u2212i$ color gradient of galaxies with $-\u221219.5 > Mr > -\u221220.5$ with respect to the pair separation $r_p$ and the large-scale background density $\\rho_{20}$. In each column, target galaxies are divided into four cases: the E-e, E-l, L-e, and L-l galaxies, respectively. Dots are galaxies belonging to each subset. At each location of the $r_p-\\rho_{20}$ space, the median value of the physical parameter is found from those of galaxies within a certain distance from the location. Curves are the constant-parameter contours}\\label{fig6}\n\\end{figure*}\n\n\n\n\\subsection{\\it Structure Parameters}\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=5.50in]{fig7.png}\n\\end{center}\n\\caption{Structural parameters of our target galaxies with $-\u221219.5>M_r > -\u221220.5$ and their dependence on the separation between the target galaxies and their nearest neighbor galaxy. The physical parameters considered here are the inverse concentration index $c_{in}$, central velocity dispersion $\\sigma$, and Petrosian radius $R_{Pet}$. Left panels\nare for early types and right panels for late types. Cases are further divided into early-type neighbor (magenta median curves, red dots) and late-type neighbor (green curves, blue dots) cases.}\n\\label{fig7}\n\\end{figure*}\n\nConcentration indices provide a quantitative determination of the radial distribution of light in\nspecified passbands that is both useful in itself, and also provides a more objective indicator of galaxy type than is provided by, for\nexample, Hubble classifications \\citep{2001AJ....122.1238S,2001AJ....122.1861S}. Central velocity dispersion refers to the velocity dispersion of the interior regions of an extended object like galaxy or cluster of galaxies. Taking into account finite resolution of the SDSS spectrograph, the most probable velocity dispersion is restricted to galaxies with $\\sigma >$ 40~km~s$^{-1}$\\citep{2007ApJ...658..884C}. Physical parameter representing size of the galaxy is the Petrosian radius obtained from $i-$band images of galaxies.\n\nTop row in figure \\ref{fig7} shows that radial dependence of concentration on morphology of the target galaxy. For early type target galaxies (left panel) the concentration is almost constant as a function of separation from neighbor whereas for late type targets (right panel) it increases ($c_{in}$ decreases) as the target galaxy approaches the neighbor within its virial radius. This may be due to smaller size and compactness of the early type galaxy resulting in them being tidally more stable. The larger size and lower concentration of late type galaxies compared to their early type counterparts makes the former tidally more vulnerable, resulting in a higher concentration of late type galaxies as they approach their neighbors within the virial radius. The significant drop in inverse concentration ratio in case of late type galaxies with late type neighbors can also result from relatively smaller velocity difference between the target and neighbor galaxy, resulting in larger tidal energy deposits. Variation in central velocity dispersion as a function of neighbor separation follows similar pattern as concentration of galaxies. As indicated by middle row in figure \\ref{fig7}, $\\sigma$ is almost independent of neighbor separation for early type galaxies (left panel) and increases as the late type galaxies approach their neighbor well within their virial radius (right panel). This behavior can once again be explained by compact, fast and tidally more stable nature of early type galaxies. The large deposits of tidal energy in case of late-late galaxy pairs can give rise to an increase in velocity dispersion as the pairs come closer \\citep{1987gady.book.....B}. The size dependence of a galaxy in terms of neighbor separation is shown in the bottom row of figure \\ref{fig7}. The inclination and seeing corrected size of galaxies located in low density regions of the Universe does not show any variation as a function of neighbor separation for different combinations of target and neighbor morphology.\n\nA three dimensional dependence of various structural parameters on morphology, large scale background density $\\rho_{20}$ and projected separation from the neighbor $r_p$ is shown in figure \\ref{fig8}. The left column shows the variation of $i-$band inverse concentration ratio of galaxies. As expected we observe a very weak decrease of inverse concentration as the target elliptical galaxy approaches its neighbor galaxy attaining a weak maximum for $E-e$ case. For late type target galaxies, however, the decrease of $c_{in}$ is comparatively stronger for decreasing $r_p$. For different combinations of target and neighbor galaxies, we find $c_{in}$ to be independent of $\\rho_{20}$. The decrease in $c_{in}$ of late type galaxies at smaller neighbor separation has also been reported in \\citet{2009ApJ...699.1595P} who analyzed a sample of galaxies in relatively denser environment. Effect of various environments on central velocity dispersion $\\sigma$ is shown in middle column of Figure \\ref{fig8}. We observe a weak dependence of $\\sigma$ on $\\rho_{20}$ when $\\rho_{20} < \\bar\\rho$. For early type galaxies, $\\sigma$ slightly increases with $\\rho_{20}$ in the low and intermediate density regions at $r_p > r_{vir,nei}$. This behavior is consistent with similar findings using galaxies from earlier SDSS data release \\citep{2007ApJ...658..898P,2009ApJ...691.1828P}. However, in the high density regions we do not find any dependence of $\\sigma$ on $\\rho_{20}$. The E - e panel shows an increase in $\\sigma$ as galaxies come closer to each other at fixed $\\rho_{20}$ while in case of E - l we don't find any dependence of $\\sigma$ on $r_p$. For late type target galaxies we report a $\\sigma$ that is independent of $r_p$ and $\\rho_{20}$ at separations larger than $\\sim$ 0.1 $r_{vir,nei}$. At closer separations $\\sigma$ increases as the separation decreases particularly for L - l galaxies. \nThe right column of Figure \\ref{fig8} shows the dependence of galaxies size on neighbor separation and background density. For E - e panel we notice that the size of the galaxy slightly increases as it moves in a higher density region at separation larger than the virial radius of its neighbor. Inside the virial radius we report a fixed size for E - e galaxies. \\citet{2009ApJ...691.1828P} have reported a rapid increase in size of E - e galaxies at separations below 0.01$ r_{vir,nei}$ but in our dataset such a merging sample of early type galaxies is absent. For the remaining cases also the size of galaxies does not show a significant change as the galaxy traverses the full $r_p$ - $\\rho_{20}$ parameter space. \\citet{2007ApJ...658..884C} reported a strong dependence of galaxy size on morphology and luminosity of the galaxy population. In this work we have fixed the morphology, restricted the change in magnitude to 1 mag and ommited galaxies from high density regions of the universe resulting into the measurements of petrosian radius that is essentially fixed across a range of background density and neighbor separation. \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=6.0in]{fig8.png}\n\\end{center}\n\\caption{Three dimensional (morphology, $r_p$ and $\\rho_{20}$) environment dependence of the inverse concentration index $c_{in}$, central velocity dispersion $\\sigma$ and Petrosian radius $R_{pet}$ of galaxies with $-\u221219.5 > Mr > -\u221220.5$. In each column, galaxies are divided into four cases: the E-e, E-l, L-e, and L-l galaxies, respectively. Dots are galaxies belonging to each subset. At each location of the $r_p-\\rho_{20}$ space, the median value of the physical parameter is found from those of galaxies within a certain distance from the location. Curves are the constant-parameter contours}\\label{fig8}\n\\end{figure*}\n\n\n\n\\section{Discussion and Conclusions}\\label{discuss}\nIn this paper we have studied various physical properties of mainly passively evolving field galaxies from low density regions of SDSS. Effects of the nearest neighbor distance, the nearest neighbor's morphology, and the large-scale background density are examined. In order to better understand the effect of the nearest neighbor galaxy, we have removed all those galaxies from our analysis that lie within the virial radius of their second nearest neighbor. We have hoped to remove the biases in galaxy properties due to morphology misclassification by using an accurate automated morphology classifier \\citep{2005ApJ...635L..29P} combined with careful visual inspection. Mass transfer between interacting galaxies \\citep{1977egsp.conf..401T} can help us better understand the reason for decrease in early type fraction of galaxies as the target galaxies approach a late type neighbor galaxy (see Figure \\ref{fig2}) within their virial radius. In such a scenario, the early type galaxy can acquire cold gas from the late type neighbor enabling the former to form a disk and transform itself to a late type galaxy \\citep{2008ApJ...674..784P}. Figure \\ref{fig5} indicates that at fixed background density the isolated galaxies are comparatively brighter and among isolated galaxies the brighter ones lie in higher density environments. It can serve as the evidence of transformation of the galaxy luminosity class through the merger process. Our analysis reconfirms the importance of radius of the galaxy plus dark halo systems as a distance scale inside which most of the galaxy properties start to be sensitive to both the nearest neighbor's distance and its morphology. The gravitationally bound pair of galaxies orbit each other within the virial radius resulting in repeated hydrodynamic interactions contributing to change in properties of the orbiting galaxies. Our study emphasize the importance of neighbor galaxy's morphology in either enhancing or suppressing of star formation activity (see e.g. middle column of Figure \\ref{fig6}) of target galaxy which has long been assumed to only increase due to the internal mass perturbed by the tidal force of the neighbor \\citep{1987AJ.....93.1011K,2004MNRAS.355..874N,2011MNRAS.412..591P}. Such a relation between morphology and star formation activity can be attributed to hydrodynamic interactions between approaching galaxies. Ram pressure effects experienced due to the collision with the hot gas of the neighbor elliptical galaxy can explain the change in star formation activity of the late type galaxy. In order to analyze the robustness of our findings against the choice of neighbor selection parameter($\\Delta M_r$), we restudied the behavior of equivalent width of $H\\alpha$ line using different samples of galaxies that are constrained to have brighter as well as fainter neighbors than themselves. Figure \\ref{fig9} represents a scenarios in which the two distinct target galaxy populations exhibit similar behavior of $W(H\\alpha)$ as a function of neighbor separation. Well inside the virial radius, $W(H\\alpha)$ decreases slightly as we traverse the galaxy populations having fainter neighbors (solid cyan line, Figure \\ref{fig9}) to those having brighter neighbors(dotted magenta line). We can fit the $W(H\\alpha)$ as a function of $r_p\/r_{vir,nei}$ using the function \n\\begin{equation}\nW(H\\alpha) = (31.37\\pm5.50)exp((-0.70\\pm0.46)\\frac{r_p}{r_{vir,nei}}) + C\n\\end{equation} where C is the measure of the value outside the virial radius of the neighbor. The slight decrease in $W(H\\alpha)$ as a function of neighbor brightness for passive galaxies may be related to exchange of gas between interacting galaxies.\n\n The additional role played by gravitational effects in the evolution of galaxies is evident by morphology independent structural parameter changes in Figure \\ref{fig7} as a function of neighbor separation. Figure \\ref{fig7} presents a comparatively larger variation in $c_{in}$ and $\\sigma$ for galaxy pairs that have smaller relative velocity, suggesting an inverse correlation between tidal energy deposits in a galaxy and velocity difference between pairs \\citep{2000ApJ...530..660B,2009ApJ...691.1828P}. The overall small variation in structural properties as a function of environments as evident in Figure \\ref{fig8} suggests these are less closely related to their ''environments'' than are their masses and star formation histories \\citep[see e.g. ][]{2005ApJ...631..208B,2008ApJ...675L..13V}. \n\\begin{figure*}\n\\includegraphics[width=3.0in]{fig9.png}\n\\caption{Dependence of $W(H\\alpha)$ - $r_p\/r_{vir,nei}$ relation on the absolute magnitude of the neighbor galaxy for late type galaxies with late type neighbors. The solid {\\it cyan } line is for the target galaxies having neighbors that are up to one magnitude fainter ($\\Delta M_r = 1$), and dotted {\\it magenta} line is for galaxies having neighbors that are up to one magnitude brighter ($\\Delta M_r = -1$) than the target itself. The $r_p$-space is uniformly binned in the logarithmic scale and in each bin, late type galaxies with axis ratio of $b\/a \\ge 0.6$ are selected for the median $W(H\\alpha)$ curve.}\\label{fig9}\n\\end{figure*}\n\n\n\nMajor outcomes of our studies are as follows.\n\\begin{itemize}\n\\item The properties of passive galaxies that are separated more than the one virial radius of the nearest neighbor are independent of the separation. Initial dependence of physical property on the separation starts as soon as the galaxy enters the neighbor's virial radius with another change occurring at about 0.1 $r_{vir,nei}$, which corresponds to $20-30h^{-1}$kpc for galaxies in our sample.\n\n\\item We do not observe a morphology density relation \\citep{1980ApJ...236..351D} for passive galaxies in low density environments as indicated by nearly horizontal, constant $f_E$ contours in Figure \\ref{fig2}. The galaxy morphology mainly depends on the pair separation combined with morphology of the neighbor galaxy.\n\n\\item Only luminosity of the passive galaxies show a dependence on $\\rho_{20}$ (Figure \\ref{fig4}) especially at separations larger than virial radius of the neighbor. This corroborates the evidence regarding transformation of luminosities of passive galaxies through mergers \\citep{2009ApJ...691.1828P}.\n\n\\item At fixed morphology and luminosity, both SF activity as well as structural parameter of galaxies show a weak or negligible dependence on $\\rho_{20}$. Variations in neighbor absolute magnitude does not vary the star formation activity of the target galaxy considerably (Figure \\ref{fig9}). The weak residual dependence on $\\rho_{20}$ of the $u - r$ color of galaxies can be attributed to the presence of hotter and denser halo gas in galaxies in high density environments. \n\n\\end{itemize}\n\n\n\n\n\\section*{Acknowledgments}\nJKY thanks Changbom Park and Yun-Young Choi for many useful discussions and suggestions. JKY acknowledges financial support from Chinese Academy of Sciences for fellowship assistance during this work. JKY and XC acknowledge support from NSFC via project number 11250110510 and 11373030 respectively.\nFunding for the SDSS and SDSS-II was provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS was managed by the Astrophysical Research Consortium for the Participating Institutions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\nThe Minimal Model Program (MMP) predicts that every projective pair with mild singularities is birationally built out of three classes of pairs: those whose log canonical classes are ample, numerically trivial or anti-ample.\n\nMore precisely, let $(X,\\Delta)$ be a log canonical pair. Then there should exist a birational contraction $\\varphi\\colon (X,\\Delta)\\dashrightarrow (X_{\\min},\\Delta_{\\min})$ together with a fibration $f\\colon (X_{\\min},\\Delta_{\\min})\\to X_\\mathrm{can}$ so that $K_{X_{\\min}}+\\Delta_{\\min}\\sim_\\mathbb{Q} f^*A$, for a suitable ample $\\mathbb{Q}$-divisor $A$ on $X_\\mathrm{can}$. Note that the Iitaka dimension of $K_X+\\Delta$ restricted to a general fibre of the composed map $f\\circ\\varphi$ is zero.\n\nIt is a natural and important question to determine whether singularities of the MMP are preserved in this process. The singularities of $(X_{\\min},\\Delta_{\\min})$ are the same as those of $(X,\\Delta)$. On the other hand, it remains an open problem whether there exists a boundary divisor $\\Delta_\\mathrm{can}$ on $X_\\mathrm{can}$ such that $(X_\\mathrm{can},\\Delta_\\mathrm{can})$ is log canonical and $K_{X_{\\min}}+\\Delta_{\\min}\\sim_\\mathbb{Q} f^*(X_\\mathrm{can}+\\Delta_\\mathrm{can})$; in other words, whether singularities of $(X,\\Delta)$ \\emph{descend} to the canonical model $X_\\mathrm{can}$.\n\nWhen the singularities of $(X,\\Delta)$ are klt, it is known by a work of Ambro and Kawamata that such a divisor exists; this result has already had numerous consequences in birational geometry. However, the proof is not constructive: more precisely, one loses control of the coefficients of $\\Delta$ at the last step. It is desirable that the singularities of $(X_\\mathrm{can},\\Delta_\\mathrm{can})$ reflect in a canonical way the singularities of $(X,\\Delta)$.\n\nIn general, with notation as above, it is known that \n$$A\\sim_\\mathbb{Q} K_{X_\\mathrm{can}}+B_{X_\\mathrm{can}}+M_{X_\\mathrm{can}},$$ \nwhere $B_{X_\\mathrm{can}}$ -- the \\emph{discriminant} -- is closely related to the singularities of $f$, and the divisor $M_{X_\\mathrm{can}}$ -- the \\emph{moduli divisor} -- conjecturally carries information on the birational variation of the fibres of $f$. A formula of this form is called the \\emph{canonical bundle formula}. \n\nThis paper is an attempt to give an account of all the known results on the canonical bundle formula, and to serve as a guide to those wishing to study this important subject.\n\n\n\n\\section{Lc-trivial fibrations}\n\nWe work over $\\mathbb{C}$. We denote by $\\equiv$, $\\sim$ and $\\sim_\\mathbb{Q}$ the numerical, linear and $\\mathbb{Q}$-linear equivalence of divisors respectively. \n\nFor a Weil $\\mathbb{Q}$-divisor $D=\\sum d_i D_i$, for a real number $r$ we denote $D^{\\leq r}:=\\sum_{d_i\\leq r}d_i D_i$. If $f\\colon X\\to Y$ is a proper surjective morphism between normal varieties and $D$ is a Weil $\\mathbb{R}$-divisor on $X$, then $D_v$ and $D_h$ denote the vertical and the horizontal part of $D$ with respect to $f$. In this setup, we say that $D$ is \\emph{$f$-exceptional} if $\\codim_Y\\Supp f(D)\\geq2$.\n\nIn this section we introduce the main topic of this survey -- \\emph{lc-trivial fibrations}. In this section we define them and give some examples which will accompany us through the paper.\n\nWe first need to introduce singularities of pairs. This is by now a standard topic in higher dimensional birational geometry, and good references are \\cite{KM98} and \\cite{Kol13}.\n\nA \\emph{pair} $(X,\\Delta)$ consists of a normal variety $X$ and a Weil $\\mathbb{Q}$-divisor $\\Delta$ such that $K_X+\\Delta$ is $\\mathbb{Q}$-Cartier. A pair $(X,\\Delta)$ is \\emph{log smooth} if $X$ is smooth and the support of $\\Delta$ is a simple normal crossings divisor.\n\nA \\emph{log resolution} of a pair $(X,\\Delta)$ is a birational morphism $f\\colon Y\\to X$ such that the exceptional locus $\\Exc(f)$ is a divisor and the pair $\\big(Y,\\Supp(f_*^{-1}\\Delta+\\Exc(f)\\big)$ is log smooth.\n\nIf $(X,\\Delta)$ be a pair and if $\\pi\\colon Y\\to X$ is a birational morphism with $Y$ is normal, we can write\n$$K_Y\\sim_\\mathbb{Q} \\pi^*(K_X+\\Delta)+\\sum a(E_i,X,\\Delta)\\cdot E_i,$$\nwhere $E_i\\subseteq Y$ are distinct prime divisors and the numbers $a(E_i,X,\\Delta)\\in\\mathbb{Q}$ are called \\emph{discrepancies}. The order of vanishing at the generic point of each $E_i$ defines a \\emph{geometric valuation} on $\\mathbb{C}(X)$. The pair $(X,\\Delta)$ is \\emph{klt}, respectively \\emph{log canonical}, if $a(E,X,\\Delta)>-1$, respectively $a(E,X,\\Delta) \\geq -1$, for every geometric valuation $E$ over $X$. \n\nMuch of what we say in this paper can be generalised to pairs $(X,\\Delta)$, where $\\Delta$ is allowed to have real coefficients. We stick to rational divisors mostly for reasons of clarity and simplicity.\n\n\\subsection{Definition and first examples}\n\nThe objects for which we can write a canonical bundle formula are called \\emph{lc-trivial fibrations}. \n\n\\begin{dfn}\\label{dfn:lctrivial}\nLet $(X,\\Delta)$ be a pair. A morphism $f \\colon (X,\\Delta) \\rightarrow Y$ to a normal projective variety $Y$ is a \\emph{klt-trivial}, respectively \\emph{lc-trivial}, fibration if:\n\\begin{enumerate}\n\\item[(a)] $f$ is a surjective morphism with connected fibres,\n\\item[(b)] $(X,\\Delta)$ has klt, respectively log canonical, singularities over the ge\\-ne\\-ric point of $Y$,\n\\item[(c)] there exists a $\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisor $D$ on $Y$ such that\n$$K_X+\\Delta\\sim_\\mathbb{Q} f^*D,$$\n\\item[(d)] there exists a log resolution $\\pi'\\colon X'\\rightarrow X$ of $(X,\\Delta)$ such that, if $\\mathcal E$ is the set of all geometric valuations over $X$ which are defined by a prime divisor $E$ on $X'$ such that $a(E,X,\\Delta)>-1$, and if we denote $\\Xi'=\\sum\\limits_{E\\in\\mathcal E} a(E,X,\\Delta)\\cdot E$, then\n$$\\rk (f\\circ\\pi')_*\\mathcal{O}_{X'}(\\lceil \\Xi'\\rceil) = 1.$$\n\\end{enumerate} \n\\end{dfn}\n\n\\begin{ter}\nIn \\cite{Amb04}, klt-trivial fibrations as in Definition \\ref{dfn:lctrivial} are called lc-trivial fibrations.\n\\end{ter}\n\n\n\\begin{rem}\nWe make a few comments on the condition (d) in Definition \\ref{dfn:lctrivial}. For simplicity, assume that the pair $(X,\\Delta)$ is klt. Note that then\n$$\\Xi'\\sim_\\mathbb{Q} K_{X'}-\\pi'^*(K_X+\\Delta).$$\nThe divisor $\\lceil \\Xi'\\rceil$ is effective on the generic fibre of $f\\circ\\pi'$ by (b), hence $\\rk (f\\circ\\pi')_*\\mathcal{O}_{X'}(\\lceil \\Xi'\\rceil) \\geq 1$, hence the point of (d) is the opposite inequality. \n\nThe most important case to keep in mind is that when the divisor $\\Delta$ is effective on the generic fibre: indeed, in that case the divisor $\\lceil \\Xi'\\rceil$ is an effective exceptional divisor on the generic fibre of $f\\circ\\pi'$, and the condition (d) is immediate. However, in order to be able to study lc-trivial fibrations by applying basic operations of birational geometry in \\S\\ref{subsec:basechange}, it is crucial to allow divisors $\\Delta$ with negative coefficients.\n\nOne more thing to notice is that if (d) holds for a log resolution $\\pi'$, then it holds on any log resolution $\\pi''\\colon X''\\to X$ which factors through $\\pi'$. Define the divisor $\\Xi''$ on $X''$ analogously as in Definition \\ref{dfn:lctrivial}, and let $\\theta\\colon X''\\to X'$ be the induced morphism. Since $X'$ and $X''$ are smooth, there exists an integral effective divisor $E$ such that $K_{X''}\\sim \\theta^*K_{X'}+E$. Thus,\n\\begin{align*}\nf_*\\pi''_*\\mathcal{O}_{X''}(\\lceil \\Xi''\\rceil)&=f_*\\pi''_*\\mathcal{O}_{X''}(\\lceil \\theta^*\\Xi'+E\\rceil)=f_*\\pi''_*\\mathcal{O}_{X''}(\\lceil \\theta^*\\Xi'\\rceil+E)\\\\\n&\\subseteq f_*\\pi''_*\\mathcal{O}_{X''}(\\theta^*\\lceil \\Xi'\\rceil+E)=f_*\\pi'_*\\mathcal{O}_{X'}(\\lceil \\Xi'\\rceil),\n\\end{align*}\nwhere we used that $\\lceil \\theta^*\\Xi'\\rceil\\leq \\theta^*\\lceil \\Xi'\\rceil$. Therefore, $\\rk (f\\circ\\pi'')_*\\mathcal{O}_{X''}(\\lceil \\Xi''\\rceil)=1$.\n\nIn general, one can show that (d) is independent of the choice of the resolution by using \\cite[Lemma 2.7]{Fuj10}.\n\\end{rem}\n\nNow we can formulate the canonical bundle formula associated to an lc-trivial fibration.\n\n\\begin{dfn}\\label{dfn:cbf}\nLet $f\\colon (X,\\Delta)\\to Y$ be an lc-trivial fibration such that $K_X+\\Delta\\sim_\\mathbb{Q} f^*D$ for some $\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisor $D$ on $Y$. If $P\\subseteq Y$ is a prime divisor, the \\emph{log canonical threshold} of $f^*P$ with respect to $(X,\\Delta)$ is\n$$\\gamma_P=\\sup\\{t\\in\\mathbb{R}\\mid (X,\\Delta+tf^*P) \\textrm{ is log canonical over the generic point of } P\\}.$$\nThe condition that $(X,\\Delta+tf^*P)$ is \\emph{log canonical over the generic point of $P$} means that for every geometric valuation $E$ over $X$ which surjects onto $P$, we have $a(E,X,\\Delta+tf^*P)\\geq-1$. \nThe \\emph{discriminant} of $f$ is\n$$\\textstyle B_Y=\\sum_P(1-\\gamma_P)P.$$\nFix $\\varphi\\in\\mathbb{C}(X)$ and the smallest positive integer $r$ such that $K_X + \\Delta +\\frac{1}{r}\\ddiv\\varphi = f^*D$. Then there exists a unique Weil $\\mathbb{Q}$-divisor $M_Y$, the \\emph{moduli part} of $f$, such that \n$$\\textstyle K_X + \\Delta +\\frac{1}{r}\\ddiv\\varphi = f^*(K_Y+B_Y+M_Y).$$\nThis formula is the \\emph{canonical bundle formula} associated to $f$.\n\\end{dfn}\n\n\\begin{rem}\nThe definition of the discriminant as above first appeared in \\cite[Theorem 2]{Kaw98}. The discriminant is a Weil $\\mathbb{Q}$-divisor on $Y$, and it is effective if $\\Delta$ is effective. We notice that the discriminant is defined as an actual divisor, while the moduli part is defined only up to $\\mathbb{Q}$-linear equivalence: it depends on the choice of $D$. Further, if $G$ is a $\\mathbb{Q}$-divisor on $Y$, then $f\\colon (X,\\Delta+f^*G)\\to Y$ is an lc-trivial fibration with discriminant $B_Y+G$ and moduli divisor $M_Y$.\n\nNote that if we are interested in proving properties of the moduli divisor of an lc-trivial fibrations as above, we may always assume that the pair $(X,\\Delta)$ is log canonical by \\cite[Remark 3.6]{FL18}, although we may not assume that $\\Delta$ is effective.\n\\end{rem}\n\n\\begin{exa}\nAssume that $X$ is smooth, that $\\Delta=0$, that $Y$ is a curve, that $f^{-1}(P)$ is smooth and that $f^*P=mf^{-1}(P)$ is a multiple fibre. Then $\\gamma_P=\\frac{1}{m}$.\n\\end{exa}\n\n\\begin{exa}\\label{ellfibr}\nThis example is historically the first example of a canonical bundle formula. Let $f\\colon X\\to C$ be an elliptic fibration, that is, $X$ is a smooth surface, $C$ is a smooth curve and a general fibre of $f$ is a smooth elliptic curve. We assume furthermore that $f$ is relatively minimal: there are no $(-1)$-curves contained in the fibres of $f$. Kodaira in \\cite[Theorem 6.2]{Ko60} classified the singular fibres of $f$. Kodaira's canonical bundle formula reads as\n$$K_X\\sim f^*(K_C+B_C+M_C),$$\nwhere $B_C$ is defined in terms of the classification of the singular fibres and by \\cite{Ko60, Ue73} we have $12M_C=j^*\\mathcal{O}_{\\mathbb{P}^1}(1)$, with $j\\colon C\\to \\mathbb{P}^1$ being the $j$-invariant. For a detailed account on Kodaira's canonical bundle formula see \\cite[Chapter V, \\S 7--\\S13]{BPV}.\n\\end{exa}\n\n\\begin{exa}\nLet $X=\\mathbb{P}^1\\times\\mathbb{P}^1$ and let $D$ be a reduced divisor of bidegree $(d,k)$ with $d\\geq 2$. Let $\\Delta=\\frac{2}{d}D$ and let $f\\colon (X,\\Delta)\\to \\mathbb{P}^1$ be the projection onto the second factor. Then $f$ is an lc-trivial fibration. Indeed, $K_X+\\Delta$ has bidegree $(-2,-2)+\\frac{2}{d}(d,k)=(0,-2+\\frac{2k}{d})$ and therefore is the pullback of a divisor from $\\mathbb{P}^1$.\n\\end{exa}\n\n\n\\subsection{Base change property}\\label{subsec:basechange}\n\nIn this subsection we investigate how canonical bundle formulas behave under base change. This will help improve the properties of the moduli part of a canonical bundle formula, at least on a sufficiently high birational model.\n\nIf $f \\colon(X,\\Delta)\\to Y$ is a klt-trivial fibration (respectively lc-trivial), and if we consider a base change diagram\n\\begin{equation}\\label{eq:diag}\n\\begin{gathered}\n\\xymatrix{\n(X',\\Delta') \\ar[r]^\\tau \\ar[d]_{f'} & (X,\\Delta)\\ar[d]^{f}\\\\\nY' \\ar[r]_{\\rho}&Y,\n}\n\\end{gathered}\n\\end{equation}\nwhere $\\rho$ is a proper generically finite morphism, $X'$ is the normalisation of the fibre product and $\\Delta'$ is defined so that we have\n$$K_{X'}+\\Delta'=\\tau^*(K_X+\\Delta),$$\nthen $f' \\colon(X',\\Delta')\\to Y'$ is also a klt-trivial (respectively lc-trivial) fibration. In the rest of the paper, we implicitly refer to this klt-trivial fibration when writing $M_{Y'}$ and $B_{Y'}$ for the moduli part and discriminant.\n\nIf $\\rho$ is birational, then $\\rho_*M_{Y'}=M_Y$ and $\\rho_*B_{Y'}=B_Y$; in other words, these collections of divisors form \\emph{b-divisors} see for instance \\cite[Section 1.2]{Amb04}. \n\nThe following is the \\emph{base change property} of canonical bundle formulas.\n\n\\begin{thm}\\label{nefness}\nLet $f \\colon(X,\\Delta)\\to Y$ be an lc-trivial fibration. Then there exists a proper birational morphism $Y'\\to Y$ such that for every proper birational morphism $\\pi \\colon Y''\\to Y'$ we have:\\begin{enumerate}\n\\item[(i)] $K_{Y'}+B_{Y'}$ is a $\\mathbb{Q}$-Cartier divisor and $K_{Y''}+B_{Y''}=\\pi^*(K_{Y'}+B_{Y'})$,\n\\item[(ii)] $M_{Y'}$ is a nef $\\mathbb{Q}$-Cartier divisor and $M_{Y''}=\\pi^*M_{Y'}$.\n\\end{enumerate}\n\\end{thm}\n\nThe first version of Theorem \\ref{nefness} is \\cite[Theorem 2]{Kaw98}, which essentially shows the nefness of the moduli part; this is also the point of the proof where the condition (d) in Definition \\ref{dfn:lctrivial} is used. Theorem \\ref{nefness} has been proved in this form for klt-trivial fibrations by Ambro \\cite[Theorem 0.2]{Amb04}. For lc-trivial fibrations, it was proved by Koll\\'ar \\cite[Theorem 8.3.7]{Kol07} and \\cite[Theorem 3.6]{FG14}, with an alternative proof in \\cite{Flo14a}. \n\n\\medskip\n\nIn the context of the previous theorem, we say that $M_{Y'}$ \\emph{descends} to $Y'$, and we call any such $Y'$, where additionally $B_{Y'}$ has simple normal crossings support, an \\emph{Ambro model} for $f$. \n\n\\medskip\n\nWe give a brief sketch of the proof of the nefness of the moduli divisor in the previous theorem; very good references are \\cite[Lemma 5.2]{Amb04} and especially \\cite[Theorem 8.5.1 and \\S8.10]{Kol07}, where many more details are given. By the proof of \\cite[Theorem 8.5.1]{Kol07}, it suffices to show the claim after making a suitable generically finite base change and taking the cyclic cover of $X$ associated to $\\sqrt[r]{\\varphi}$. The base change as in \\eqref{eq:diag} that we are aiming for is a composition of a log resolution with a Kawamata cover such that, on an open subset of $U'\\subseteq Y'$ whose complement has codimension at least $2$ in $Y'$, the local systems $R^if'_*\\mathbb{C}_{X'}|_{U'}$ have unipotent monodromies; this is the content of \\cite[ 8.10.7--8.10.10]{Kol07}. We may also assume that $M_{Y'}$ is a Cartier divisor. If $f$ is a klt-trivial fibration, then by \\cite[Theorem 8.5.1]{Kol07} the line bundle $\\mathcal{O}_{Y'}(M_{Y'})$ is a quotient of the locally free sheaf $f'_*\\omega_{X'\/Y'}$. Then one applies \\cite[Theorem 5]{Kaw81}, which asserts that $f'_*\\omega_{X'\/Y'}$ is the canonical extension of the bottom piece of the Hodge filtration of $R^{\\dim X-\\dim Y}f'_*\\mathbb{C}_{X'}|_{U'}$, and hence its quotients are nef by the same result. A similar statement holds also for lc-trivial fibrations. \n\nRecently, a stronger statement was shown in \\cite[Theorem 1.1]{FF17}. The result shows that \\cite[Theorem 5]{Kaw81} can be improved to show that not only any quotient of $f'_*\\omega_{X'\/Y'}$ is nef, but moreover, it carries a singular metric whose Lelong numbers are all zero. This is much stronger than being nef, as it implies, in particular, that the multiplier ideal associated to this metric is trivial. \n\nWe summarise this in the following result.\n\n\\begin{thm}\nLet $f \\colon(X,\\Delta)\\to Y$ be a klt-trivial fibration. Then there exists a proper birational morphism $Y'\\to Y$ such that for every proper birational morphism $\\pi \\colon Y''\\to Y'$ we have:\n\\begin{enumerate}\n\\item[(i)] $K_{Y'}+B_{Y'}$ is a $\\mathbb{Q}$-Cartier divisor and $K_{Y''}+B_{Y''}=\\pi^*(K_{Y'}+B_{Y'})$,\n\\item[(ii)] $M_{Y'}$ is a $\\mathbb{Q}$-Cartier divisor carrying a singular metric whose all Lelong numbers are zero, and $M_{Y''}=\\pi^*M_{Y'}$.\n\\end{enumerate}\n\\end{thm}\n\nThe proof of part (ii) of the theorem is the same as the sketch of the proof of Theorem \\ref{nefness}(ii) above, since a $\\mathbb{Q}$-divisor carries a singular metric whose all Lelong numbers are zero if and only if its pullback by a proper surjective map carries a singular metric whose all Lelong numbers are zero by \\cite[Corollary 4]{Fav99}.\n\n\\subsection{Inversion of adjunction}\n\nIn order to appreciate the following result and to see why base change property is important, let us go back to the construction of the canonical bundle formula. Recall that the discriminant divisor was constructed in terms of \\emph{local} log canonical thresholds, that is, log canonical thresholds over the generic point of a prime divisor; in particular, with notation from Definition \\ref{dfn:cbf}, for some prime divisor $P$ on $Y$, the pair $(X,\\Delta+\\gamma_P f^*P)$ does not have to be \\emph{globally} log canonical. However, the following \\emph{inversion of adjunction} \\cite[Theorem 3.1]{Amb04} states that this is precisely what happens on an Ambro model.\n\n\\begin{thm}\\label{thm:invAdjunction}\nLet $f\\colon(X,\\Delta)\\rightarrow Y$ be an lc-trivial fibration, and assume that $Y$ is an Ambro model for $f$. Then $(Y,B_Y)$ has klt, respectively log canonical, singularities in a neighbourhood of a point $y\\in Y$ if and only if $(X,\\Delta)$ has klt, respectively log canonical, singularities in a neighbourhood of $f^{-1}(y)$.\n\\end{thm}\n\nNote that Theorem \\ref{thm:invAdjunction} is stated for klt-trivial fibrations in \\cite{Amb04}, but the proof extends verbatim to the lc-trivial case by using Theorem \\ref{nefness}.\n\nWe finish this subsection with the following nice result \\cite[Proposition 3.1]{Amb05a} which we will apply in the proof of Theorem \\ref{thm:kltdescent} below.\n\n\\begin{thm}\\label{thm:pullbackAmbro}\nLet $f\\colon(X,\\Delta)\\rightarrow Y$ be a klt-trivial fibration, and assume that $Y$ is an Ambro model for $f$. Then for every base change by a surjective morphism $w\\colon W\\to Y$ we have $K_W+B_W\\sim_\\mathbb{Q} w^*(K_Y+B_Y)$ and $M_W\\sim_\\mathbb{Q} w^*M_Y$.\n\\end{thm}\n\n\n\\subsection{Coefficients of the moduli divisor}\n\nOften in applications one needs to bound the denominators of $M_Y$. For instance, such bounds were used in \\cite[Theorem 1.2]{Flo14}.\n\n\\begin{thm}\nFor each nonnegative integer $b$ there exists an integer $N$ depending on $b$ such that the following holds.\n\nLet $f\\colon(X,\\Delta)\\to Y$ be a klt-trivial fibration with a general fibre $F$, and let $r$ be the smallest positive integer such that $r(K_F+\\Delta|_F)\\sim0$. Let $E\\to F$ be the associated $r$-th cyclic cover and let $\\overline{E}$ be a resolution of singularities of $E$. If $\\dim H^{\\dim\\overline{E}}(\\overline{E},\\mathbb{C})=b$, then the divisor $NM_Y$ is integral.\n\\end{thm}\n\nThe result was proved in \\cite[Theorem 3.1]{FM00} when $\\Delta=0$, but the same proof works for klt-trivial fibrations \\cite[Theorem 5.1]{Flo14}.\n\nA more refined result holds when a general fibre is a rational curve, \\cite[Theorem 1.6(2)]{Flo13}.\n\n\\begin{thm}\nFix a positive integer $r$. For a prime number $q$ set $s(q) = \\max\\{s \\mid q^s \\leq 2r\\}$ and define\n$$N=\\prod_{q \\text{ prime}}q^{s(q)}.$$\n\\begin{enumerate}\n\\item[(a)] Let $f\\colon(X,\\Delta)\\to Y$ be an lc-trivial fibration whose general fibre $F$ is a rational curve, and assume that $r$ is the smallest positive integer such that $r(K_F+\\Delta|_F)\\sim0$. Then the divisor $NM_Y$ is integral. \n\\item[(b)] Assume $r$ is odd. Then there exists an lc-trivial fibration $f\\colon(X,\\Delta)\\to Y$ such that if $v$ is the smallest integer for which the divisor $v M_Y$ is integral, then $v = N\/r$.\n\\end{enumerate}\n\\end{thm}\n\n\\subsection{Goodness of moduli divisors}\n\nNow we come to the central topic of this survey, already announced in the introduction: the descent of singularities. Since we already know the nefness of the moduli divisor by Theorem \\ref{nefness}, if it were additionally big, then this would allow to conclude in many cases. Bigness is too much to ask; however, the following result of Ambro \\cite[Theorem 3.3 and Proposition 4.4]{Amb05a} turn out to be almost as good.\n\n\\begin{thm}\\label{ambro1}\nLet $f\\colon(X,\\Delta)\\to Y$ be a klt-trivial fibration between normal projective varieties such that $\\Delta$ is effective over the generic point of $Y$. Then there exists a diagram\n$$\n\\xymatrix{\n(X,\\Delta)\\ar[d]_f && (X^+,\\Delta^+)\\ar[d]^{f^+}\\\\\nY & W \\ar[l]^{\\tau}\\ar[r]_{\\rho}&Y^+\n}\n$$\nsuch that:\n\\begin{enumerate}\n\\item[(i)] $f^+\\colon(X^+,\\Delta^+)\\rightarrow Y^+$ is a klt-trivial fibration,\n\\item[(ii)] $\\tau$ is generically finite and surjective, and $\\rho$ is surjective,\n\\item[(iii)] if $M_Y$ and $M_{Y^+}$ are the moduli divisors of $f$ and $f^+$ respectively, then $M_{Y^+}$ is big and, after possibly a birational base change, we have $\\tau^*M_Y=\\rho^*M_{Y^+}$,\n\\item[(iv)] there exists a non-empty open set $U\\subseteq W$ and an isomorphism\n$$\n\\xymatrix{\n(X,\\Delta)\\times_{Y} U\\ar[rd]\\ar[rr]^{\\simeq\\quad}&&(X^+,\\Delta^+)\\times_{Y^+} U\\ar[ld]\\\\\n&U,&\n}\n$$\n\\item[(v)] if there exists an isomorphism\n$$\\Phi\\colon (X,\\Delta)\\times_Y U\\to (F, \\Delta_F)\\times U$$\nover a non-empty open subset $U\\subseteq Y$, then $\\Phi$ extends to an isomorphism over \n$$Y^0=Y\\setminus \\big(\\Supp B_Y\\cup \\Sing (Y)\\cup f(\\Supp \\Delta_v^{\\leq0})\\big).$$\n\\end{enumerate}\n\\end{thm}\n\n\nNote that in (v) one does not need that $\\Delta$ is effective on the generic fibre of $f$.\n\nFor us, the most important part of this result is (iii). Its immediate consequence is the descent of klt singularities \\cite[Theorem 0.2]{Amb05a}.\n\n\\begin{thm}\\label{thm:kltdescent}\nLet $(X,\\Delta)$ be a projective klt pair with $\\Delta$ effective, and let $f\\colon X\\to Y$ be a surjective morphism to a normal projective variety such that $K_X+\\Delta\\sim_\\mathbb{Q} f^*D$ for some $\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisor $D$ on $Y$. Then there exists an effective $\\mathbb{Q}$-divisor $\\Delta_Y$ on $Y$ such that the pair $(Y,\\Delta_Y)$ is klt and \n$$K_X+\\Delta\\sim_\\mathbb{Q} f^*(K_Y+\\Delta_Y).$$\n\\end{thm}\n\n\\begin{proof}\nWe use the notation from Theorem \\ref{ambro1}, which clearly applies in our situation. We have the base change diagram\n$$\n\\xymatrix{\n(W,\\Delta_W) \\ar[r]^{w} \\ar[d]_{f_W} & (X,\\Delta)\\ar[d]^{f}\\\\\nW \\ar[r]_{\\tau}&Y.\n}\n$$\nWe may additionally assume that $\\tau$ factors through an Ambro model $\\pi\\colon Y'\\to Y$ of $f$, and denote by $\\sigma\\colon W\\to Y'$ the induced morphism. By replacing $W$ by a suitable log resolution, by an easy argument with the Stein factorisation of $\\sigma$ together with Theorem \\ref{thm:pullbackAmbro} we may assume that $W$ is an Ambro model of $f_W$. \n\nNow, $\\tau^*D\\sim_\\mathbb{Q} K_W+B_W+M_W$, and by the inversion of adjunction, Theorem \\ref{thm:invAdjunction}, the pair $(W,B_W)$ is klt. Since the divisor $M_{Y^+}$ is nef and big, by using a version of Kodaira's trick \\cite[Proposition 2.61]{KM98} together with Bertini's theorem, we may find an effective $\\mathbb{Q}$-divisor $E_W$ on $W$ such that $M_W\\sim_\\mathbb{Q} E_W$ and such that the pair $(W,B_W+E_W)$ is klt. \n\nBy Theorem \\ref{thm:pullbackAmbro} we have $K_W+B_W\\sim_\\mathbb{Q} \\sigma^*(K_{Y'}+B_{Y'})$ and $M_W\\sim_\\mathbb{Q} \\sigma^*M_{Y'}$. Hence, if we denote $E_{Y'}=\\frac{1}{\\deg\\sigma}\\sigma_*E$, we have $M_{Y'}\\sim_\\mathbb{Q} E_{Y'}$ and\n$$K_W+B_W+E\\sim_\\mathbb{Q} \\sigma^*(K_{Y'}+B_{Y'}+E_{Y'}).$$\nThen the pair $(Y',B_{Y'}+E_{Y'})$ is klt by \\cite[Proposition 5.20]{KM98}. Setting \n$$\\Delta_Y=\\pi_*(B_{Y'}+E_{Y'})=B_Y+\\pi_*E,$$\nwe have $K_Y+\\Delta_Y\\sim_\\mathbb{Q} D$, and $(Y,\\Delta_Y)$ is a klt pair. Since $\\Delta$ is effective, the divisor $B_Y$ is effective, thus $\\Delta_Y$ is effective.\n\\end{proof}\n\nFinally, we mention that Theorem \\ref{ambro1}(i)-(iii) was generalised to lc-trivial fibrations with $\\Delta\\geq0$ and $(X,\\Delta)$ is log canonical in \\cite[Lemma 1.1]{FG14}. The proof involves running a careful MMP in order to reduce to a situation where one has a klt-trivial fibration.\n\n\n\\begin{exa}\nTheorem \\ref{ambro1}(iv) does not hold for lc-trivial fibrations. For instance, let $X=\\mathbb{P}^1\\times\\mathbb{P}^1$, let $f$ be the second projection, let $\\delta$ be the diagonal and set $\\Delta=\\delta+\\frac{1}{2}\\{0\\}\\times\\mathbb{P}^1+\\frac{1}{2}\\{\\infty\\}\\times\\mathbb{P}^1$. Then $f\\colon(X,\\Delta)\\to \\mathbb{P}^1$ is an lc-trivial fibration with discriminant supported on $\\{0\\}\\cup\\{\\infty\\}$. By considering log resolutions, one calculates that the discriminant is equal to $\\frac{1}{2}0+\\frac{1}{2}\\infty$, hence the moduli divisor is torsion. Indeed, we have\n$$\\textstyle K_X+\\Delta\\sim_{\\mathbb{Q}}f^*\\big(K_{\\mathbb{P}^1}+\\frac{1}{2}0+\\frac{1}{2}\\infty+M_{\\mathbb{P}^1}\\big),$$\nbut the divisor $K_X+\\Delta$ has bi-degree $(0,-1)$.\n \nHowever, $f$ is not birational to a product. Indeed, it induces a family of rational curves with 4 marked points parametrised by $\\mathbb{P}^1$: for $t\\in \\mathbb{P}^1$, set $p_1(t)=0$, $p_2(t)=1$, $p_3(t)=p_4(t)=t$. This family of marked curves is not trivial, therefore the fibration cannot be locally a product.\n\\end{exa}\n\n\\section{B-Semiampleness conjectures}\n\nWe saw above in Theorem \\ref{thm:kltdescent} that klt singularities descend along a klt-trivial fibration. However, even if one had the full analogue of Theorem \\ref{ambro1} in the log canonical setting one could not follow the same strategy to show that log canonical singularities descend. Moreover, one sees that the use of Kodaira's trick in the proof of Theorem \\ref{thm:kltdescent} obliterated the connection of the coefficients of the divisor $\\Delta_Y$ to that of the divisor $\\Delta$. In order to remedy the situation, something more is needed.\n\nThe main conjecture on the canonical bundle formula predicts something much stronger: that the moduli part is \\emph{semiample} on an Ambro model of the fibration. We discuss it in this section.\n\nThere are two versions of the conjecture; the stronger one was proposed in \\cite[Conjecture 7.13.3]{PS09}. We start with the stronger, \\emph{effective} version.\n\n\\begin{EffbSemCon}\\label{effbsemi}\nFix positive integers $d$ and $r$. Then there exists an integer $m$ depending only on $d$ and $r$, such that for any lc-trivial fibration $f\\colon (X,B)\\rightarrow Y$ with the generic fibre $F$, if $\\dim F=d$ and $r$ is the smallest positive integer such that $r(K_F+B|_F)\\sim 0$, there exists an Ambro model $Y'$ of $f$ such that $mM_{Y'}$ is base point free. \n\\end{EffbSemCon}\n\nMore generally, any conjecture as above, in which $m$ depends on some invariants of the generic fibre of the fibration, goes under the name of \\emph{effective b-semiampleness}. \n\nIn the original statement \\cite[Conjecture 7.13.3]{PS09}, the constant $m$ depended on $\\dim X$ and $r$. The main result of \\cite{Flo14} is that it suffices to show the Effective B-Semiampleness Conjecture in the case where $Y$ is a curve.\nThis led to the formulation of the Effective B-Semiampleness Conjecture above.\n\nThe conjecture is widely open. We list below the cases where it is known. They all make use of some notion of moduli space for the fibres.\n\n\\begin{thm}\\label{thm:eff}\nThe Effective B-Semiampleness Conjecture holds in the following cases:\n\\begin{enumerate}\n\\item if general fibres are elliptic curves by \\cite{Ko60, Ue73}; we have $m=12$;\n\\item if general fibres are rational curves \\cite[Theorem 8.1]{PS09}; \n\\item if general fibres are K3 surfaces or abelian varieties of dimension $d$ by \\cite[Theorem 1.2]{Fuj03}; then we have $m=19k$, respectively $m=k(d+1)$, where $k$ is a weight associated to the Baily-Borel-Satake compactification of the period domain.\n\\end{enumerate}\n\\end{thm}\n\nIn the weaker version of the conjecture we lose control on the constant $m$.\n\n\\begin{bSemCon}\\label{bsemi}\nLet $f\\colon (X,B)\\rightarrow Y$ be an lc-trivial fibration. Then there exists an Ambro model $Y'$ of $f$ such that $M_{Y'}$ is semiample. \n\\end{bSemCon}\n\nMore is known about this conjecture than about its effective version above, although the progress has been limited to the cases where either the bases $Y$ or the fibres of $f$ are low dimensional. We summarise the situation for low dimensional fibres in the following result.\n\n\\begin{thm}\nApart from the cases listed in Theorem \\ref{thm:eff}, the B-Semi\\-am\\-ple\\-ness Conjecture holds in the following cases:\n\\begin{enumerate}\n\\item if the fibres are surfaces of Kodaira dimension 0 by \\cite[Lemma 4.1 and Corollary 6.4]{Fuj03} and \\cite[Part I, (5.15.9)(ii)]{Mo87};\n\\item if $f$ is a klt-trivial fibration and the generic fibre is a uniruled surface not isomorphic to $\\mathbb{P}^2$ by \\cite[Theorem 1.7]{Fil18}.\n \\end{enumerate} \n \\end{thm}\n\nThe B-Semiampleness Conjecture holds for another important family of fibrations:\n\n\\begin{thm}\\label{thm:torsionAmbro}\nLet $f\\colon(X,\\Delta)\\to Y$ be an lc-trivial fibration, and assume that the moduli part $M_Y$ descends to $Y$. If $M_Y\\equiv0$, then $M_Y\\sim_\\mathbb{Q} 0$.\n\nIn particular, if $\\dim Y=1$, then $M_Y$ is semiample.\n\\end{thm}\n\nTheorem \\ref{thm:torsionAmbro} is \\cite[Theorem 3.5]{Amb05a} for klt-trivial fibrations and \\cite[Theorem 1.3]{Flo14} for lc-trivial fibration. Theorem 1.2 in \\cite{Flo14} states that Effective B-Semiampleness Conjecture is true for klt-trivial fibrations with numerically trivial moduli part.\n\nAnother partial result is \\cite[Theorem 3.2]{BC16}. They prove that a small perturbation of the moduli part in a specific direction is semiample, under some effectivity hypotheses for $K_Y+B_Y$.\n\n\\subsection{Restrictions to divisors}\n\nAs we saw in the previous subsection, the progress on the B-Semiampleness Conjecture has been concentrated on the cases of either the low dimension of the base $Y$ or the low dimension of the generic fibre of the fibration $f$. \n\nIn \\cite[Theorem A]{FL18} we obtained the following result towards the conjecture valid in every dimension. Note that the phrase \\emph{the B-semiampleness Conjecture in dimension $n$} means that we consider the conjecture in the case when the dimension of the base $Y$ is $n$.\n\n\\begin{thm}\\label{thm:main}\nAssume the B-Semiampleness Conjecture in dimension $n-1$.\n\nLet $(X,\\Delta)$ be a log canonical pair and let $f\\colon (X,\\Delta)\\to Y$ be an lc-trivial fibration to an $n$-dimensional variety $Y$, where the divisor $\\Delta$ is effective over the generic point of $Y$. Assume that $Y$ is an Ambro model for $f$. \n\nThen for every birational model $\\pi\\colon Y'\\to Y$ and for every prime divisor $T$ on $Y'$ with the normalisation $T^\\nu$ and the induced morphism $\\nu\\colon T^\\nu\\to Y'$, the divisor $\\nu^*\\pi^*M_Y$ is semiample on $T^\\nu$.\n\\end{thm}\n\nAs a corollary, combining with Theorem \\ref{thm:torsionAmbro}, we obtain that the restriction of the moduli part to every prime divisor on every sufficiently high birational model of $Y$ is semiample if $Y$ is a surface.\n\nWe comment on the proof of Theorem \\ref{thm:main}, as it will be useful in the following subsection. We first apply a base change to $Y$ and modify $(X,\\Delta)$ by blowing up suitably, but we try to remember $(X,\\Delta)$ along the proof. We then run a suitable relative MMP over $Y$, which contracts many ``bad'' components of $\\Delta$ (in particular, those with negative coefficients); as a result, we obtain a new lc-trivial fibration $g\\colon (W,\\Delta_W)\\to Y$ with $\\Delta_W\\geq0$ and with the same moduli divisor $M_Y$. Choosing a minimal log canonical centre $S$ of $(W,\\Delta_W)$ which surjects onto $T$, we obtain an induced \\emph{klt}-trivial fibration $g|_S\\colon(S,\\Delta_S)\\to T'$, where $T'$ is obtained from the Stein factorisation of the morphism $S\\to T$. Then we first show that $M_Y|_{T'}$ is \\emph{almost} $M_{T'}$. Even though at this step we may not deduce equality between these two divisors, we can control their difference in a very precise manner. After a suitable further blowup of $Y$, we can force this difference to disappear and we conclude by induction on the dimension.\n\n\\subsection{Reduction result}\n\nAs we mentioned above, in the setup of lc-trivial fibrations $f\\colon(X,\\Delta)\\to Y$ one does not assume that $\\Delta$ is effective. Furthermore, often it is much more difficult to work with lc-trivial fibrations than with klt-trivial fibrations.\n\nIn \\cite{FL18} the B-Semiampleness Conjecture is reduced to the following conjecture with much weaker hypotheses.\n\n\\begin{con}\\label{con:weakbsemi}\nLet $(X,\\Delta)$ be a log canonical pair and let $f\\colon (X,\\Delta)\\to Y$ be a klt-trivial fibration to an $n$-dimensional variety $Y$. If $Y$ is an Ambro model of $f$ and if the moduli divisor $M_Y$ is big, then $M_Y$ is semiample.\n\\end{con}\n\nWe show in \\cite[Theorem E]{FL18}:\n\n\\begin{thm}\\label{KLTimplLC}\nAssume Conjecture \\ref{con:weakbsemi} in dimensions at most $n$. Then the B-Semiampleness Conjecture holds in dimension $n$.\n\\end{thm}\n\nThe proof is similar to the proof of Theorem \\ref{thm:main} sketched above.\n\n\\subsection{Generalisation}\n\nIn the papers \\cite{Fuj18,FFL18} the authors consider slc-trivial fibrations. Those are completely analogous to lc-trivial fibrations, the difference being that the ambient space $X$ is not irreducible, but the pair $(X,\\Delta)$ is \\emph{slc} on the generic fibre of the fibration; such a setup appears occasionally in inductive proofs. The precise statement is \\cite[Definition 4.1]{Fuj18}.\n\nThen one can define, as in the case of lc-trivial fibrations, the moduli divisor and the discriminant. Then \\cite[Theorem 1.2]{Fuj18} proves the analogue of Theorem \\ref{nefness} for slc-trivial fibrations, and \\cite[Theorem 1.3]{FFL18} shows the analogue of Theorem \\ref{thm:torsionAmbro} in this context.\n\n\\section{Parabolic fibrations}\n\nFinally, in this section we discuss a more general situation than that of lc-trivial fibrations.\n\nLet $g\\colon (X,\\Delta)\\to Z$ be a surjective morphism, where $(X,\\Delta)$ is a klt projective pair and $Z$ is a projective variety. Assume that $g_*\\mathcal{O}_X\\big(m(K_X+\\Delta)\\big)\\neq0$ for some positive integer $m$, and consider the relative Iitaka fibration $f\\colon X\\dashrightarrow Y$ associated to $K_X+\\Delta$. Possibly by blowing up further, one may assume that $(X,\\Delta)$ is log smooth and that $f$ is a morphism. Then if $F$ is a general fibre of $f$, we have $\\kappa\\big(F,(K_X+\\Delta)|_F\\big)=0$, however $K_X+\\Delta$ is not necessarily a pullback from $Y$. One still wonders if there is a canonical bundle formula for the map $f$.\n\nThe resulting formula is the canonical bundle formula of Fujino and Mori \\cite{FM00}. We first need a definition, which is justified from the setup above.\n\n\\begin{dfn}\nA \\emph{parabolic fibration} is a fibration $f\\colon (X,\\Delta)\\to Y$, where $(X,\\Delta)$ is a projective klt pair, $Y$ is a smooth projective variety and if $F$ is the generic fibre of $f$, then $\\kappa\\big(F,(K_X+\\Delta)|_F\\big)=0$.\n\\end{dfn}\n\nThe following is \\cite[Section 4]{FM00}, the \\emph{canonical bundle formula} of Fujino and Mori.\n\n\\begin{thm}\\label{thm:FM}\nLet $f\\colon (X,\\Delta)\\to Y$ be a parabolic fibration, where $\\Delta$ is effective. Then there is a commutative diagram\n$$\n\\xymatrix{\nX' \\ar[r]^{\\tau'} \\ar[d]_{f'} & X\\ar[d]^{f}\\\\\nY' \\ar[r]_{\\tau}&Y,\n}\n$$\nwhere $\\tau$ and $\\tau'$ are birational, $X'$ and $Y'$ are smooth, and $f'$ has connected fibres, such that the following holds.\n\nThere exist effective $\\mathbb{Q}$-divisors $B^+$ and $B^-$ on $X'$ without common components, a $\\mathbb{Q}$-divisor $\\Delta'\\geq0$ on $X'$ and $\\mathbb{Q}$-divisors $B_{Y'}$ and $M_{Y'}$ on $Y$ such that\n$$K_{X'}+\\Delta'+B^-\\sim_\\mathbb{Q} f'^*(K_{Y'}+B_{Y'}+M_{Y'})+B^+,$$\nwith the following properties:\n\\begin{enumerate}\n\\item[(i)] the pair $(X',\\Delta')$ is klt and log smooth, and there exists an effective exceptional divisor $E$ on $X'$ such that\n$$K_{X'}+\\Delta'\\sim_\\mathbb{Q}\\tau'^*(K_X+\\Delta)+E,$$ \n\\item[(ii)] $f'_*\\mathcal{O}_{X'}(\\lfloor nB^+\\rfloor)\\simeq\\mathcal{O}_{Y'}$ for all $n\\in\\mathbb{N}$,\n\\item[(iii)] $B^-$ is $f'$-exceptional and $\\tau'$-exceptional,\n\\item[(iv)] the induced map $f'\\colon(X',\\Delta'+B^--B^+)\\to Y'$ is a klt-trivial fibration, and $B_{Y'}$ and $M_{Y'}$ are the corresponding discriminant and moduli divisors,\n\\item[(v)] the pair $(Y',B_{Y'})$ is klt, $B_{Y'}\\geq0$ and $M_{Y'}$ is nef,\n\\item[(vi)] for every $n\\in\\mathbb{N}$ sufficiently divisible we have \n$$H^0\\big(X,n(K_X+\\Delta)\\big)\\simeq H^0\\big(X',n(K_{X'}+\\Delta')\\big)\\simeq H^0\\big(Y',n(K_{Y'}+B_{Y'}+M_{Y'})\\big).$$\n\\end{enumerate}\n\\end{thm}\n \nThere are several non-trivial parts of this formula which do not follow from considerations in the previous sections: the existence of divisors $B^+$ and $B^-$ with the properties (ii) and (iii) above, as well as the fact that $B_{Y'}$ is effective.\n\n\\medskip\n\nPart (vi) follows immediately from (i), (ii) and (iii). We sketch how (iv) follows from (i), (ii) and (iii), following \\cite[Lemma 4.2]{Amb04a}. Let $F'$ be a general fibre of $f'$, and we define the divisor $\\Xi'$ with respect to $f'\\colon(X',\\Delta'+B^--B^+)\\to Y'$ as in Definition \\ref{dfn:lctrivial}(d). We may assume that $\\Xi'=-\\Delta'-B^-+B^+$, and $B^-|_{F'}=0$ by (iii).\n\nWe have $(K_{X'}+\\Delta'+B^--B^+)|_{F'}\\sim_\\Q0$ by construction and $\\kappa\\big(F',(K_{X'}+\\Delta')|_{F'}\\big)=0$ by (i), hence \n$$\\kappa(F',B^+|_{F'})=\\kappa\\big(F',(B^+-B^-)|_{F'}\\big)=0.$$\nSince there exists a positive integer $b$ such that $\\lceil B^+\\rceil |_{F'}\\leq b B^+|_{F'}$, this implies $\\kappa\\big(F',\\lceil B^+\\rceil|_{F'}\\big)=0$. Therefore,\n\\begin{align*}\n\\kappa\\big(F',\\lceil\\Xi'\\rceil |_{F'}\\big)&=\\kappa\\big(F',\\lceil-\\Delta'-B^-+B^+\\rceil |_{F'}\\big)\\\\\n&\\leq\\kappa\\big(F',\\lceil-\\Delta'\\rceil |_{F'}+\\lceil-B^-\\rceil |_{F'}+\\lceil B^+\\rceil |_{F'}\\big)\\\\\n&\\leq \\kappa\\big(F',\\lceil B^+\\rceil |_{F'}\\big)=0.\n\\end{align*}\nThis shows part (d) of Definition \\ref{dfn:lctrivial}, and the rest is easy.\n\n\\medskip\n\nIn order to apply the previous result, we recall that for a log canonical pair $(X,\\Delta)$, the ring\n$$R(X,K_X+\\Delta)=\\bigoplus_{n\\in\\mathbb{N}}H^0\\big(X,\\lfloor n(K_X+\\Delta)\\rfloor\\big)$$\nis the \\emph{canonical ring} of $(X,\\Delta)$. We also recall that for a graded ring $R=\\bigoplus\\limits_{n\\in\\mathbb{N}}R_n$, the \\emph{$d$-th Veronese subring} of $R$ is defined as $R^{(d)}:=\\bigoplus\\limits_{n\\in\\mathbb{N}}R_{dn}$.\n\nAn immediate consequence of Theorem \\ref{thm:FM} is the following result \\cite[Theorem 5.2]{FM00}, which has widespread use in the Minimal Model Program. It often allows to pass from a pair $(X,\\Delta)$ with $\\kappa(X,K_X+\\Delta)\\geq0$ to a pair $(X',\\Delta')$ on which $K_{X'}+\\Delta'$ is big.\n\n\\begin{thm}\nLet $(X,\\Delta)$ be a projective klt pair with $\\kappa(K_X+\\Delta)=\\ell\\geq0$. Then there exist an $\\ell$-dimensional klt pair $(X',\\Delta')$ with $\\kappa(X',K_{X'}+\\Delta')=\\ell$ and positive integers $d$ and $d'$ such that\n$$R(X, K_X + \\Delta)^{(d)} \\simeq R(X', K_{X'}+\\Delta')^{(d')}.$$\n\\end{thm}\n\nThe proof follows immediately from Theorem \\ref{thm:FM}(vi), by combining it with the proof of Theorem \\ref{thm:kltdescent}; see also the proof of Theorem \\ref{thm:cbf} below.\n\nAssume now that $(X,\\Delta)$ has simple normal crossings and let $\\Delta^+$ and $\\Delta^-$ be effective divisors without common components such that $\\Delta=\\Delta^+-\\Delta^-$. Then if $\\kappa\\big(F,(K_X+\\Delta^+)|_F\\big)=0$ for a general fibre $F$ of $f$, and if there exists a good model of $(F,\\Delta^+|_F)$, then it was shown in \\cite[Theorem 3.13]{Fuj15} that the moduli b-divisor is b-nef and good in the sense of \\cite[Definition 3.2]{Amb05a}; this is an application of MMP techniques from \\cite{FG14} and \\cite[Theorem 3.3]{Amb05a}. \n\n\\medskip\n\nWe finish the paper with the following result, which can sometimes be used in order to avoid running a Minimal Model Program; for instance, compare the proofs of \\cite[Lemma 4.4]{GL13} and \\cite[Theorem 5.3]{LP18a}. Note that $\\nu(X,L)$ denotes the \\emph{numerical dimension} of a divisor $L$ on a projective variety $X$, see for instance \\cite[\\S2.2]{LP18a} for basic properties and related references.\n\n\\begin{thm}\\label{thm:cbf}\nLet $(X,\\Delta)$ be a projective klt pair and let $f\\colon (X,\\Delta)\\to Y$ be a parabolic fibration such that $\\nu\\big(F,(K_X+\\Delta)|_F\\big)=0$ for a general fibre $F$ of $f$. Then there exists a commutative diagram\n\\[\n\\xymatrix{ \nX' \\ar[r]^{\\pi'} \\ar[d]_{f'} & X \\ar[d]^{f}\\\\\nY'\\ar[r]_{\\pi} & Y,\n}\n\\]\nwhere $X'$ and $Y'$ are smooth, $f'$ has connected fibres, $\\pi$ and $\\pi'$ are birational, and such that, if we write\n$$K_{X'}+\\Delta'\\sim_\\mathbb{Q} \\pi'^*(K_X+\\Delta)+E',$$\nwhere $\\Delta'$ and $E'$ have no common components, then:\n\\begin{enumerate}\n\\item[(i)] we have\n$$K_{X'}+\\Delta'+B^-\\sim_\\mathbb{Q} f'^*(K_{Y'}+\\Delta_{Y'})+B^+,$$\nwhere the pair $(Y',\\Delta_{Y'})$ is klt, and the divisors $B^+$ and $B^-$ are effective and have no common components,\n\\item[(ii)] $B^-$ is $\\pi'$-exceptional and $f'$-exceptional,\n\\item[(iii)] we have $f'_*\\mathcal{O}_{X'}(\\lfloor\\ell B^+\\rfloor)\\simeq\\mathcal{O}_{Y'}$ for all positive integers $\\ell$.\n\\end{enumerate}\nMoreover, if $K_X+\\Delta$ is pseudoeffective, then $K_{Y'}+\\Delta_{Y'}$ is pseudoeffective.\n\\end{thm}\n\n\\begin{proof}\nBy \\cite[Corollaire 3.4]{Dru11} and \\cite[Corollary V.4.9]{Nak04} we have\n\\begin{equation}\\label{eq:1}\n\\kappa\\big(F,(K_X+\\Delta)|_F\\big)=\\nu\\big(F,(K_X+\\Delta)|_F\\big)=0.\n\\end{equation} \nBy Theorem \\ref{thm:FM} there exists a diagram as in the theorem such that (ii) and (iii) hold, as well as\n$$K_{X'}+\\Delta'+B^-\\sim_\\mathbb{Q} f'^*(K_{Y'}+B_{Y'}+M_{Y'})+B^+,$$\nwhere $(Y',B_{Y'})$ is klt and $M_{Y'}$ is the moduli part of the associated klt-trivial fibration $f'\\colon (X',\\Delta'+B^--B^+)\\to Y'$. Then analogously as in the proof of Theorem \\ref{thm:kltdescent} one shows that there exists an effective $\\mathbb{Q}$-divisor $\\Delta_{Y'}\\sim_\\mathbb{Q} B_{Y'}+M_{Y'}$ such that the pair $(Y',\\Delta_{Y'})$ is klt, which gives (i).\n\nFinally, if $F'$ is a general fibre of $f'$, we have $\\nu\\big(F',(K_{X'}+\\Delta')|_{F'}\\big)=0$ by \\eqref{eq:1} and by \\cite[Lemma 2.3]{LP18a}. Therefore, there exists a good model of $(F',\\Delta'|_{F'})$ by \\cite[Corollaire 3.4]{Dru11} and \\cite[Corollary V.4.9]{Nak04}, hence $(K_{X'}+\\Delta')|_{F'}$ is geometrically abundant in the sense of \\cite[Definition V.2.23]{Nak04}. Then by \\cite[Lemma V.2.27]{Nak04} for an ample divisor $A$ on $Y'$ and for any positive rational number $\\varepsilon$, the divisor $K_{X'}+\\Delta'+\\varepsilon f'^*A$ is geometrically abundant. In particular, $\\kappa(X',K_{X'}+\\Delta'+\\varepsilon f'^*A)\\geq0$, and hence by (i) and by (iii) we have\n\\begin{align*}\n\\kappa\\big(Y',K_{Y'}+\\Delta_{Y'}+\\varepsilon A)&=\\kappa\\big(X',f'^*(K_{Y'}+\\Delta_{Y'}+\\varepsilon A)+B^+\\big)\\\\\n&=\\kappa(X',K_{X'}+\\Delta'+B^-+\\varepsilon f'^*A)\\geq0.\n\\end{align*}\nSince this holds for any positive rational number $\\varepsilon$, we conclude that $K_{Y'}+\\Delta_{Y'}$ is pseudoeffective, as desired.\n\\end{proof}\n\n\\bibliographystyle{amsalpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\subsection{Introductory remarks and statements of the main theorems}\n\nIn recent years there has been a push towards understanding the mechanisms connecting various well-known asymptotic stability results in topology and algebra. For instance, let $\\N$ be a connected oriented manifold of dimension $\\geq 2$, which is the interior of a manifold with boundary, and write $\\Conf_n(\\N)$ for the $n$-strand configuration space\n\\[\n\\Conf_n(\\N) := \\{(x_1,\\ldots,x_n) \\in \\N^n \\mid x_i \\neq x_j\\}.\n\\]\nThere is a natural action on $\\Conf_n(\\N)$ by the symmetric group $\\Sn_n$, and we may therefore define the $n$-strand unordered configuration space\n\\[\n\\UConf_n(\\N) := \\Conf_n(\\N)\/\\Sn_n.\n\\]\nA classical theorem of McDuff \\cite[Theorem 1.2]{McD} implies that for each $i$, and any $n \\gg i$, the group $H_i(\\UConf_n(\\N))$ is independent of $n$. In contrast to the work of McDuff, it can be seen that the analogous statement is not true for the ordered configuration spaces $\\Conf_n(\\N)$. What is true, however, is perhaps the next best thing. It follows from work of Church, Ellenberg, and Farb \\cite[Theorem 6.4.3]{CEF} that for any $i$, there is a polynomial $P \\in \\Q[x]$ such that the Betti number $\\dim_\\Q(H_i(\\Conf_n(U);\\Q))$ agrees with $P(n)$ for $n \\geq 0$. Results of this type fall under the heading of what one might call asymptotic algebra.\\\\\n\nThe modern philosophy of asymptotic algebra can be roughly stated as follows: a family of algebraic objects which display asymptotic stability phenomena can often times be encoded into a single object, which is finitely generated in an appropriate abelian category. In the case of McDuff, for each $i$ and $n$ the group $H_i(\\UConf_n(\\N))$ can be realized as the $n$-graded piece of some finitely generated graded module over $\\Z[x]$. The result of Church, Ellenberg, and Farb involves showing that the $\\Sn_n$-representations $H_i(\\Conf_n(U);\\Q)$ are each constituents of some finitely generated representation of the category $\\FI$, of finite sets and injections (see \\cite{CEF} for more on this notion). This philosophy is also heavily featured in Sam and Snowden's recent resolution of Stembridge's conjecture \\cite{SS}. The goal of this paper is to apply similar techniques to the homologies of the unordered configuration spaces of trees. Note that this problem was considered by L\\\"utgehetmann in his Master's thesis \\cite{L}. The results of that work are disjoint from the current work.\\\\\n\nIn this paper, a \\textbf{graph} will always refer to a connected, compact CW-complex of dimension 1. A \\textbf{tree} is a graph which is contractible as a topological space. An \\textbf{essential vertex} of $G$ is a vertex of degree, or valency, at least 3, while an \\textbf{essential edge} of $G$ is a connected component of the space obtained by removing all essential vertices from $G$. Note that both the essential edges and vertices of a graph are unaffected by subdivision of edges, and can be thought of as the topologically essential pieces of the graph.\\\\\n\nOur main result relates to the asymptotic behavior of the homologies of the braid group of a tree. To state this result, we first need to define a kind of connectivity invariant for trees.\\\\\n\nLet $G$ be a tree. Then we set\n\\[\n\\Delta_G^i := \\max_{\\{\\{v_{j_1},\\ldots,v_{j_i}\\} \\mid v_{j_k}\\text{ essential}\\}} \\{ \\dim_\\Q(H_0(G - \\{v_{j_1},\\ldots,v_{j_i}\\};\\Q))\\}.\n\\]\nIn words, $\\Delta_G^i$ is the maximum number of connected components that $G$ can be broken into by removing exactly $i$ essential vertices. Therefore $\\Delta_G^1$ is the maximum degree of a vertex in $G$, while, if $G$ has $N_G$ essential vertices, $\\Delta_G^{N_G}$ is the number of essential edges of $G$. By convention, $\\Delta_G^0 = 1$, while $\\Delta_G^i = 0$ for $i > N_G$.\\\\\n\n\\begin{thmab}\\label{bettipolyrefine}\nLet $G$ be a tree, and write $B_nG$ to denote the braid group $B_nG := \\pi_1(\\UConf_n(G))$. Then for each $i$ there is a polynomial $P_i \\in \\Q[x]$ of degree $\\Delta_G^i-1$, such that for all $n \\geq 0$,\n\\[\nP_i(n) = \\dim_\\Q(H_i(B_nG;\\Q)).\n\\]\n\\text{}\\\\\n\\end{thmab}\n\n\\begin{remark}\nIt follows from a Theorem of Ghrist \\cite[Theorem 3.3]{Gh} that given any graph $G$ not homeomorphic to $S^1$, $H_i(B_nG) = 0$ for all $i$ strictly greater than the number of essential vertices of $G$. This is realized in the case where $G$ is a tree in Theorem \\ref{bettipolyrefine} by the fact that $\\Delta_G^i - 1 = -1$ in these cases.\\\\\n\\end{remark}\n\nThe polynomial $P_i$ of Theorem \\ref{bettipolyrefine} is explicitly computed throughout the course of this work (see Theorem \\ref{explicitpoly}). This computation implies something somewhat surprising about these homology groups.\\\\\n\n\\begin{corab}\\label{degreesequence}\nLet $G$ be a tree, and let $i \\geq 0$. Then the homology groups $H_i(B_nG)$ depend only on $i, n$, and the degree sequence of $G$.\\\\\n\\end{corab}\n\nIt is interesting to note that the rank of $H_i(B_nG)$ agrees with a polynomial for all $n \\geq 0$, as opposed to only agreeing for $n$ sufficiently large. In the case of configuration spaces of manifolds, a result of this kind does have precedent. We have already discussed the result of Church, Ellenberg, and Farb which states that if $\\N$ is an oriented manifold, which is the interior of a manifold with non-empty boundary, then for any $i$ the dimension of $H_i(\\Conf_n(\\N);\\Q)$ agrees with a polynomial for all $n$ \\cite[Theorem 6.4.3]{CEF}. It is perhaps an interesting observation that trees can be thought of as graphs with non-trivial boundary. It is unclear whether the connection to the work of Church, Ellenberg, and Farb goes any deeper than this, however. \\\\\n\n\\subsection{An Outline of the proof}\n\nTo prove Theorem \\ref{bettipolyrefine}, we will rely on classical techniques in commutative algebra, as well as more modern techniques in combinatorial topology. The first key ingredient is the discrete Morse theory of Forman \\cite{Fo}. Given a regular CW complex $X$ (see Definition \\ref{regcw}), write $\\K_i$ for the set of $i$-cells of $X$. A \\textbf{discrete Morse function} is a map $f$ from the cells of $X$ to $\\R$ satisfying the following two hypotheses for all cells $\\sigma \\in \\K_i$:\n\n\\begin{enumerate}\n\\item $|\\{\\tau \\in \\K_{i+1} \\mid \\sigma \\subseteq \\overline{\\tau} \\text{ and } f(\\sigma) \\geq f(\\tau)\\}| \\leq 1$; \\label{dm1}\n\\item $|\\{\\tau \\in \\K_{i-1} \\mid \\tau \\subseteq \\overline{\\sigma} \\text{ and } f(\\sigma) \\leq f(\\tau)\\}| \\leq 1$. \\label{dm2}\n\\end{enumerate}\n\nWe call $\\sigma \\in \\K_i$ a \\textbf{critical $i$-cell} with respect to $f$, if the sets of conditions $\\ref{dm1}$ and $\\ref{dm2}$ are both empty. The main consequence of discrete Morse theory is that the critical cells of $X$ determine its homotopy type. Formally, let $a \\in \\R$ and write $X(a)$ for the subcomplex of $X$ comprised of the closures of cells $\\sigma$ with $f(\\sigma) \\leq a$. If there is no critical cell $\\sigma$ with $a < f(\\sigma) < b$, then $X(a)$ is a deformation retract of $X(b)$. Otherwise, if there is a unique critical cell $\\sigma$ with $a < f(\\sigma) < b$, then $X(b)$ is obtained from $X(a)$ by attaching the cell $\\sigma$. Moreover, there is a complex\n\\begin{eqnarray}\n\\ldots \\rightarrow \\M_i \\rightarrow \\ldots \\stackrel{\\widetilde{\\partial}}\\rightarrow \\M_1 \\rightarrow \\M_0 \\rightarrow 0\\label{mc}\n\\end{eqnarray}\nwhere $\\M_i$ is a free $\\Z$-module on the critical $i$-cells of $X$, whose homology is the homology of the space $X$. We call the differential $\\widetilde{\\partial}$ the \\textbf{Morse differential} (see Definition \\ref{morsediff}).\\\\\n\nUsing work of Abrams \\cite{A}, Farley and Sabalka were able to impose a discrete Morse structure on the spaces $\\UConf_n(G)$, where $G$ is any graph \\cite{FS}. We will spend a good amount of time recounting the construction of Farley and Sabalka in Section \\ref{confg}. Once we have accomplished this, our strategy will be to develop a strong understanding of the critical cells. More specifically, we will work towards understanding the changing behaviors of the critical cells as $n$ varies.\\\\\n\nLet $G$ be a graph with $E$ essential edges, and write $S_G := \\Z[x_1,\\ldots,x_E]$ for the integral polynomial ring in $E$-variables. We will prove the following in Section \\ref{morsecomplexismodule}.\\\\\n\n\\renewcommand{\\labelitemi}{(\\textasteriskcentered)}\n\\begin{itemize}\n\\item For each $i$, there exists a finitely generated graded $S_G$-module $\\M_{i,\\dt}$ for which $\\M_{i,n}$ is a free $\\Z$-module with basis vectors indexed by the critical $i$-cells of $\\UConf_n(G)$.\\\\\n\\end{itemize}\n\nSpecializing to the case where $G$ is a tree, work of Farley \\cite{Fa} implies that the Morse differential is always trivial. Using the complex (\\ref{mc}) we obtain the following.\\\\\n\n\\begin{thmab}\\label{homologyismodule}\nLet $G$ be a tree with $E$ essential edges, and let $S_G$ denote the integral polynomial ring in $E$ variables. Then for each $i$ and $n$, the action of $S_G$ on the critical cells of $\\UConf_n(G)$ described in Section \\ref{morsecomplexismodule} imposes the structure of a finitely generated graded $S_G$-module on the abelian group $\\bigoplus_{n \\geq 0} H_i(\\UConf_n(G))$. In particular, there exists a finitely generated graded $S_G$-module $\\mathcal{H}_i$ such that,\n\\[\n(\\mathcal{H}_i)_n \\cong H_i(\\UConf_n(G)).\n\\]\n\\text{}\\\\\n\\end{thmab}\n\nA result of Ghrist and Abrams (see Theorem \\ref{asp}) implies the spaces $\\UConf_n(G)$ are aspherical. It follows immediately from this that $H_i(\\UConf_n(G)) \\cong H_i(B_nG)$. With this in mind, the first part of Theorem \\ref{bettipolyrefine} simply follows from the existence of the Hilbert polynomial. Of course, the above theorem does not tell us anything about the degree of the Hilbert polynomial, nor does it bound its obstruction. To accomplish this, we must first prove a structure theorem about the modules $\\mathcal{H}_i$.\\\\\n\nTo state this theorem, we first recall the definition of a \\textbf{squarefree monomial ideal}. We say an ideal $I \\subseteq \\Q[x_1,\\ldots,x_d]$ is a squarefree monomial ideal, if it contains a generating set of monomials, none of which are divisible by a square in $\\Q[x_1,\\ldots,x_d]$. These ideals are the subject of Stanley-Reisner theory, and have many desirable properties. For instance, much is known about their Hilbert polynomial (see \\cite{MS} for a reference on the subject).\\\\\n\n\\begin{thmab}\\label{sqfr}\nLet $G$ be a tree, and let $\\mathcal{H}_i$ denote the $S_G$-module of Theorem \\ref{homologyismodule}. Then $\\mathcal{H}_i \\otimes_\\Z \\Q$ is isomorphic to a direct sum of graded twists of squarefree monomial ideals, each having dimension at most $\\Delta^i_G$.\\\\\n\\end{thmab}\n\nWe will find that this theorem implies the conclusions of Theorem \\ref{bettipolyrefine}. In fact, we will be able to compute the polynomial $P_i$ associated to $\\mathcal{H}_i$ explicitly in terms of invariants of the tree $G$ (see Theorem \\ref{explicitpoly}).\\\\\n\n\\subsection{An overview of the paper}\n\nIn the next section, we will spend time developing necessary background. This includes short summaries of discrete Morse theory (Section \\ref{dmorsethy}), and configuration spaces of graphs (Sections \\ref{confg} and \\ref{dmorse}). Following this, we use the machinery developed in these preliminary sections to prove the statement (\\textasteriskcentered) (Section \\ref{morsecomplexismodule}). Finally, we specialize the the case of trees, and use enumerative combinatorial methods to prove Theorem \\ref{bettipolyrefine} via an explicit computation of the Hilbert polynomial (Sections \\ref{caseoftrees} and \\ref{hilbertcomp}).\\\\\n\nTo finish the paper, we briefly consider the case of a general graph $G$. Note that while most of the explicit results in this paper are limited to the case where $G$ is a tree, the statement (\\textasteriskcentered) will hold for any graph $G$. A result like Theorem \\ref{homologyismodule} will therefore hold for general graphs so long as we know that the Morse differential commutes with the action of $S_G$ on $\\M_{i,\\dt}$. It is the belief of the author that the action of $S_G$, or perhaps a slight alteration there of, will indeed commute with this differential. Unfortunately, it is known that this differential can become tremendously complicated as $G$ increases in complexity (see \\cite{KP}). In any case, the methods in this work therefore provide at least a strategy for proving stability results for more general graphs. In the final sections, we discuss some implications of this.\\\\\n\n\\section*{Acknowledgments}\nThe author would like to send thanks to Jordan Ellenberg for his many insights, as well as his help in editing this paper. The author would also like to send special thanks to Steven Sam for his vital advice in approaching the primary problem of this work. A great amount of gratitude must also be sent to Daniel Farley, whose expertise in the field was invaluable to the author during his learning of the material. Finally, the author would like the acknowledge the generous support of the National Science Foundation through NSF-RTG grant DMS-502553.\\\\\n\n\n\\section{Preliminaries}\n\\subsection{Discrete Morse theory}\\label{dmorsethy}\n\nWe now take the time to briefly summarize the key points in Forman's discrete Morse theory \\cite{Fo}. We will largely be following the exposition of Forman \\cite{Fo}\\cite{Fo2}, Farley and Sabalka \\cite{FS}, and Ko and Park \\cite{KP}.\\\\\n\nIn the introduction, we spent some time discussing the notion of discrete Morse function. One thing that should stand out about this definition is that the literal values of the function are immaterial. Namely, the classification of critical cells is unchanged by composition with any strictly monotone function $\\R \\rightarrow \\R$. In many cases it is often easier to construct the relationships between the cells, rather than the discrete Morse function itself. This hints towards the construction of what are known as discrete vector fields. We will use this approach during the exposition of this, and all future sections.\\\\\n\nTo begin, we first must place certain light restrictions on the spaces we will be working with.\\\\\n\n\\begin{definition}\\label{regcw}\nLet $X$ be a CW complex. A \\textbf{cell} of $X$ will always refer to an open cell in $X$. Given a cell $\\sigma$ of dimension $i$, we will often write $\\sigma^{(i)}$ to indicate that $\\sigma$ has dimension $i$. We will write $\\K$ to denote the set of cells of $X$, and $\\K_i$ to denote the set of $i$-cells of $X$.\\\\\n\nA cell $\\tau^{(i)} \\subseteq \\overline{\\sigma^{i+1}}$ is said to be a \\textbf{regular face} of a cell $\\sigma^{(i+1)}$ if, given a characteristic map $\\Phi_\\sigma:D^{i+1} \\rightarrow X$ for $\\sigma^{(i+1)}$, $\\Phi_{\\sigma}^{-1}(\\tau)$ is a closed ball, and the map $\\Phi_{\\sigma}|_{\\Phi_\\sigma^{-1}(\\tau)}$ is a homeomorphism. We say that the complex $X$ is \\textbf{regular} if given a pair of cells $\\tau^{(i)} \\subseteq \\overline{\\sigma^{(i+1)}}$, $\\tau^{(i)}$ is a regular face of $\\sigma^{(i+1)}$. Equivalently, $X$ is regular if and only if the attaching map of each of its cells is a homeomorphism.\n\\end{definition}\n\nWe will assume throughout most of our exposition that $X$ is a regular CW complex. In his original paper on discrete Morse theory \\cite{Fo}, Forman proves some of his results without the requirement that $X$ be regular. The spaces $\\UConf_n(G)$, for any graph $G$, are actually cubical complexes, which are certainly regular CW complexes. Therefore, the condition that $X$ be regular is not restrictive for what we need.\n\n\\begin{definition}\nLet $X$ be a regular CW complex. A \\textbf{discrete vector field} $V$ on $X$ is a collection of partially defined functions $V_i:K_i \\rightarrow K_{i+1}$ satisfying the following three conditions for each $i$:\n\\begin{enumerate}\n\\item $V_i$ is injective;\n\\item the image of $V_i$ is disjoint from the domain of $V_{i+1}$;\n\\item for any $\\sigma^{(i)}$ in the domain of $V_i$, $\\sigma^{(i)}$ is a face of $V_i(\\sigma^{(i)})$.\\\\\n\\end{enumerate}\n\nGiven a regular CW complex $X$ equipped with a discrete vector field $V$, a \\textbf{cellular path} between two cells $\\alpha^{(i)}$ and $\\beta^{(i)}$ is a finite sequence of $i$-cells\n\\[\n\\alpha^{(i)} = \\alpha_0^{(i)}, \\alpha_1^{(i)}, \\ldots, \\alpha_{l-1}^{(i)}, \\alpha_l^{(i)} = \\beta^{(i)}\n\\]\nsuch that $\\alpha_{j+1}^{(i)}$ is a face of $V_i(\\alpha_{j}^{(i)})$. We say that the path is \\textbf{closed} if $\\alpha^{(i)} = \\beta^{(i)}$, and we say it is \\textbf{trivial} if $\\alpha_j^{(i)} = \\alpha_{k}^{(i)}$ for all $j,k$.\\\\\n\nA discrete vector field $V$ is said to be a \\textbf{discrete gradient vector field} if it admits no non-trivial closed cellular paths.\\\\\n\nIf $V$ is a discrete gradient vector field on a regular CW complex $X$, then we call a cell $\\sigma$ of $X$ \\textbf{redundant} if $\\sigma$ is in the domain of $V_i$ for some $i$, \\textbf{collapsible} if it is in the image of $V_i$ for some $i$, and \\textbf{critical} otherwise.\\\\\n\\end{definition}\n\n\\begin{proposition}[\\cite{FS} Proposition 2.2, \\cite{Fo} Theorem 3.4]\\label{VFdecomp}\nLet $X$ be a regular CW complex equipped with a discrete gradient vector field $V$. Consider the filtration\n\\[\n\\emptyset = X_0'' \\subseteq X_0' \\subseteq X_1'' \\subseteq X_1' \\subseteq \\ldots \\subseteq X_n'' \\subseteq X_n' \\subseteq \\ldots\n\\]\nwhere $X_i'$ is the $i$-skeleton of $X$ with the redundant cells removed, and $X_i''$ is the $i$-skeleton of $X$ with both the redundant and critical cells removed. Then:\n\\begin{enumerate}\n\\item For any $i$, $X'_i$ is obtained from $X''_i$ by attaching $m_i$ $i$-cells to $X_n''$ along their boundaries, where $m_i$ is the number of critical $i$-cells of the discrete gradient vector field $V$.\n\\item For any $i$, $X_{i+1}''$ deformation retracts onto $X_i'$.\\\\\n\\end{enumerate}\n\\end{proposition}\n\nThe above proposition leads to one notable corollary, which we record now.\\\\\n\n\\begin{corollary}[\\cite{FS} Proposition 2.3, \\cite{Fo} Corollary 3.5]\\label{critdecomp}\nLet $X$ be a regular CW complex equipped with a discrete gradient vector field $V$. Then $X$ is homotopy equivalent to a CW complex with precisely $m_i$ $i$-cells for each $i$, where $m_i$ is the number of critical $i$-cells of $V$.\\\\\n\\end{corollary}\n\nJust as is the case with traditional Morse theory, the decomposition of the space $X$ given by Corollary \\ref{critdecomp} can be used to compute the homology groups of $X$. For simplicity, we will state the construction for cubical complexes, although the general case is similar.\\\\\n\n\\begin{definition}\\label{morsediff}\nLet $X$ be a cubical complex, and write $C_i(X)$ for the free abelian group of $i$-cells of $X$. If $c \\cong I_1 \\times \\ldots \\times I_i$ is an $i$-cell of $X$, with $I_j \\cong [0,1]$, then we define\n\\[\nc^j_{\\tau} = I_1 \\times \\ldots \\times I_{j-1} \\times \\{0\\} \\times I_{j+1} \\times \\ldots \\times I_i, \\hspace{.5cm} c^j_{\\iota} = I_1 \\times \\ldots \\times I_{j-1} \\times \\{1\\} \\times I_{j+1} \\times \\ldots \\times I_i.\n\\]\nThis allows us to define a boundary morphism\n\\[\n\\partial: C_i(X) \\rightarrow C_{i-1}(X)\n\\]\ngiven by\n\\[\n\\partial(c) := \\sum_j c^j_\\tau - c^j_\\iota,\n\\]\nturning $C_\\dt(X)$ into a chain complex. It is a well known fact that the homology of this chain complex is the usual homology of the space $X$.\\\\\n\nFurther assume that $X$ is equipped with a discrete gradient vector field $V$. Then we have a map $R:C_i(X) \\rightarrow C_i(X)$ defined by\n\\[\nR(c) = \\begin{cases} 0 &\\text{ if $c$ is collapsible}\\\\ c &\\text{ if $c$ is critical}\\\\ \\pm \\partial(V_i(c)) + c &\\text{ otherwise,}\\end{cases}\n\\]\nwhere the sign of $\\partial(V_i(c))$ in the above definition is chosen so that $c$ has a negative coefficient. The property that $V$ has no non-trivial closed paths implies that $R^m(c) = R^{m+1}(c)$ for all $m \\gg 0$ and all $i$-cells $c$ \\cite{Fo}. We set $R^\\infty(c)$ to be this stable value.\\\\\n\nFor each $i$, let $\\M_i$ denote the free abelian group with basis indexed by the critical $i$-cells of $V$. Then the \\textbf{Morse complex} associated to $V$ is defined to be\n\\[\n\\M_\\dt : \\ldots \\rightarrow \\M_n \\rightarrow \\ldots \\rightarrow \\M_1 \\stackrel{\\widetilde{\\partial}}\\rightarrow \\M_0 \\rightarrow 0,\n\\]\nwhere boundary map $\\widetilde{\\partial}$ is given by\n\\[\n\\widetilde{\\partial}(c) := R^\\infty(\\partial(c))\n\\]\nThe map $\\widetilde{\\partial}$ is known as the \\textbf{Morse differential}.\\\\\n\\end{definition}\n\n\n\\begin{theorem}[Theorem 8.2 \\cite{Fo}, Theorem 7.3 \\cite{Fo2}]\\label{morsediffcomp}\nFor all $i$ there are isomorphisms,\n\\[\nH_i(X) \\cong H_i(\\M_\\dt)\n\\]\n\\text{}\\\\\n\\end{theorem}\n\n\\subsection{The configuration spaces of graphs}\\label{confg}\n\nIn this section we review necessary facts about configuration spaces of graphs. In the next section, we will explain how the techniques of discrete Morse theory apply to these spaces.\\\\\n\n\\begin{definition}\nA \\textbf{graph} is any compact, connected CW complex of dimension one. A \\textbf{tree} is a topologically contractible graph. Given a graph $G$ with vertex $v$, we will write $\\mu(v)$ to denote the degree of the vertex $v$. Note that any loop on $v$ contributes 2 to its degree count. We say that $v$ is \\textbf{essential} if $\\mu(v) \\geq 3$. If $\\mu(v) = 1$, we say that $v$ is a \\textbf{boundary} vertex, and the unique edge connected to $v$ is called a \\textbf{boundary edge}.\\\\\n\nThe \\textbf{configuration space on $n$ points} of $G$ is the topological space,\n\\[\n\\Conf_n(G) := \\{(x_1,\\ldots,x_n) \\in G^n \\mid x_i \\neq x_j \\text{ if } i \\neq j\\}.\n\\]\nWe note that there is a natural fixed-point-free action on $\\Conf_n(G)$ by the symmetric group $\\Sn_n$. The \\textbf{unordered configuration space on $n$ points} of $G$ is the quotient space,\n\\[\n\\UConf_n(G) := \\Conf_n(G)\/\\Sn_n.\n\\]\n\\text{}\\\\\n\\end{definition}\n\nFor the majority of this paper, we will work with the spaces $\\UConf_n(G)$. Note that many of the structural theorems discussed in this section will apply to both spaces.\\\\\n\nIn order to apply discrete Morse theory to questions about these configuration spaces, we will first need to place a CW complex structure on them. To accomplish this, we use a theorem of Abrams \\cite{A}.\\\\\n\n\\begin{definition}\nThe \\textbf{discretized configuration space on $n$ points} over $G$, $D_n(G)$, is the CW subcomplex of $G^n$ spanned by cells of the form\n\\[\n\\sigma_1 \\times \\ldots \\times \\sigma_n\n\\]\nwhere $\\sigma_i \\subseteq G$ is a cell (i.e. an edge or vertex) of $G$, and $\\overline{\\sigma_i} \\cap \\overline{\\sigma_j} = \\emptyset$ whenever $i \\neq j$. We write $UD_n(G)$ to denote the quotient of $D_n(G)$ by the action of the symmetric group.\\\\\n\\end{definition}\n\n\\begin{theorem}[\\cite{A} Theorem 2.1, \\cite{KKP} Theorem 2.4, \\cite{PS}]\\label{cellstructure}\nLet $G$ be a graph, and assume that $G$ satisfies the following two properties:\n\\begin{enumerate}\n\\item each path connecting distinct vertices of degree $\\neq 2$ has length at least $n-1$;\n\\item each homotopically essential path connecting a vertex to itself has length at least $n+1$.\n\\end{enumerate}\nThen the inclusions $D_n(G) \\hookrightarrow \\Conf_n(G)$, and $UD_n(G) \\hookrightarrow \\UConf_n(G)$ are homotopy equivalences.\\\\\n\\end{theorem}\n\n\\begin{remark}\nNote that in the first cited source, Abrams states the theorem assuming that each path connecting distinct vertices of degree $\\neq 2$ has length at least $n+1$. It is noted after the proof that the version of the theorem stated above is true, and a brief argument is given for how it is proven. In the second source, Kim, Ko, and Park give a formal argument for this improvement. In the third source, Prue and Scrimshaw provide a discrete Morse theory argument, which is independent of the first two sources. For our purposes, the exact number of vertices needed is unimportant, as we can always just subdivide the edges of $G$ more if needed.\\\\\n\\end{remark}\n\nNote that subdividing edges of a graph $G$ does not impact the configuration spaces $\\Conf_n(G)$ and $\\UConf_n(G)$. For much of what follows, we will often just assume, without explicit mention, $G$ is subdivided enough so that the homotopy equivalence of Theorem \\ref{cellstructure} holds.\\\\\n\nTheorem \\ref{cellstructure} implies that $\\Conf_n(G)$ and $\\UConf_n(G)$ are homotopy equivalent to cubical complexes of dimension $n$. In fact, we will be able to do better than this.\\\\\n\n\\begin{theorem}[\\cite{Gh} Theorem 3.3]\\label{dimconf}\nLet $G$ be a graph which is not homeomorphic to $S^1$. Then $\\Conf_n(G)$ and $\\UConf_n(G)$ are homotopy equivalent to CW complexes of dimension $N_G$, where $N_G$ is the number of essential vertices of $G$.\\\\\n\\end{theorem}\n\n\\begin{remark}\nIf $G$ is homeomorphic to $S^1$, then $\\UConf_n(G)$ is easily seen to be homotopy equivalent to a circle for all $n$. Throughout this work, and the literature in general, it is a recurring theme that certain theorems only apply to graphs which are neither $S^1$ nor the interval. It is interesting to observe that these two graphs are precisely those which are homeomorphic to compact manifolds.\\\\\n\\end{remark}\n\nNote that Ghrist originally proved Theorem \\ref{dimconf} using more classical topological means. We will later see it naturally falls out of the discrete Morse structure that Farley and Sabalka placed on $UD_n(G)$. This was first noted by Farley and Sabalka in \\cite{FS}.\\\\\n\nOne remarkable thing to note about Ghrist's theorem is that the dimension of the configuration spaces of graphs is independent of $n$. This behavior is in stark contrast to the behavior of configuration spaces of smooth manifolds of dimension $\\geq 2$.\\\\\n\nIn his paper \\cite{Gh}, Ghrist also proves that configuration spaces of graphs are, in fact, aspherical. This result was later reproven by Abrams \\cite[Theorem 3.10]{A}, where he shows that both $D_n(G)$ and $UD_n(G)$ are universally covered by CAT(0) complexes.\\\\\n\n\\begin{theorem}[\\cite{Gh} Theorem 3.1, \\cite{A} Corollary 3.11]\\label{asp}\nLet $G$ be a graph. Then $\\Conf_n(G)$, and hence $\\UConf_n(G)$, are aspherical. That is to say, $\\pi_k(\\Conf_n(G)) = 0$ for $k > 1$.\\\\\n\\end{theorem}\n\nNote that this theorem is analogous to that which says the configuration spaces of the plane are aspherical. In that case, the fundamental groups of the ordered and unordered spaces are the Artin pure braid groups, and the Artin braid groups, respectively. We borrow this terminology for our context as well.\\\\\n\n\\begin{definition}\nLet $G$ be a graph. The \\textbf{braid group on $n$ strands} of $G$ is defined to be\n\\[\nB_nG := \\pi_1(\\UConf_n(G)).\n\\]\nWe similarly define the \\textbf{pure braid group on $n$ strands} as\n\\[\nP_nG := \\pi_1(\\Conf_n(G))\n\\]\n\\text{}\\\\\n\\end{definition}\n\nThe study of the braid groups of graphs is still a very active area of research. See \\cite{KKP}\\cite{KP}\\cite{FS2} for more on these groups.\\\\\n\nAs an immediate corollary to Theorem \\ref{asp}, we obtain the following.\\\\\n\n\\begin{corollary}\nLet $G$ be a graph. then there are isomorphisms\n\\[\nH_i(\\Conf_n(G)) \\cong H_i(P_nG), H_i(\\UConf_n(G)) \\cong H_i(B_nG).\n\\]\n\\text{}\\\\\n\\end{corollary}\n\nThe goal for this paper is to establish a methodology for understanding stability phenomena in the groups $H_i(B_nG)$, in the spirit of modern trends of asymptotic algebra. Note that we will spend very little time considering pure braid groups. In fact, there doesn't seem to be much literature about the homology of these groups, as they are vastly more complicated than the braid groups \\cite{KP}\\cite{BF}.\\\\\n\nThe following theorem of Farley exactly computes the homology groups when $G$ is a tree.\\\\\n\n\\begin{theorem}[\\cite{Fa}]\\label{treefree}\nLet $G$ be a tree. Then the group $H_i(B_nG)$ is free for all $n \\geq 0$.\n\\end{theorem}\n\nNote that the theorem of \\cite{Fa} is slightly more general than this. We will discuss the more general version in later sections.\\\\\n\nExpanding upon the work of Farley, the following theorem of Ko and Park suggests that the tree case is indicative of a more general phenomenon.\\\\\n\n\\begin{theorem}[\\cite{KP} Theorem 3.6]\\label{kkpthm}\nLet $G$ be a graph. Then $G$ is planar if and only if $H_1(B_nG)$ is torsion free for any, and therefore all, $n\\geq 2$. Moreover, in the case where $G$ is not planar, all torsion is 2-torsion.\\\\\n\\end{theorem}\n\nKim, Ko, and Park proved that $G$ is planar if and only if $H_1(B_2G,\\Z)$ is torsion free in \\cite[Theorem 5.5]{KKP}. It is also conjectured in that work that Theorem \\ref{kkpthm} is true. It should be noted that \\cite[Theorem 3.6]{KP} is far stronger than what we have written above. In fact, their result explicitly computes the groups $H_1(B_nG,\\Z)$ in terms of combinatorial invariants of the graph $G$. It follows from their computation that the amount 2-torsion of the group is unvarying in $n$. Moreover, if $G$ is biconnected - that is, $G$ requires the removal of at least 2 vertices to disconnect it - then $H_1(B_nG,\\Z) \\cong H_1(B_2G,\\Z)$ for all $n \\geq 2$ \\cite[Lemma 3.12]{KP}. In the final section of this paper we will conjecture an extension of this fact to the higher homologies (see Conjecture \\ref{mainconj}).\\\\\n\nTo finish this section, we record a result of Gal on the Euler characteristic of these spaces. Note that Gal proves a more general theorem for computing the Euler characteristic of the configuration space of any simplicial complex.\\\\\n\n\\begin{theorem}[\\cite{Ga} Theorem 2]\\label{eulercharacteristic}\nLet $G$ be a graph, and set\n\\[\n\\mathfrak{e}(t) := \\sum_{n \\geq 0} \\frac{\\chi(\\Conf_n(G))}{n!}t^n.\n\\]\nThen,\n\\[\n\\mathfrak{e}(t) = \\frac{\\prod_{v} (1-(1-\\mu(v))t)}{(1-t)^E}\n\\]\nwhere $E$ is the number of edges of $G$.\\\\\n\\end{theorem}\n\n\\begin{remark}\n$\\Conf_n(G)$ is an $n!$-fold cover of $\\UConf_n(G)$. It follows from this that the above formula can be easily used to compute the Euler characteristic of $\\UConf_n(G)$ as well.\\\\\n\\end{remark}\n\n\\begin{definition}\nLet $G$ be a graph which has at least one essential vertex, and let $\\mathcal{E}$ denote the set of essential vertices of $G$. Then an \\textbf{essential edge} of $G$ is a connected component of $G - \\mathcal{E}$.\\\\\n\\end{definition}\n\n\\begin{corollary}\\label{eulerpolynomial}\nLet $G$ be a graph with at least one essential vertex, and let $E$ denote the number of essential edges of $G$. Then there is a polynomial $P$ of degree $E-1$, for all $n \\geq 0$, such that the function\n\\[\nn \\mapsto \\chi(\\UConf_n(G))\n\\]\nis equal $P(n)$.\n\\end{corollary}\n\n\\begin{proof}\nWe first note that smoothing the degree 2 vertices of $G$ does not impact the spaces $\\UConf_n(G)$. We may therefore assume without loss of generality that $G$ does not have any such vertices. In this case, the Euler characteristic $\\chi(\\UConf_n(G))$ is the coefficient of $t^n$ in the power series expansion of\n\\[\n\\frac{\\prod_{v \\text{ essential}} (1-(1-\\mu(v)t))}{(1-t)^{E}}\n\\]\nA straight forward enumerative combinatorics argument implies that the $n$-th coefficient of the power series expansion of this expression is a polynomial of degree exactly $E-1$ for $n$ sufficiently large. It remains to show that this agreement begins at $n = 0$.\\\\\n\nAssume that $G$ has $N_G$ essential vertices. It is a standard fact from enumerative combinatorics that the coefficients of the power series expansion of a rational function of the form $\\frac{f(x)}{(1-t)^l}$ will agree with a polynomial for all $n$ so long as $\\deg(f) < l$. This is the case in our specific instance, so long as $N_G > 1$.\n\\end{proof}\n\nTheorem \\ref{eulercharacteristic} was largely the inspiration for this work. It suggests that the asymptotically the Betti numbers of $H_i(B_nG)$ are polynomial. In this work we will prove this suggestion in the case where $G$ is a tree. We will also provide a setup in the case where $G$ is a general graph, which hopefully will be able to illustrate this behavior in that case as well.\\\\\n\n\\subsection{The discrete Morse theory of $\\UConf_n(G)$} \\label{dmorse}\n\nIn this section we will outline the discrete Morse structure on $\\UConf_n(G)$ developed in the work of Farley and Sabalka \\cite{FS}. We begin by fixing $n$, and assuming the graph $G$ is sufficiently subdivided for Theorem \\ref{cellstructure} to hold for $\\UConf_n(G)$.\\\\\n\nFix a spanning tree $T$ for $G$, as well as an embedding of $T$ into the plane. We will label the vertices of $T$ by applying a depth-first search. Concretely, begin by choosing a boundary vertex $v_0$ to be the root of $T$, and label it with the number 0. Continue down the boundary edge adjacent to $v_0$, labeling any vertices encountered along the way with increasing labels. If at any point an essential vertex is encountered, then travel down the leftmost - relative to the current direction of travel - edge whose vertices have not been labeled. If a boundary vertex is encountered, then one returns to the most recently passed essential vertex. An example of a correctly labeled tree is given in Figure \\ref{labeledtree}.\\\\\n\n\\begin{figure}\n\\begin{tikzpicture}\n \\tikzstyle{every node}=[draw,circle,fill=white,minimum size=4pt,\n inner sep=0pt]\n \\draw (0,0) node (0) [label=above:$0$] {}\n\t\t --(.5,0) node (1) [label=above:$1$]{}\n\t\t\t\t\t--(1,0) node (2) [label=right:$2$]{}\n\t\t\t\t\t--(1,.5) node (3) [label=left:$3$]{}\n\t\t\t\t\t--(1,1) node (4) [label=left:$4$]{}\n\t\t\t\t\t--(1,1.5) node (5) [label=left:$5$]{}\n\t\t\t\t\t--(1,2) node (6) [label=left:$6$]{}\n \t\t\t(4) -- (1.5,1) node (7) [label=above:$7$]{}\n\t\t\t\t\t--(2,1) node (8) [label=above:$8$]{}\n\t\t\t\t\t(2) -- (1,-.5) node (9) [label=left:$9$]{}\n\t\t\t\t\t--(1,-1) node (10) [label=left: $10$]{};\n\n\\end{tikzpicture}\n\\caption{A tree which is properly labeled. Note that this tree is also sufficiently subdivided to apply Theorem \\ref{cellstructure} with $n = 3$.}\\label{labeledtree}\n\\end{figure}\n\nFor the remainder of this section, $G$ and $T$ will be as in the previous paragraphs. As before we will write $\\K$ to denote the set of cells of $\\UConf_n(G)$, while $\\K_i$ will denote the set of $i$-cells of $\\UConf_n(G)$.\\\\\n\n\\begin{definition}\\label{dgvf}\nGiven an edge $e$ of $G$, we write $\\iota(e)$ to denote its largest, with respect to the given labeling, vertex, and $\\tau(e)$ to denote its smallest vertex. If $v$ is a vertex of $G$, which is not $v_0$, then we write $e(v)$ to denote the unique edge of $T$ for which $\\iota(e(v)) = v$. We note that $e(v)$ will be the first edge on the unique path within $T$ from $v$ to $v_0$.\\\\\n\nLet $c = \\{v_{j_1},\\ldots, v_{j_{n-i}}, e_{r_1},\\ldots, e_{r_i}\\}$ be an $i$-cell of $UD_n(G)$. If\n\\[\n\\tau(e(v_j)) \\cap \\{v_{j_1},\\ldots, v_{j_{n-i}}, \\iota(e_{r_1}),\\tau(e_{r_1}),\\ldots, \\iota(e_{r_i}),\\tau(e_{r_i})\\} \\neq \\emptyset,\n\\]\nthen we say that $v_j$ is \\textbf{blocked} in $c$. Note that by convention $v_0$ is always blocked.\\\\\n\nWe define a partial function $V_i: \\K_i \\rightarrow \\K_{i+1}$ inductively in the following way. Given an $i$-cell $c = \\{v_{j_1},\\ldots, v_{j_{n-i}}, e_{r_1},\\ldots, e_{r_i}\\}$, assume without loss of generality that $v_{j_1}$ is an unblocked vertex of $c$ which is smallest with respect to the labeling of $T$. Assuming that $c$ is not in the image of $V_{i-1}$, we set\n\\[\nV_i(\\{v_{j_1},\\ldots, v_{j_{n-i}}, e_{r_1},\\ldots, e_{r_i}\\}) := \\{v_{j_2},\\ldots, v_{j_{n-i}}, e_{r_1},\\ldots, e_{r_i}, e(v_{j_1}))\\}\n\\]\nIf $c$ does not have any unblocked vertices, or if $c$ is in the image of $V_{i-1}$, then $V_i(c)$ is undefined.\\\\\n\\end{definition}\n\n\\begin{theorem}[\\cite{FS} Sections 3.1 and 3.2]\nThe collection of partial functions $V_i:\\K_i \\rightarrow \\K_{i+1}$ form a discrete gradient vector field on $UD_n(G)$.\\\\\n\\end{theorem}\n\nWhile the spaces $\\UConf_n(G)$ can be high dimensional, we can still visualize their cells. Indeed, it is often useful to think of the cells of $\\UConf_n(G)$ as being subsets of edges and vertices of the graph $G$ itself. In Figure \\ref{cellexample} we see examples of a critical, a collapsible, and a redundant 1-cell of V, where $G$ is taken to be the tree in Figure \\ref{labeledtree} and $n = 3$. In these examples, the cell $\\{v_0,v_1,e_{9,10}\\}$ is collapsible, as\n\\[\n\\{v_0,v_1,e_{9,10}\\} = V_0(\\{v_0,v_1,v_{10}\\}),\n\\]\nand $v_{10}$ is the smallest unblocked vertex. The cell $\\{v_1,v_5,e_{4,7}\\}$ is redundant, as $v_1$ is not blocked, and it is not in the image of $V_0$. Indeed, the only cell that could possibly map to it would be $\\{v_1,v_5,v_7\\}$. However, in this case $v_1$ is the smallest unblocked vertex. Finally, the cell $\\{v_0,v_5,e_{4,7}\\}$ is critical, as its every vertex is blocked, and it is not collapsible.\\\\\n\n\\begin{figure}\n\\begin{tikzpicture}\n\\tikzstyle{every node}=[draw,circle,fill=white,minimum size=4pt,\n inner sep=0pt]\n \\draw (0,0) node (0) [fill = black, label=above:$0$] {}\n\t\t --(.5,0) node (1) [label=above:$1$]{}\n\t\t\t\t\t--(1,0) node (2) [label=right:$2$]{}\n\t\t\t\t\t--(1,.5) node (3) [label=left:$3$]{}\n\t\t\t\t\t--(1,1) node (4) [fill = black, label=left:$4$]{}\n\t\t\t\t\t--(1,1.5) node (5) [fill = black, label=left:$5$]{}\n\t\t\t\t\t--(1,2) node (6) [label=left:$6$]{}\n \t\t\t(4) -- (1.5,1) node (7) [fill = black, label=above:$7$]{}\n\t\t\t\t\t--(2,1) node (8) [label=above:$8$]{}\n\t\t\t\t\t(2) -- (1,-.5) node (9) [label=left:$9$]{}\n\t\t\t\t\t--(1,-1) node (10) [label=left: $10$]{};\n\t\t \\draw [line width=1mm] (4) -- (7);\n\\end{tikzpicture}\n\\begin{tikzpicture}\n\\tikzstyle{every node}=[draw,circle,fill=white,minimum size=4pt,\n inner sep=0pt]\n \\draw (0,0) node (0) [fill = black, label=above:$0$] {}\n\t\t --(.5,0) node (1) [fill = black, label=above:$1$]{}\n\t\t\t\t\t--(1,0) node (2) [label=right:$2$]{}\n\t\t\t\t\t--(1,.5) node (3) [label=left:$3$]{}\n\t\t\t\t\t--(1,1) node (4) [label=left:$4$]{}\n\t\t\t\t\t--(1,1.5) node (5) [label=left:$5$]{}\n\t\t\t\t\t--(1,2) node (6) [label=left:$6$]{}\n \t\t\t(4) -- (1.5,1) node (7) [label=above:$7$]{}\n\t\t\t\t\t--(2,1) node (8) [label=above:$8$]{}\n\t\t\t\t\t(2) -- (1,-.5) node (9) [fill = black, label=left:$9$]{}\n\t\t\t\t\t--(1,-1) node (10) [fill = black, label=left: $10$]{};\n\t\t \\draw [line width=1mm] (9) -- (10);\n\\end{tikzpicture}\n\\begin{tikzpicture}\n\\tikzstyle{every node}=[draw,circle,fill=white,minimum size=4pt,\n inner sep=0pt]\n \\draw (0,0) node (0) [label=above:$0$] {}\n\t\t --(.5,0) node (1) [fill = black, label=above:$1$]{}\n\t\t\t\t\t--(1,0) node (2) [label=right:$2$]{}\n\t\t\t\t\t--(1,.5) node (3) [label=left:$3$]{}\n\t\t\t\t\t--(1,1) node (4) [fill = black, label=left:$4$]{}\n\t\t\t\t\t--(1,1.5) node (5) [fill = black, label=left:$5$]{}\n\t\t\t\t\t--(1,2) node (6) [label=left:$6$]{}\n \t\t\t(4) -- (1.5,1) node (7) [fill = black, label=above:$7$]{}\n\t\t\t\t\t--(2,1) node (8) [label=above:$8$]{}\n\t\t\t\t\t(2) -- (1,-.5) node (9) [label=left:$9$]{}\n\t\t\t\t\t--(1,-1) node (10) [label=left: $10$]{};\n\t\t \\draw [line width=1mm] (4) -- (7);\n\\end{tikzpicture}\n\\caption{The critical 1-cell $\\{v_0,v_5,e_{4,7}\\}$, the collapsible 1-cell $\\{v_0,v_1,e_{9,10}\\}$, and the redundant 1-cell $\\{v_1,v_5,e_{4,7}\\}$, respectively.}\\label{cellexample}\n\\end{figure}\n\nFrom these examples, one can imagine there existing a way to classify all cells of the three types. To write down such a classification, we first need a bit of nomenclature.\\\\\n\n\\begin{definition}\nLet $c = \\{v_{j_1},\\ldots, v_{j_{n-i}}, e_{r_1},\\ldots, e_{r_i}\\}$ be an $i$-cell of $UD_n(G)$. We say that an edge $e \\in c$ is \\textbf{order respecting in $c$} if $e$ is in $T$, and for all $v \\in c$ such that $\\tau(e(v)) = \\tau(e)$, the label of $v$ in $T$ is larger than the label of $\\iota(e)$.\\\\\n\\end{definition}\n\n\\begin{remark}\nNote that edges $e \\in G-T$ - which we call \\textbf{deleted edges} - are, by definition, never order respecting.\\\\\n\\end{remark}\n\n\\begin{theorem}[The Classification Theorem, \\cite{FS} Theorem 3.6]\nLet $c$ be a cell of $UD_n(G)$, and let $T$ be a choice of spanning tree of $G$ equipped with a planar embedding. Then:\n\\begin{enumerate}\n\\item $c$ is critical if and only if it contains no order respecting edges and all of its vertices are blocked;\n\\item $c$ is redundant if and only if\n\\begin{enumerate}\n\\item it contains no order respecting edges and at least one of its vertices is unblocked or,\n\\item it contains an order respecting edge (and thus a minimal order respecting edge e) and there is some unblocked vertex $v$ such that the label of $v$ in $T$ is less than that of $\\iota(e)$;\n\\end{enumerate}\n\\item $c$ is collapsible if and only if it contains an order respecting edge (and thus a minimal order respecting edge $e$) and, for any vertex $v$ such that the label of $v$ in $T$ is less than that of $\\iota(e)$, $v$ is blocked.\\\\\n\\end{enumerate}\n\\end{theorem}\n\nIt is not hard to check that the previous examples satisfy the conditions of the Classification Theorem.\\\\\n\nOne useful fact about the discrete gradient vector field $V$ is that its critical cells are, in some sense, the most restrictive of the three possible types. Indeed, we may expand upon the classification of critical cells in the following way.\\\\\n\n\\begin{lemma}\\label{critlemma}\nLet $c = \\{v_{j_1},\\ldots, v_{j_{n-i}}, e_{r_1},\\ldots, e_{r_i}\\}$ be a critical $i$-cell of $V$.\n\\begin{enumerate}\n\\item For each $k$ in $\\{1,\\ldots,i\\}$, either $e_{r_k}$ is a deleted edge, or $\\tau(e_{r_k})$ is an essential vertex of $T$.\n\\item Let $c'$ be a critical $i$-cell with edges $e_{r_1},\\ldots,e_{r_i}$, such that the number of its vertices in each component of $T - \\{\\overline{e_{r_1}},\\ldots,\\overline{e_{r_i}}\\}$ agrees with the number of vertices of $c$ in these same components. Then, $c = c'$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFor the first statement, let $e$ be an edge of $c$ which is not deleted. If $\\tau(e) = v_0$, then it is clear that $e$ is order respecting. We note that $\\tau(e)$ is not a boundary vertex in any other case by how the labeling on $T$ was defined. It therefore remains to show that $\\tau(e)$ does not have degree 2. Indeed, if this were the case, then for any vertex $v$ of $G$, $\\tau(e(v)) \\neq \\tau(e)$. In particular, $e$ must be order preserving. This proves the first claim.\\\\\n\nFor the second claim, one uses the fact that the vertices of $c$ must all be blocked.\\\\\n\\end{proof}\n\nNote that the observations made in the above lemma were originally pointed out by Farley and Sabalka when they defined the vector field $V$ \\cite[Section 3.3]{FS}. We collected these observations in Lemma \\ref{critlemma} so that they could be easily referred back to during future sections. We will find that this lemma is critical in defining the polynomial ring structure on $\\bigoplus_n H_i(B_nG)$. In the next section, we will find that the deleted edges of $G$ can be chosen so that they remain unchanged as $n$ increases and we repeatedly subdivide the graph. The above lemma also suggests that those edges of critical cells contained in the tree $T$ are unchanging in $n$. While this is not literally the case, as all edges of $T$ are being subdivided as $n$ increases, Lemma \\ref{critlemma} tells us that the important information encoded by such an edge is the essential vertex of its tail, as well as the direction it is leaving this essential vertex. For this reason we will often be a bit loose with our language and claim that two critical cells with different choices of $n$ have the ``same'' edges.\\\\\n\n\\begin{definition}\nLet $c$ be a critical cell of the discrete gradient vector field, and let $e \\in T$ be an edge in $c$. The classification theorem implies that there must exist some vertex $v \\in c$ such that the label of $v$ in $T$ is smaller than that of $\\iota(e)$, while $\\tau(e(v)) = \\tau(e)$. We refer to such a vertex as a \\textbf{witness} for $e$ in $c$. If $v$ is the only witness for $e$ in $c$, then we say $v$ is \\textbf{necessary}.\\\\\n\\end{definition}\n\n\\section{The proof Theorem \\ref{bettipolyrefine}}\n\n\\subsection{The setup}\n\nWe use this section to fix the notation which will be used throughout the proofs of the main theorems. Note that all of the constructions presented in the previous sections were already well established in the literature prior to this work. To the knowledge of the author, the main construction of the following sections - namely the action of $S_G$ on the critical cells of $\\UConf_n(G)$ - does not appear elsewhere in the literature.\\\\\n\nAs was the case in all previous sections, let $G$ denote a graph, which is neither a circle nor a line segment. We will reserve $E$ to denote the number of essential edges of $G$, while $N_G$ will denote the number of essential vertices of $G$.\\\\\n\nNext, we must construct our spanning tree $T$, as in Section \\ref{dmorse}. To do this, we first subdivide every edge of $G$, which connects two essential vertices, once. Note that this includes any loops of $G$. Once this is done, we choose a spanning tree $T$ of $G$ which satisfies the following:\n\n\\begin{eqnarray}\n\\text{Every edge of $G$ not in $T$ is adjacent to an essential vertex of $G$.}\\label{treecondition}\n\\end{eqnarray}\n\nNote that it is not entirely obvious that such a $T$ exists for an arbitrary graph $G$. This fact was proven by Farley and Sabalka during the proof of \\cite[Theorem 4.4]{FS}.\\\\\n\n\\begin{remark}\nIt is noted by Farley and Sabalka that if one chooses $T$ to satisfy (\\ref{treecondition}), then Lemma \\ref{critlemma} and Corollary \\ref{critdecomp} imply Theorem \\ref{dimconf}.\\\\\n\\end{remark}\n\nHaving chosen our spanning tree $T$ to satisfy (\\ref{treecondition}), we observe that for $n \\geq 2$, we can sufficiently subdivide $G$ for Theorem \\ref{cellstructure} by subdividing $T$. This follows from the fact that we began by subdividing all edges of $G$ which connected essential vertices. We will often not differentiate the spanning trees chosen for each $n$ for this very reason.\\\\\n\n\\subsection{The modules $\\M_{i,\\dt}$}\\label{morsecomplexismodule}\n\nRecall from Section \\ref{dmorsethy} the Morse complex\n\\[\n\\ldots \\rightarrow \\M_i \\rightarrow \\ldots \\rightarrow \\M_1 \\rightarrow \\M_0 \\rightarrow 0\n\\]\nwhere $\\M_i$ is the free $\\Z$-module with basis indexed by critical cells. In our case, we write $\\M_{i,n}$ to denote the free $\\Z$-module with basis indexed by the critical cells of the Farley-Sabalka discrete gradient vector field of $UD_n(G)$.\\\\\n\nLet $\\mathfrak{e}_1,\\ldots,\\mathfrak{e}_E$ denote the essential edges of the graph $G$. Then we set\n\\[\nS_G := \\Z[x_{\\mathfrak{e}_1},\\ldots,x_{\\mathfrak{e}_{E}}]\n\\]\n\nOur goal for the remainder of this section will be to argue, for each $i$, that $\\M_{i,\\dt}$ has the structure of a finitely generated graded $S_G$-module.\\\\\n\n\\begin{definition}\nFix $i$ and $n$, and let $\\mathfrak{e}$ denote an essential edge of $G$. Given a critical $i$-cell $c = \\{v_{j_1},\\ldots, v_{j_{n-i}}, e_{r_1},\\ldots, e_{r_i}\\}$ of $UD_n(G)$, we define $x_{\\mathfrak{e}} \\cdot c$ to be the unique critical $i$-cell of $UD_n(G)$ obtained from $c$ by adding a vertex to the connected component of $T - \\{\\overline{e_{r_1}},\\ldots,\\overline{e_{r_i}}\\}$ containing $\\mathfrak{e}$. More precisely, $x_{\\mathfrak{e}}\\cdot c$ is obtained from $c$ by adding the smallest vertex on the connected component containing $\\mathfrak{e}$. This is well defined by Lemma \\ref{critlemma}, as well as the choice of the tree $T$, as $T$ will contain a unique representative of each essential edge of $G$ by construction. This turns the collection $\\{\\M_{i,n}\\}_n$ into a graded $S_G$-module, which we denote $\\M_{i,\\dt}$.\\\\\n\\end{definition}\n\nOne can visualize the action of $S_G$ described in the above definition in the following way. The labeling on $T$ induces a natural flow on $T$. Namely, all edges flow towards the root of $T$. Let $c = \\{v_{j_1},\\ldots, v_{j_{n-i}}, e_{r_1},\\ldots, e_{r_i}\\}$ be a critical cell. We imagine the vertices $v_j$ as being particles drifting in the direction of the flow, while the edges $e_j$, along with their endpoints, are stationary blockades. For any essential edge $\\mathfrak{e}$, the action of $x_{\\mathfrak{e}}$ on $c$ involves placing a new particle somewhere on $\\mathfrak{e}$, and allowing it to flow until it too is blocked.\\\\\n\nWe provide an illustration of the action of $S_G$ in Figure \\ref{multiplication}.\\\\\n\n\\begin{figure}\n\\begin{tikzpicture}\n\\tikzstyle{every node}=[draw,circle,fill=white,minimum size=4pt,\n inner sep=0pt]\n \\draw (0,0) node (0) [fill = black, label=above:$0$] {}\n\t\t --(.5,0) node (1) [label=above:$1$]{}\n\t\t\t\t\t--(1,0) node (2) [label=right:$2$]{}\n\t\t\t\t\t--(1,.5) node (3) [label=left:$3$]{}\n\t\t\t\t\t--(1,1) node (4) [fill = black, label=left:$4$]{}\n\t\t\t\t\t--(1,1.5) node (5) [fill = black, label=left:$5$]{}\n\t\t\t\t\t--(1,2) node (6) [label=left:$6$]{}\n \t\t\t(4) -- (1.5,1) node (7) [fill = black, label=above:$7$]{}\n\t\t\t\t\t--(2,1) node (8) [label=above:$8$]{}\n\t\t\t\t\t(2) -- (1,-.5) node (9) [label=left:$9$]{}\n\t\t\t\t\t--(1,-1) node (10) [label=left: $10$]{};\n\t\t\t\t\t\n\t \\draw [line width=1mm] \n\t\t\t\t (4) -- (7);\n\t\t\t\t\n\t \\draw[->, line width=.5mm]\n\t\t (3, .5) -- (5,.5);\n\t\t\t\t\n\t \\draw (6,0) node (0) [fill = black, label=above:$0$] {}\n\t\t --(6.5,0) node (1) [label=above:$1$]{}\n\t\t\t\t\t--(7,0) node (2) [label=above:$2$]{}\n\t\t\t\t\t--(7.5,0) node(3)[label = right:$3$]{}\n\t\t\t\t\t--(7.5,.5) node (4) [label=left:$4$]{}\n\t\t\t\t\t--(7.5,1) node (5) [label=left:$5$]{}\n\t\t\t\t\t--(7.5,1.5) node (6) [fill = black, label=left:$6$]{}\n\t\t\t\t\t--(7.5,2) node (7) [fill = black,label = left:$7$]{}\n\t\t\t\t\t--(7.5,2.5) node (8) [fill = black,label = left:$8$]{}\n\t\t\t\t\t--(7.5,3) node (9) [label = left:$9$]{}\n\t\t\t\t\t(6) -- (8,1.5) node (10) [fill = black, label = above:$10$]{}\n\t\t\t\t\t--(8.5,1.5) node (11) [label = above: $11$]{}\n\t\t\t\t\t--(9,1.5) node (12) [label = above: $12$]{}\n\t\t\t\t\t(3) -- (7.5,-.5) node (13) [label = left:$13$]{}\n\t\t\t\t\t-- (7.5,-1) node (14) [label = left:$14$]{}\n\t\t\t\t\t-- (7.5,-1.5) node (15) [label = left:$15$]{};\n\t\t\\draw [line width = 1mm]\n\t\t (6)--(10);\n\t\t\n\\end{tikzpicture}\n\\caption{An illustration of multiplication by $x_{\\mathfrak{e}}$, where $\\mathfrak{e}$ is the essential edge containing the boundary vertex $v_6$ of the tree $G$ of Figure \\ref{labeledtree}.}\\label{multiplication}\n\\end{figure}\n\n\\begin{lemma}\\label{fgcomplex}\nFor each $i$, the module $\\M_{i,\\dt}$ is finitely generated over $S_G$.\\\\\n\\end{lemma}\n\n\\begin{proof}\nWe claim that every critical cell in $\\M_{i,n}$, with $n > 2i$, can be obtained from a critical cell in $\\M_{i,n-1}$. Indeed, if $c$ is a critical cell in $\\M_{i,n}$, then at least one vertex is not a necessary witness of some edge in $e$. Remove the vertex, among those which are not necessary witness vertices, which occupies the maximal position with respect to the labeling on $T$. Removing this vertex leaves us with a critical cell $c'$ in $\\M_{i,n-1}$, as this removal cannot create unblocked vertices nor order respecting edges. It follows that $c$ is the image of $c'$ under the action of the appropriate essential edge.\\\\\n\\end{proof}\n\n\\begin{remark}\\label{slowgrowth}\nIt follows from the above lemma that the number of critical $i$-cells grows, as a function of $n$, like a polynomial of degree $\\leq E-1$. In fact, if $i < N_G$, we claim that this polynomial must have degree strictly less than $E-1$. First, we note that the action of $S_G$ on a critical cell $c$ does not affect the edges of $c$. Because of this it follows that $\\M_{i,\\dt}$ can be expressed as a direct sum of graded $S_G$-modules, where each summand corresponds to a choice of edges in our critical $i$-cell. For any fixed critical $i$-cell $c$, one observes that two variables $x_{\\mathfrak{e}}$ and $x_{\\mathfrak{e}'}$ will act identically whenever $\\mathfrak{e}$ and $\\mathfrak{e'}$ are on the same connected component of $G$ with the edges of $c$ removed. It follows that the summand of $\\M_{i,\\dt}$ containing this cell has Hilbert polynomial of degree strictly less than $E$ whenever $i < N_G$. The Hilbert polynomial of the whole of $\\M_{i,\\dt}$ is a sum of such polynomials, and therefore also has degree strictly less than $E$. This will be used in the final section of the paper.\\\\\n\\end{remark}\n\nLemma \\ref{fgcomplex}, and Theorem \\ref{morsediffcomp} now imply some asymptotic data about the homology groups $H_i(B_nG)$.\\\\\n\n\\begin{theorem}\nLet $G$ be a graph with $E$ essential edges. Then for all $i \\geq 0$, there exists a polynomial $P_i \\in \\Q[t]$, such that for all $n \\gg i$\n\\[\n\\dim_\\Q(H_i(B_nG;\\Q)) \\leq P_i(n).\n\\]\n\\text{}\\\\\n\\end{theorem}\n\n\\subsection{The case of trees}\\label{caseoftrees}\n\nIn this section, we begin to explore the specific case where $G$ is a tree. The work of Farley \\cite{Fa} will allow us to conclude quite a bit more than we are able to in the general case. To begin with, we state a refined version of Theorem \\ref{treefree}.\\\\\n\n\\begin{theorem}[\\cite{Fa}]\\label{treefree2}\nLet $G$ be a tree. Then for each $n$ the Morse complex $\\M_{i,n}$ has trivial differential. In particular,\n\\[\n\\mathcal{H}_i \\cong \\M_{i,\\dt},\\\\\n\\]\nas $S_G$-modules, where $\\mathcal{H}_i$ is the module of Theorem \\ref{homologyismodule}.\\\\\n\\end{theorem}\n\nTo state our main stability theorem, we begin with the following definition.\\\\\n\n\\begin{definition}\\label{deltaig}\nLet $G$ be a tree. Then we define the quantity\n\\[\n\\Delta_G^i := \\max_{\\{\\{v_{j_1},\\ldots,v_{j_i}\\} \\mid v_{j_k} \\text{ essential}\\}} \\{ \\dim_\\Q(H_0(G - \\{v_{j_1},\\ldots,v_{j_i}\\};\\Q))\\}.\n\\]\nFor example, $\\Delta_G^1$ is the maximum degree of $G$, while $\\Delta_{G}^{N_G}$ is $E$, the number of essential edges. By convention, $\\Delta_G^0 = 1$ while $\\Delta_G^i = 0$ for $i > N_G$.\\\\\n\\end{definition}\n\n\\begin{theorem}\\label{treedeg}\nLet $G$ be a tree, and fix $i\\geq 0$. Then the Betti number $b_i(n) := \\dim_\\Q(H_i(B_n(G);\\Q))$ is equal to a polynomial of degree $\\Delta_G^i - 1$ for $n \\gg i$.\\\\\n\\end{theorem}\n\n\\begin{proof}\nThe Theorem \\ref{treefree2} and the work of the previous sections imply that $b_i(n)$ is eventually a polynomial; it remains only to show that this polynomial has the claimed degree.\\\\\n\nWe begin by partitioning the basis vectors of $\\M_{i,n}$ according to the collection of edges which appear. The action of $S_G$ does not impact the edges, and so each of these partitions will correspond to a summand of $\\M_{i,n}$. Moreover, considering any one of these summands, we observe that the variables induce at most $\\Delta_G^i$ distinct operators on $\\M_{i,n}$. It follows that the degree of the Hilbert polynomial of $\\M_{i,\\dt}$ is at most $\\Delta_G^i-1$. To finish the proof, we will show that $\\M_{i,\\dt}$ has a summand whose Hilbert polynomial has degree $\\Delta_G^i-1$.\\\\\n\nFix the $i$ essential vertices of $G$ whose absence realizes the maximum in Definition \\ref{deltaig}. For each essential vertex in this fixed list, let $e$ be an edge adjacent to it in the second leftmost direction, relative to the direction of the root. Note that any critical cell containing these edges has a unique essential edge which can house a witness vertex. There is therefore a critical $i$-cell in $\\M_{i,2i}$ constructed by choosing each of these edges, along with the unique witness vertex for each. Denote this cell by $c$. The submodule generated by $c$ is a summand of $\\M_{i,\\dt}$. We claim that this submodule our desired summand.\\\\\n\nAny $i$-cell of $\\M_{i,n}$ which shares the same edges of $c$ - i.e. those basis vectors in the $n$-th graded piece of $S_G\\cdot c$ - is entirely determined by distributing $n-2i$ vertices to one of $\\Delta_G^i$ components. Thus,\n\\[\n\\rank_\\Z((S_G\\cdot c)_n) = \\binom{n-2i + \\Delta_G^i - 1}{\\Delta_G^i-1}.\n\\]\nas desired.\\\\\n\\end{proof}\n\nThe only piece of Theorem \\ref{bettipolyrefine} which remains to be proven is in showing that $b_i(n)$ agrees with a polynomial for all $n \\geq 0$. To accomplish this, we must first prove Theorem \\ref{sqfr}. As a first simplification, we note that our answer is unaffected by changing basis to $\\Q$, and may therefore assume we are working over a rational polynomial ring. This grants us access to many classical approaches to solving the problem. We will also make heavy use of the direct sum decomposition of $\\M_{i,\\dt}$ described in the proof of Theorem \\ref{treedeg}.\\\\\n\n\\begin{definition}\\label{esspair}\nLet $G$ be a tree. The notation\n\\[\n\\overline{v} := (v_{j_1},\\ldots,v_{j_i})\n\\]\nwill always refer to an $i$-tuple of essential vertices of $G$, appearing in the order induced by the labeling on $G$. We write $\\mu(\\overline{v})$ to denote the number of connected components of $G-\\{v_{j_1},\\ldots,v_{j_i}\\}$. Given $\\overline{v}$, let $\\overline{l} := (l_1,\\ldots,l_i)$ denote an $i$-tuple of positive integers such that\n\\[\n1 \\leq l_k \\leq \\mu(v_{j_k})-2 \\text{ for all $k$}.\n\\]\nWe will also set\n\\[\n|\\overline{l}| := \\sum_k l_k.\n\\]\nGiven a pair $(\\overline{v},\\overline{l})$, we associate to it the summand of $\\M_{i,\\dt} \\otimes_\\Z \\Q$ generated in degree $2i$ by critical cells $c$ satisfying the following two properties:\n\\begin{enumerate}\n\\item If $e \\in c$ is an edge, then $\\tau(e) = v_{j_k}$ for some $k$, and;\n\\item if $e \\in c$ has $\\tau(e) = v_{j_k}$, then there are precisely $l_k$ essential edges which can house witness vertices of $e$.\\\\\n\\end{enumerate}\nNote that these two pieces of data uniquely determine the edges which are allowed to appear in cells which form the bases for the graded pieces of the summand. Denote this summand by $N^{(\\overline{v},\\overline{l})}$.\\\\\n\nWe observe that two variables $x_{\\mathfrak{e}_1}, x_{\\mathfrak{e}_2}$ of $S_G$ act identically on $N^{(\\overline{v},\\overline{l})}$ if and only if $\\mathfrak{e}_1$ and $\\mathfrak{e}_2$ are contained in the same connected component of $G - \\{v_{j_1},\\ldots,v_{j_i}\\}$. Let $S^{(\\overline{v},\\overline{l})}$ denote the quotient of $S_G \\otimes_\\Z \\Q$ by $x_{\\mathfrak{e}_2} - x_{\\mathfrak{e}_2}$. In particular, $S^{(\\overline{v},\\overline{l})}$ is isomorphic to a rational polynomial ring in $\\mu(\\overline{v})$ variables, and $N^{(\\overline{v},\\overline{l})}$ is a finitely generated graded module over $S^{(\\overline{v},\\overline{l})}$.\\\\\n\\end{definition}\n\n\\begin{remark}\nIf $i = 0$, then $\\overline{v}$ and $\\overline{l}$ are empty tuples with $\\mu(\\overline{v}) = 1$. If $i > N_G$, then by convention $N^{(\\overline{v},\\overline{l})} = 0$.\\\\\n\\end{remark}\n\nnext goal will be to show that the Hilbert function\n\\[\nn \\mapsto \\dim_\\Q(N^{(\\overline{v},\\overline{l})})\n\\]\nagrees with a polynomial for all $n$. Note that we already know that this Hilbert function is a polynomial for $n$ sufficiently large, and must only show that this agreement is the case for all $n$. This will imply the same about the Hilbert function for $\\M_{i,\\dt}$. We accomplish this through the computations in the next section, although we spend time now to set some ground work in this direction. Following this, we will explicitly compute this Hilbert polynomial. We begin with the following.\\\\\n\n\\begin{proposition}\\label{monoideal}\nThe module $N^{(\\overline{v},\\overline{l})}(i)$ is isomorphic to a squarefree monomial ideal of $S^{(\\overline{v},\\overline{l})}$.\n\\end{proposition}\n\n\\begin{proof}\nWe recall that a monomial ideal is called squarefree, if it is generated by monomials, none of which are divisible by a square.\\\\\n\nBy definition, $N^{(\\overline{v},\\overline{l})}(i)_{i} = N^{(\\overline{v},\\overline{l})}_{2i}$ is spanned by critical cells containing the edges $e_1,\\ldots,e_i$ determined by the pair $(\\overline{v},\\overline{l})$ (see Definition \\ref{esspair}), along with a single witness vertex for each. To define a map $\\phi:N^{(\\overline{v},\\overline{l})}(i) \\rightarrow S^{(\\overline{v},\\overline{l})}$, it suffices to specify where each of these cells is mapped. We set for any such cell $c$\n\\begin{eqnarray}\n\\phi(c) = \\prod_{\\mathfrak{e}}x_{\\mathfrak{e}}, \\label{monodecomp}\n\\end{eqnarray}\nwhere the product is over essential edges containing a witness vertex of $c$. Note that it is impossible for a single essential edge to house a witness vertex for multiple edges, and so the above monomial is indeed squarefree. Extending this map through the action of $S^{(\\overline{v},\\overline{l})}$ defines our desired isomorphism.\\\\\n\\end{proof}\n\nSquarefree monomial ideals are some of the most well understood objects in commutative algebra. They are also the subject of the so-called Stanley-Reisner theory (see \\cite{MS} for a comprehensive reference on the subject).\\\\\n\n\\subsection{Computing the Hilbert polynomial}\\label{hilbertcomp}\n\nWe note that for $i = 1$, the polynomial describing the Betti number $b_1(n)$ has been explicitly computed in terms of invariants of the tree $G$. Indeed, this follows from the structure theorem of Ko, and Park \\cite[Theorem 3.6]{KP}, and also from the work of Farley and Sabalka \\cite{FS}. The Hilbert polynomial has also been computed for all $i$ in the case where $G$ is a tree with maximal degree 3 by Farley \\cite{Fa}. Aside from these cases, no explicit description of the polynomial is known. That being said, the work in this paper implies that computing this polynomial is no more difficult than computing the Hilbert polynomials of some squarefree monomial ideals. In other words, for any tree $G$, the work of this paper reduces the task of computing $H_i(B_nG)$ for all $n$ to a finite computation. We will proceed with this computation in this section.\\\\\n\n\\begin{remark}\nProposition \\ref{monoideal} reveals that the monomial ideals which appear as graded shifts of summands of $\\mathcal{H}_i$ are quite simple. Because of this, we will find that the computation of the Hilbert polynomial of $\\mathcal{H}_i$ falls more in the realm of enumerative combinatorics rather than Stanley-Reisner theory. It is likely that studying the cases of more general graphs will require more robust commutative algebra, and we therefore proceed with the case that $G$ is a tree using that language. Our hope is that once the case of general graphs is completed, this will aid in creating a uniform means of approaching asymptotic behaviors in the configuration spaces of graphs.\\\\\n\\end{remark}\n\nIn this section we will use the decomposition of Proposition \\ref{monoideal} to compute the Hilbert polynomial of $\\M_{i,\\dt}$. Proposition \\ref{monoideal} implies that $N^{(\\overline{v},\\overline{l})}(i)$ is isomorphic to a squarefree monomial ideal of $S^{(\\overline{v},\\overline{l})}$. To ease computations, we reorder the variables of $S^{(\\overline{v},\\overline{l})}$ to satisfy the following. Begin by labeling the essential edges adjacent to $v_{j_1}$ which house witness vertices by $x_1$ through $x_{l_1}$, chosen in the order induced by our ordering of the vertices of $G$. Next, label the essential edges housing a witness vertex adjacent to $v_{j_2}$ with the variables $x_{l_1 + 1}$ to $x_{l_1 + l_2}$, using the same rule. Continue in this fashion until $|\\overline{l}|$ essential edges have been labeled. We then use the remaining variables to label the final $\\mu(\\overline{v}) - |\\overline{l}|$ essential edges, chosen in any order. Our first observation is the following.\\\\\n\n\\begin{lemma} \\label{tensordecomp}\nThere is an isomorphism of $S^{(\\overline{v},\\overline{l})}$-modules,\n\\[\nN^{(\\overline{v},\\overline{l})}(i) \\cong (x_1,\\ldots,x_{l_1}) \\otimes_\\Q (x_{l_1 + 1},\\ldots, x_{l_1 + l_2}) \\otimes_\\Q \\ldots \\otimes_\\Q (x_{\\sum_{j=1}^{i-1} l_j + 1}, \\ldots, x_{|\\overline{l}|}) \\otimes_\\Q \\Q[x_{|\\overline{l}| + 1},\\ldots, x_{\\mu(\\overline{v})}],\n\\]\nwhere $(x_1,\\ldots,x_{l_1})$ is considered as a module over $\\Q[x_1,\\ldots,x_{l_1}]$, $(x_{l_1 + 1},\\ldots, x_{l_1 + l_2})$ a module over $\\Q[x_{l_1+1},\\ldots,x_{l_1+l_2}]$, and so on.\\\\\n\\end{lemma}\n\n\\begin{proof}\nFollows immediately from the isomorphism (\\ref{monodecomp}).\\\\\n\\end{proof}\n\n\\begin{definition}\nThe \\textbf{Hilbert series} associated to the module $N^{(\\overline{v},\\overline{l})}$ is the formal power series\n\\[\nH_{(\\overline{v},\\overline{l})}(t) := \\sum_{n \\geq 0} (\\dim_\\Q N^{(\\overline{v},\\overline{l})}_n)t^n.\n\\]\n\\text{}\\\\\n\\end{definition}\n\nOur next goal will be to express $H_{(\\overline{v},\\overline{l})}(t)$ as a rational function in $t$. From here standard enumerative combinatorics will allow us to compute the Hilbert polynomial.\\\\\n\n\\begin{proposition}\\label{hilbertseries}\nThe Hilbert series $H_{(\\overline{v},\\overline{l})}(t)$ can be expressed as the rational function,\n\\[\nH_{(\\overline{v},\\overline{l})}(t) = \\frac{t^i\\prod_j (1-(1-t)^{l_j})}{(1-t)^{\\mu(\\overline{v})}}.\n\\]\n\\text{}\\\\\n\\end{proposition}\n\n\\begin{proof}\nWe will prove this using the fact that the Hilbert series is multiplicative over tensor products. It is easy to see that the Hilbert series for the augmentation ideal of a rational polynomial ring in $d$ variables is equal to the rational function\n\\[\n\\frac{1}{(1-t)^d} - 1 = \\frac{1 - (1-t)^d}{(1-t)^d}.\n\\]\nLemma \\ref{tensordecomp} therefore implies\n\\[\nH_{(\\overline{v},\\overline{l})}(t) = t^i \\left(\\prod_j \\frac{1-(1-t)^{l_j}}{(1-t)^{l_j}}\\right)\\left(\\frac{1}{(1-t)^{\\mu(\\overline{v}) - |\\overline{l}|}}\\right) = \\frac{t^i\\prod_j (1-(1-t)^{l_j})}{(1-t)^{\\mu(\\overline{v})}},\n\\]\nwhere the factor $t^i$ arises from the graded shift of $N^{(\\overline{v},\\overline{l})}$.\\\\\n\\end{proof}\n\nThis will allow us to prove our result on the obstruction to the Hilbert polynomial of $\\M_{i,\\dt}$.\\\\\n\n\\begin{corollary}\\label{exactbound}\nLet $G$ be a tree, and fix $i \\geq 0$. Then the dimension of $\\M_{i,n} \\otimes_\\Z \\Q$ agrees with a polynomial for all $n \\geq 0$.\\\\\n\\end{corollary}\n\n\\begin{proof}\nIt suffices to prove the claim for the summands $N^{(\\overline{v},\\overline{l})}$. Proposition \\ref{hilbertseries} tells us that the Hilbert series of $N^{(\\overline{v},\\overline{l})}$ takes the form\n\\[\n\\frac{t^i \\prod_{j} (1-(1-t)^{l_j})}{(1-t)^{\\mu(\\overline{v})}}.\n\\]\nThe proof of Corollary \\ref{eulerpolynomial} implies that it suffices to show the total degree of the numerator $t^i \\prod_{j} (1-(1-t)^{l_j})$ is strictly smaller than $\\mu(\\overline{v})$. Namely, we must argue\n\\[\ni + |\\overline{l}| < \\mu(\\overline{v})\n\\]\nThis follows from the fact that each essential vertex of $\\overline{v}$, say $v_{j_k}$, is adjacent to at least one essential edge which can never house a witness vertex. Namely, the essential edge which is maximal among all essential edges adjacent to $v_{j_k}$ with respect to the ordering of the vertices of $G$. Moreover, the essential edge containing the root can never house a witness vertex for any essential vertex.\\\\\n\nPut another way,\n\\[\ni + |\\overline{l}| \\leq i + \\sum_k( \\mu(v_{j_k})-2) = i + \\mu(\\overline{v}) - i - 1 = \\mu(\\overline{v}) - 1.\n\\]\nThis concludes the proof.\\\\\n\\end{proof}\n\nWe have now proven enough to explicitly compute the Hilbert polynomial of $\\M_{i,\\dt}$, and consequently $H_i(B_nG)$. The following theorem follows from the results of this section, as well as the combinatorics of generating functions.\\\\\n\n\\begin{theorem}\\label{explicitpoly}\nLet $G$ be a tree, fix $i,r \\geq 0$, and for any pair $(\\overline{v},\\overline{l})$ write\n\\[\na^{(\\overline{v},\\overline{l})}_{i,r} := \\sum_{\\substack{d_1 + \\ldots + d_i = r \\\\ d_j \\geq 1}} (-1)^{r+i}\\binom{l_1}{d_1}\\cdots \\binom{l_i}{d_i}.\n\\]\nAlso let $P^{\\overline{v}}_i(n)$ be the polynomial\n\\[\nP^{\\overline{v}}_i(n) = \\binom{\\mu(\\overline{v})-1+n-i}{\\mu(\\overline{v}) - 1}.\n\\]\nThen for all $n \\geq 0$, the dimension of $H_i(B_n,\\Q)$ is equal to the polynomial\n\\begin{eqnarray}\n\\sum_{(\\overline{v},\\overline{l})}\\sum_{r = i}^{|\\overline{l}|} a^{(\\overline{v},\\overline{l})}_{i,r}P^{\\overline{v}}_i(n-r).\\label{explicitform}\n\\end{eqnarray}\n\\text{}\\\\\n\\end{theorem}\n\n\\begin{remark}\nOne observes from the above formulation that the polynomial $P^{\\overline{v}}_i(n)$, as well as the constants $a^{(\\overline{v},\\overline{l})}_{i,r}$, only depend on the degree sequence of the graph $G$. It follows that the homologies of the braid group of a tree only depends on its degree sequence, as we recorded in Corollary \\ref{degreesequence}\n\\end{remark}\n\nSpecializing Theorem \\ref{explicitpoly} to the case wherein $i = 1$ yields a much simpler form.\\\\\n\n\\begin{corollary}\nLet $G$ be a tree. Then for all $n \\geq 0$, the dimension of $H_1(B_nG,\\Q)$ agrees with the polynomial\n\\[\n\\sum_{v \\text{ essential}}\\sum_{l=1}^{\\mu(v)-2} \\binom{\\mu(v) - 2 + n}{\\mu(v) - 1} - \\binom{\\mu(v) - 2 + n - l}{\\mu(v)-1-l}.\n\\]\n\\text{}\\\\\n\\end{corollary}\n\n\\begin{proof}\nIn this case we note that for an essential vertex $v$, and a chosen integer $0 \\leq l \\leq \\mu(v)-2$, we have\n\\[\na^{v,l}_{1,r} = (-1)^{r+1}\\binom{l}{r}.\n\\]\nTherefore the polynomial (\\ref{explicitform}) can be written\n\\[\n\\sum_{v \\text{ essential}}\\sum_{l=1}^{\\mu(v)-2} \\sum_{r = 1}^l (-1)^{r+1}\\binom{l}{r}\\binom{\\mu(v)-2+n-r}{\\mu(v) - 1}.\n\\]\nUsing the principle of inclusion-exclusion, it can be seen that\n\\[\n\\sum_{r = 1}^l (-1)^{r+1}\\binom{l}{r}\\binom{\\mu(v)-2+n-r}{\\mu(v) - 1} = \\binom{\\mu(v) - 2 + n}{\\mu(v) - 1} - \\binom{\\mu(v) - 2 + n - l}{\\mu(v)-1-l},\n\\]\nas desired.\\\\\n\\end{proof}\n\nA result of this kind appears in the work of Farley and Sabalka, where they compute the number of generators of the group $B_nG$ \\cite{FS}. A formula for the case $i = 1$ is also found in the results of Ko and Park \\cite{KP}.\\\\\n\nAnother simple case is that in which every essential vertex of $G$ has degree 3. In this case Theorem \\ref{explicitpoly} can be used to recover the following result of Farley \\cite{Fa}.\\\\\n\n\\begin{corollary}\nLet $G$ be a tree whose every essential vertex has degree 3. Then for all $n \\geq 0$, the dimension of $H_i(B_nG;\\Q)$ agrees with the polynomial\n\\[\n\\binom{N_G}{i}\\binom{n}{2i}\n\\]\n\\text{}\\\\\n\\end{corollary}\n\n\\begin{proof}\nThe major thing to note in this case is that for any $\\overline{v}$, the only associated vector $\\overline{l}$ is $(1,\\ldots,1)$. Therefore,\n\\[\na^{(\\overline{v},\\overline{l})}_{i,r} = \\begin{cases} 1 &\\text{ if $r = i$}\\\\ 0 &\\text{ otherwise.}\\end{cases}\n\\]\nThe formula (\\ref{explicitform}) now becomes\n\\[\n\\binom{N_G}{i}P^{\\overline{v}}_i(n-i) = \\binom{N_G}{i}\\binom{\\mu(\\overline{v})-1+n-2i}{\\mu(\\overline{v}) - 1} = \\binom{N_G}{i}\\binom{2i+1-1+n-2i}{2i} = \\binom{N_G}{i}\\binom{n}{2i}.\n\\]\nThis completes the proof.\\\\\n\\end{proof}\n\n\\section{Concluding remarks}\n\nIn this section we consider what this work implies for the homology of graph braid groups when $G$ is a general graph rather than a tree.\\\\\n\nLet $G$ be a graph with $N_G$ essential vertices, and $E$ essential edges. Also assume that $G$ is homeomorphic to neither $S^1$ nor a line segment. Then the work of Section \\ref{morsecomplexismodule} implies that $\\M_{i,\\dt}$ carries the structure of a finitely generated $S_G$-module. If $G$ is not a tree, however, it is no longer the case that the Morse differential is trivial. We therefore pose the following question.\n\n\\begin{question}\nDoes the action of $S_G$ on $\\M_{i,\\dt}$ commute with the Morse differential? In other words, is the Morse differential a morphism of graded $S_G$-modules?\n\\end{question}\n\nWe note that altering our choice of spanning tree $T$ has a non-trivial influence on the Morse differential. It is therefore unclear whether the Morse differential will commute with the action of $S_G$ for all choices of $T$. We therefore refine Question 1 as follows.\n\n\\begin{question}\nDoes there exist a choice of spanning tree $T$ of $G$ such that the action of $S_G$ on $\\M_{i,\\dt}$ commutes with the Morse differential?\n\\end{question}\n\nIf the answer to these questions is affirmative, then we can immediately conclude many things about the asymptotic behavior of these homology groups. Indeed, in this case the Hilbert basis theorem would imply that there is a finitely generated $S_G$-module $\\mathcal{H}_i$ such that $(\\mathcal{H}_i)_n \\cong H_i(B_nG)$.\\\\\n\n\\begin{corollary}\\label{maincor}\nAssume that Questions 1 and 2 have an affirmative answer. For each $i,n$ write\n\\[\nH_i(B_nG) \\cong \\Z^{b_i(n)} \\oplus \\bigoplus_{p \\text{ prime}}\\bigoplus_{j \\geq 1} (\\Z\/p^j\\Z)^{m_{p,i,j}(n)}.\n\\]\nThen:\n\\begin{enumerate}\n\\item there exists a polynomial $P_i(t) \\in \\Q[t]$ of degree $\\leq E-1$ such that $P_i(n) = b_i(n)$ for all $n \\gg 0$;\n\\item there exists a positive integer $e_i$, independent of $n$, such that the exponent of $H_i(B_nG)$ is at most $e_i$;\n\\item for each prime $p$ and each positive integer $j$, there is a polynomial $P_{i,p^j}(t) \\in \\Q[t]$ such that $P_{i,p^j}(n) = m_{p,i,j}(n)$ for $n \\gg 0$. In particular, there is a finite set of primes $p_{i,1},\\ldots,p_{i,l}$, and polynomials $m_{i,p_j}(n)$ such that the order of the torsion subgroup of $H_i(B_nG)$ is precisely $p_1^{m_{p_{i,1}}(n)}\\cdots p_l^{m_{p_{i,l}}(n)}$.\\\\\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nFollows from standard facts in the study of graded modules over integral polynomial rings.\\\\\n\\end{proof}\n\nThe work of Ko and Park \\cite[Theorem 3.6]{KP} that $H_1(B_nG)$ has a torsion subgroup annihilated by 2, and that the multiplicity of this torsion is eventually constant in $n$. Their work also implies that the rank of $H_1(B_nG)$ will be constant whenever $G$ is sufficiently connected. In contrast to this, affirmative answers to Questions 1 and 2 imply that the top homology of $B_nG$ always grows as fast as possible.\\\\\n\n\\begin{theorem}\\label{tophomology}\nAssume that Questions 1 and 2 have an affirmative answer. Then the Hilbert function\n\\[\nn \\mapsto \\dim_\\Q(H_{N_G}(B_nG;\\Q))\n\\]\nis equal to a polynomial of degree $E-1$ for $n \\gg 0$.\n\\end{theorem}\n\n\\begin{proof}\nTheorem \\ref{eulerpolynomial} implies that the Euler characteristic of $\\UConf_n(G)$ is a polynomial of degree $E-1$. On the other hand, Remark \\ref{slowgrowth} implies the lower homology groups have Hilbert functions which grow strictly slower than this. It follows that the rank of $H_{N_G}(B_nG)$ must eventually grow like a polynomial of degree $E-1$.\\\\\n\\end{proof}\n\nNote that Theorem \\ref{morsediffcomp} implies that $H_{N_G}(B_nG)$ is torsion-free. We have therefore seen that $H_1(B_nG)$ can only have 2 torsion, while the top homology $H_{N_G}(B_nG)$ does not have any torsion. It is an interesting question to ask what other types of torsion can appear in the intermediate homologies.\\\\\n\nWhile an affirmative answer to Questions 1 and 2 would be a step in the right direction, the work of Ko and Park \\cite{KP} would suggest that the action of $S_G$ on $\\M_{i,\\dt}$ described in Section \\ref{morsecomplexismodule} is perhaps non-optimal. We have already discussed that the result \\cite[Theorem 3.6]{KP} implies that the braid groups of biconnected graphs have eventually constant first homologies. However, the action of $S_G$ defined in Section \\ref{morsecomplexismodule} only detects the connectivity a chosen spanning tree of $G$. In particular, is it possible that one can improve on this action by accounting for connectivity granted by the deleted edges?\n\n\\begin{question}\nLet $G$ be a graph which is neither a circle, nor a line segment. Does there exist an action of $S_G$ on $\\M_{i,\\dt}$ which interacts in a non-trivial way with the deleted edges of $G$? More specifically, can one define this action so that if $c$ is a critical $i$-cell, then there is a $\\Z$-linear relation between $x_{\\mathfrak{e}_1}\\cdot c$ and $x_{\\mathfrak{e}_2} \\cdot c$ whenever the two essential edges $\\mathfrak{e}_1$ and $\\mathfrak{e}_2$ are in the same connected component of $G$ with the edges of $c$ removed?\n\\end{question}\n\nThe existence of such an action would actually suggest something quite strong about what kind of invariants of $G$ are encoded by the homology of its braid groups. If $G$ is a graph, recall that we define $\\Delta_G^i$ to be the maximum number of connected components $G$ can be broken into by removing exactly $i$-vertices.\\\\\n\n\\begin{conjecture}\\label{mainconj}\nLet $G$ be a graph which is neither a line segment, nor a circle. Then for all $i$, and all $n \\gg i$, the Betti number\n\\[\nb_i(n) := \\dim_\\Q(H_i(B_nG;\\Q))\n\\]\nagrees with a polynomial of degree $\\leq \\Delta^i_G-1$.\\\\\n\\end{conjecture}\n\nNote that Theorem \\ref{tophomology} is a specific case of this conjecture. Also note, if $G$ is biconnected, then the above implies that $b_1(n)$ is eventually constant. This fact is proven in the work of Ko and Park \\cite[Theorem 3.6]{KP}. The above conjecture therefore proposes a generalization of one aspect of that work.\\\\\n\nFinally, we think about the ramifications of this work on the spaces $\\UConf_n(G)$. The action of $S_G$ on $\\M_{i,\\dt}$ is defined in a purely formal way. However, in the study of more classical configuration spaces, stability phenomena often arise from maps on the underlying spaces $\\UConf_n(G) \\rightarrow \\UConf_{n+1}(G)$ (see, for instance, \\cite{EW-G} and \\cite{McD}). Is that also the case here?\\\\\n\n\\begin{question}\nIf $G$ is a tree, is the action of $S_G$ on $\\M_{i,\\dt}$ induced by a map of topological spaces $\\UConf_n(G) \\rightarrow \\UConf_{n+1}(G)$? If $G$ is a general graph, and Questions 1 and 2 have affirmative answers, is the action of $S_G$ on the homologies of $B_nG$ induced in a similar way?\\\\\n\\end{question}\n\nOf course, Question 4 is highly relevant for Questions 1 and 2. If one understands the action of $S_G$ topologically, then it might make the existence of such an action on the homologies of general graph braid groups more apparent.\\\\ \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}