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+{"text":"\\section{Introduction and results}\n\\label{}\nLarge scale oceanic motions can be described - at first sight - by the linearized Saint-Venant equations for thin layers, with   Coriolis force, \ni.e. by the following two-dimensional system~ \\cite{P1}:\n\\begin{equation}\\label{syst}\n\\partial_tU+\\left(\\begin{array}{ccc}\n0&\\partial_{x_1}&\\partial_{x_2}\\\\\n\\partial_{x_1}&0&-B(x_2)\\\\\n\\partial_{x_2}&B(x_2)&0\\end{array}\\right)U=0,\\ \\mbox{where}\\  U=U(t,x_1,x_2)=\\left(\\begin{array}{c}\\rho\\\\u_1\\\\u_2\\end{array}\\right)\n\\end{equation}\n\n\n\\noindent where $B$ is the local vertical component of the Earth rotation vector (depending only on the latitude $x_2$),~$\\rho$ denotes \nthe fluctuation of water height and $u$  the horizontal velocity field. For the sake of simplicity, we assume that $(x_1,x_2) \\in {\\mathbb R}\\times\\mathbb T$, \nmeaning that we neglect the influence of the lateral boundaries.\n\nWaves associated to that linear system are usually classified in two ~\\\\families~: First, Poincar\\'e waves, which are fast dispersive \ngravity waves; Second, Rossby waves, due to the inhomogeneities of $B$,  which propagate much slower  \\cite{DU}, \\cite{GA}.\n\nIn this Note, we are interested in the propagation of Rossby waves. We show, in agreement with physical observations \\cite{P2}, that these waves \ncan be trapped in some regions, called ventilation zones, which are not influenced by external dynamics and sources such as continental \nrecirculation for instance.\n\n\n\n\n\nMore precisely, in the limit of large values of $B=b\/\\epsilon$, we   construct  a co-dimension $1$ submanifold~$\\Lambda$ of~$T^*({\\mathbb R}\\times\\mathbb T)$. \nThis set contains the $\\epsilon$-microlocalization\nregion inside which, essentially, an initial condition of (\\ref{syst})  remains trapped. \nLet us first recall that the $\\epsilon$-frequency set of a function $u$ \\cite{MA} is the (closed) subset of~$T^*({\\mathbb R}\\times\\mathbb T)$, complement of the set of points $(\\underline x,\\underline\\xi)$ such that there exists a $C^\\infty$ function $\\chi$, with $\\chi(\\underline x,\n\\underline \\xi)=1$, such that\n\\[\n\\left\\Vert\\int\\chi(\\frac{x+y}2,\\xi)e^{i\\frac{\\xi(x-y)}\\epsilon}u(y)dyd\\xi\\right\\Vert_{L^2}=O(\\epsilon^\\infty).\n\\]\n\n\\begin{theorem}\\label{main}\nLet us consider the system \n\\begin{equation}\\label{systempsilon}\n\\partial_tU+\\left(\\begin{array}{ccc}\n0&\\partial_{x_1}&\\partial_{x_2}\\\\\n\\partial_{x_1}&0&-\\frac{b(x_2)}\\epsilon\\\\\n\\partial_{x_2}&\\frac{b(x_2)}\\epsilon&0\\end{array}\\right)U=0\n\\ \\mbox{with an initial condition}\\ \n\\end{equation}\n\\begin{equation}\nU|_{t=0}=\\left(\\begin{array}{c}\n\\rho^0\\\\u_1^0\\\\u_2^0\\end{array}\\right)\n\\end{equation}\nwith $L^2({\\mathbb R}\\times \\mathbb T,dx_1dx_2)$ conditions (periodic in $x_2$). Let $\\Lambda=\\{F(\\xi_1,x_2,\\xi_2)=0\\}\\subset T^*{\\mathbb R}\\times \\mathbb T$, where $F$ is defined in Lemma\n\\ref{pm}.\n\nWe    suppose that the $\\epsilon$-frequency set of  $U|_{t=0}$ is contained in a compact set $\\mathcal C$ satisfying:\n\\begin{equation}\\label{cond}\n\\mathcal C\\cap\\{\\xi_1=0\\} =\\mathcal C\\cap\\{\\xi_2^2+b(x_2)^2=0\\}=\\emptyset.\n\\end{equation}\nLet us fix a compact set $\\Omega$ of ${\\mathbb R}\\times\\mathbb T$. Then there exists ($\\epsilon$-)pseudo-differential operators $P_\\rho^0, P_1^0, P_2^0$ of principal symbols\n$p_\\rho^0=ib(x_2)\\xi_1(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1}$, $p_1^0=-\\xi_1\\xi_2(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1}$, $p_2^0=\\xi_1^2(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1}$, such that:\n\n\n\\begin{enumerate}\n\\item if the $\\epsilon$-frequency set  of $P_\\rho^0\\rho^0+ P_1^0u_1^0+ P_2^0u_2^0$ does not intersect $\\Lambda\\cap T^*\\Omega$, then  $\\exists\nT>0$ such that:\n\\[\n||U(\\frac t{\\epsilon})||_{L^2(\\Omega)}=O(\\epsilon^\\infty)\\ \\mbox{for}\\ t\\geq  T .\n\\]\n\n\n\\item  if the $\\epsilon$-frequency set of $P_\\rho\\rho^0+ P_1u_1^0+ P_2u_2^0$ does  intersect $\\Lambda\\cap T^*\\Omega$, then, $\\forall  t\\geq 0 $,\n$\n||U(\\frac t{\\epsilon})||_{L^2(\\Omega)}\\neq O(\\epsilon^\\infty)\n$\n(in other words  the frequency set of $U(\\frac t{\\epsilon})$ intersects $T^*\\Omega$).\n\\end{enumerate}\n\nIn the case of a WKB initial condition the conclusion is more precise. Consider\n\\[\nU|_{t=0}=\\left(\\begin{array}{c}\nR^0(x)\\\\U_1^0(x)\\\\U_2^0(x)\\end{array}\\right)e^{i\\frac{S(x)}\\epsilon}.\n\\]\nSuppose than  the (Lagrangian) manifold \\\\$\\{(x,\\nabla S(x)),(p_\\rho^0 R^0+p_1^0U_1^0+p_2^0U_2^0)(x,\\nabla S(x)) \\neq 0\\}$\nintersects $\\Lambda\\cap T^*\\Omega$, then:\n\n\\begin{equation}\\label{wkb}\n||U(\\frac t{\\epsilon})||_{L^2(\\Omega)}\n\\sim C(t)+O(\\epsilon),\\ C(t)>0.\n\\end{equation}\n\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Reduction to a scalar situation}\\label{reduction}\nPerforming first a Fourier analysis in $x_1$ (as the system does not contain explicitly this variable) and looking secondly at modes of the system (\\ref{systempsilon}), we can\nprove the following Proposition, heart of our results:\n\\begin{proposition}\\label{reduc}\nThere exist three pseudo-differential operators $T_\\pm,\\ T_0$, of leading symbols \\\\ $\\tau_{\\pm}= \\pm\\sqrt{\\xi_1^2+\\xi_2^2+b^2(x_2)},\\\n\\tau_0=\\epsilon\\frac{b'(x_2)\\xi_1}{\\xi_1^2+\\xi_2^2+b^2(x_2)}$, roots of (\\ref{the}), such that, if $u_{2,j}^0$ is microlocalized outside $\\xi_1^2-\\tau_j^2=0$, we have, $\\mbox{for}\\  j\\in\\{-,0,+\\}$:\n$$\\epsilon\\partial_tu_{2,j}=iT_ju_{2,j}   \\Longrightarrow \nU_j:=\\left(\\begin{array}{c}(i \\epsilon\\partial_{x_2}T_j+ \\epsilon\\partial_{x_1}b(x_2))(\\epsilon^2\\partial_{x_1}^2+T_j^2)^{-1}\\\\\n-(\\epsilon^2\\partial_{x_1}\\partial_{x_2}+ib(x_2)T_j)(\\epsilon^2\\partial_{x_1}^2+T_j^2)^{-1}\\\\\n\\mathbb Id\\end{array}\\right)u_{2,j}$$ $$\\mbox{ satisfies (\\ref{systempsilon}) up to } O(\\epsilon^\\infty).\n$$\n\\end{proposition}\nThe proof is a consequence of the result contained in the Appendix.\n\nThe following result shows that any initial condition of (\\ref{systempsilon}) can be decomposed on the three modes of the last section.\n\\begin{proposition}\n$\\forall \\rho,u_1, u_2,\\ \\exists  u_{2,j},\\ j\\in\\{+,0,-\\}$ such that:\n\\begin{equation}\\label{decomp}\n\\left(\\begin{array}{c}\\rho\\\\u_1\\\\u_2\\end{array}\\right)=\\sum_{j} \\left(\\begin{array}{c}(i \\epsilon\\partial_{x_2}T_j+ \\epsilon\\partial_{x_1}b(x_2))(\\epsilon^2\\partial_{x_1}^2+T_j^2)^{-1}\\\\\n-(\\epsilon^2\\partial_{x_1} \\partial_{x_2}+ib(x_2)T_j)(\\epsilon^2\\partial_{x_1}^2+T_j^2)^{-1}\\\\\n\\mathbb Id\\end{array}\\right)u_{2,j}+O(\\epsilon^\\infty)\n\\end{equation}\n\\[=:\\sum_{j} {\\mathbb Q}^j u_{2,j}+O(\\epsilon^\\infty) .\n\\]\n\\end{proposition}\nTo prove the Proposition one has just to invert the matrix:\n$(\n{\\mathbb Q}^- \\, {\\mathbb Q}^0 \\, {\\mathbb Q}^+ ).$\nSemiclassically it is enough to show that the matrix of the leading order symbol is invertible\n\\begin{equation}\\label{matrix}\\scriptsize\n\\left(\\begin{array}{ccc}\n{\\xi_2\\sqrt{\\xi_1^2+\\xi_2^2+b(x_2)^2} + i\\xi_1b(x_2)\\over \\xi_2^2+b(x_2)^2}&-{ i b(x_2)\\over \\xi_1}\n&{-\\xi_2\\sqrt{\\xi_1^2+\\xi_2^2+b(x_2)^2} + i\\xi_1b(x_2)\\over \\xi_2^2+b(x_2)^2}\\\\\n{\\xi_1\\xi_2+ib(x_2) \\sqrt{\\xi_1^2+\\xi_2^2+b(x_2)^2} \\over \\xi_2^2+b(x_2)^2}&-{\\xi_2 \\over \\xi_1}\n&{\\xi_1\\xi_2-ib(x_2) \\sqrt{\\xi_1^2+\\xi_2^2+b(x_2)^2} \\over \\xi_2^2+b(x_2)^2}\\\\1&1&1\n\\end{array}\\right).\n\\end{equation}\nA simple computation shows that the  jacobian\n$\nJ=\\frac{2(\\xi_1^2+\\xi_2^2+b^2(x_2))^{3\/2}}{(\\xi_2^2+b^2(x_2))|\\xi_1|}\\geq 2$.\nIn particular the inversion of the matrix $(\n{\\mathbb Q}^- \\, {\\mathbb Q}^0 \\, {\\mathbb Q}^+ )$ can be done symbolically at any order and gives the leading order symbols of the operators $\\mathbb\nP^j:=(P_\\rho^j,P_1^j,P_2^j)$\n such\nthat $u_{2,j}=P_\\rho^j\\rho+P_1^ju_1+P_2^ju_2,\\ j\\in\\{-,0,+\\}$. One gets~:\n$p_\\rho^0=ib(x_2)\\xi_1(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1},\\ p_1^0=-\\xi_1\\xi_2(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1},\\  p_2^0=\\xi_1^2(\\xi_2^2+\\xi_1^2+b^2(x_2))^{-1}$.\n\n\n\n\n\n\n\n\\section{Propagation under the Rossby Hamiltonian $\\tau_0$}\\label{lambda}\nThanks to  Proposition \\ref{reduc} it is enough, as far as  the Rossby mode is concerned, to look at propagation with respect to the Hamiltonian $\\tau_0\/\\epsilon$.\nThis Hamiltonian being  independent of $x_1$, $\\xi_1$ will be conserved. The flow is periodic in the variables $x_2,\\xi_2$ (one degree of freedom). \nTherefore, since \n$\\dot{x}_1=\\frac{b'(x_2)(\\xi_2^2-\\xi_1^2+b^2(x_2))}{(\\xi_2^2+\\xi_1^2+b^2(x_2))^2}\\mbox{ is periodic (the case of infinite and zero period is treated}$\n in \\cite{CGPS}),$\\ x_1(t)$ \nwill contain a part, linear in time except if\n$$F(\\xi_1,x_2(0),\\xi_2(0)):=\\int_0^{period}\\frac{b'(x_2(t))(\\xi_2(t)^2-\\xi_1^2+b^2(x_2(t)))}{(\\xi_2(t)^2+\\xi_1^2+b^2(x_2(t)))^2}dt=0.$$\n\\begin{lemma}\\label{pm}\nAs $b'(x_2)\\neq 0,\\ b'(x_2)F(\\xi_1,x_2,\\xi_2)>0\\mbox{ as } \\xi_1\\to\\pm\\infty,\\ <0\\mbox{ as }$\\\\$ \\xi_1\\to 0$ and is invariant under the flow of  $\\frac{\\tau_0}\\epsilon$.\n\nDefine \n$\nE(\\xi_1,x_2,\\xi_2)=\\frac{b'(x_2)\\xi_1}{\\xi_2^2+\\xi_1^2+b^2(x_2)}\\ \\ .\n$\n\nThen\n$\\displaystyle\n\\left\\vert F(\\xi_1,x_2,\\xi_2)\\right\\vert=\\left\\vert\\int_{x_-}^{x_+}\n\\frac{\\frac{b'(x)}{E(\\xi_1,x_2,\\xi_2)}-2\\xi_1}{\\sqrt{\\frac{b'(x)\\xi_1}{E(\\xi_1,x_2,\\xi_2)}-\\xi_1^2-b^2(x)}}dx\\right\\vert,\n$\nwhere\n~$]x_-,x_+[$\nis the largest interval of~${\\mathbb T}$ containing $x_2$ in which~$\\frac{b'(x)\\xi_1}{E(\\xi_1,x_2,\\xi_2)}-\\xi_1^2-b^2(x)>0$.\n\\end{lemma}\nWe define  \n$\\Lambda:=\\{(x_1,x_2;\\xi_1,\\xi_2)\/\\ F(\\xi_1,x_2,\\xi_2)=0\\}.$\n\n$\\mbox{ Thanks to Lemma \\ref{pm} } \\Lambda\\neq\\emptyset\\mbox{ and }dim\\Lambda=3$.\n\\begin{corollary}\nSuppose~$b'(x_2) \\neq 0$. Then $\\vert F(\\xi_1, x_2,\\xi_2)\\vert  \\geq\\frac{C(x_2,\\xi_2)}{\\xi_1}$ as $\\xi_1\\to 0$, with~$C(x_2,\\xi_2)>0$. \nThis implies that the trapping phenomenon will take place only with initial conditions oscillating enough in $x_1$.\n\\end{corollary}\n\n\\medskip\n\\begin{remark}: Since the Hamiltonian $E$ does not depend on $x_1$, we can  express it, for each value of $\\xi_1$, on the action variable \n$A$: $E(\\xi_1,x_2,\\xi_2)=H(A,\\xi_1)$. This allows to define the function\n  $A(\\xi_1,x_2,\\xi_2)$ by $E(\\xi_1,x_2,\\xi_2)=H(A(\\xi_1,x_2,\\xi_2),\\xi_1)$. One can easily show that:\n\\[F(\\xi_1,x_2,\\xi_2)=\\frac{\\partial_{\\xi_1}H(A,\\xi_1)}{\\partial_{A}H(A,\\xi_1)}|_{A=A(\\xi_1,x_2,\\xi_2)},\\]\n  and the following variational characterization of $\\Lambda$:\nLet us fix the energy to $E$ and let $\\Gamma(\\xi_1,E)$ be the energy shell $\\{E(\\xi_1,x_2,\\xi_2)=E\\}$. Then\n\n\\[\\Lambda=\\bigcup_{E,i}\\{(\\Gamma(\\xi_1^i,E),\\xi_1^i\\},\\]\\[\n\\mbox{ where $\\xi_1^i$ is such that the area inside $\\Gamma(\\xi_1^i,E)$ is extremal.}\\]\n\\end{remark}\n\n\n\n\n\n\n\n\n\\section{Dispersion of Poincar\\'e waves and proof of the Theorem}\nThe proof of the Theorem  involves, for the Rossby modes ($j=0$), the standard result of propagation of the frequency set. If\nthe initial  frequency set  is such that part of the Rossby mode is  trapped, in particular  if it intersects $\\{ b'(x_2)=0\\}$, this concludes the proof.\n\nIf not we have to prove some  dispersion for the Poincar\\'e modes ($j=\\pm$) for  times of the order $\\frac 1 \\epsilon$, for which the theorem of propagation of the frequency set is not enough. But, the system \nbeing integrable, \nwe can perform an expansion on (Bohr-Sommerfeld) eigenvalues of the Hamiltonians  $T_\\pm$  and a decomposition of~$U|_{t=0}$ on a compact set of\ncoherent states (thanks to the condition on the microlocalization of the initial datum) \\cite{PA}.\n\n\n\n\n\n\n\n\nSince  $\\tau_\\pm=\\pm\\sqrt{\\xi_2^2+b^2(x_2)+\\xi_1^2 }+O(\\epsilon)$ and we are microlocalized far away from $\\xi_2^2+b^2(x_2)+\\xi_1^2=0 $, \nwe can find  pseudo-differential operators $H_{2\\pm}$ of principal symbols \n$\\xi_2^2+b^2(x_2)$ such that $T_\\pm=\\pm\\sqrt{H_{2\\pm}+\\xi_1^2 }$. The Bohr-Sommerfeld\nquantization condition (with subsymbol) gives that the eigenvalues of $H_{2\\pm}$ are of the form:\n\\[\n\\lambda_\\pm^k=\\lambda_\\pm\\left((k+\\frac 1 2 )\\epsilon\\right)+\\epsilon\\mu^{k}_\\pm(\\xi_1)+O(\\epsilon^2),\n\\]\nwhere $\\lambda_\\pm$ is the energy $\\xi_2^2+b^2(x_2)$ defined on action variable, and $\\epsilon\\mu^{k}_\\pm(\\xi_1)\\in C^\\infty$ is the correction due to the subsymbol.\nPropagating at time $t= s \/\\epsilon$ a function, product of a coherent state at $(q,p)$ (in $x_1$) and  \nan eigenfunction of $T_\\pm$ (in $x_2 $),  gives rise to expressions of the type:\n\\[\n\\int \\exp{i\\frac{\\epsilon(x_1-q)\\xi_1\\pm(\\lambda_\\pm^k+\\xi^2_1)^{\\frac 1 2}s+i\\epsilon(\\xi_1-p)^2}{\\epsilon}}d\\xi_1.\n\\]\nThe stationary phase lemma then  gives that this integral is $O(\\epsilon^\\infty)$ except if there exists a stationary point,  given by the conditions:\n\\[\n\\xi_1=p\\ \\mbox{and}\\  \\epsilon(x_1-q)\\pm\\frac{(2\\xi_1 +\\epsilon\\partial_{\\xi_1}\\mu^{k}_\\pm)}{2\\sqrt{\\lambda_\\pm^k+\\xi^2_1}}s=0 . \\]\nThe second condition gives:\n$\n2\\sqrt{\\lambda_\\pm^k+\\xi^2_1}(x_1-q)=\\mp\\left(\\frac{2p}\\epsilon+\\partial_{\\xi_1}\\mu^{k}_\\pm\\right)s.\n$\nTherefore, since $p\\neq 0$ and the $\\lambda^k$\\ 's are bounded by the above condition (\\ref{cond}) on $\\mathcal C$, there is no critical point for $x_1$ in a compact set.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intr}\n Anomaly detection, also known as outlier detection, is the process of discovering patterns in a given data set that do not conform to expected behavior \\citep{Akyildiz}. Anomaly detection is to find the events that happen relatively infrequently, which has been extensively used in a wide variety of applications, including fraud detection for credit cards, insurance, health care, intrusion detection for cyber-security, fault detection in safety critical systems, and many others \\citep{Akyildiz}. Smart meter analytics attracts the growing research effort \\eg, \\citep{sdewes2015,icde2015,energy2016,edbt2015}, due to the wide installation of smart meters. Anomaly detection can be applied for analyze live smart meter data, which aims to help energy consumers identify unusual behaviors, e.g., forgetting to turn off stoves after cooking; and to help utilities detect extraordinary events, e.g., energy leakage and theft. Since abnormal consumption may also result from user activities, such as using inefficient appliances, or over-lighting and working overtime in office buildings, anomalous feedback can be used to warn energy consumers to minimize energy usage and to help them identify inefficient appliances or over-lighting. Furthermore, anomaly detection can help utilities to establish the baseline for providing more accurate demand-response programs to their customers \\citep{zhang2011}. Abnormal energy consumption detection is related to finding patterns in data, and the statistical and data mining techniques are used to detect the patterns, e.g., \\citep{zhang2011,nadai2015,liu2011,Chou2014}, which can perform close to or better than domain experts. \n \nToday, smart meters are widely used worldwide. Smart meters are the digital devices that can record energy consumption at the interval of an hour or fewer \\citep{Depuru2011}. Smart meters record detailed consumption readings in real-time or near real-time, which provide the opportunity to monitor timely unusual events or consumption behaviors. However, the enabling anomaly detection for smart meters typically uses data mining technologies, which require large amounts of training data sets, as well as significantly complex systems. In typical applications of data mining to anomaly detection, the detection models are produced off-line because the learning algorithms have to process tremendous amounts of data \\citep{lee2001}. The generated models are naturally used for offline anomaly detection, i.e., analyzing consumption data after being loaded into an energy management system (EMS). Effective anomaly detection, however, should happen in real-time in order to minimize the compromises to the use of energy. The efficiency of updating the detection model and the accuracy of the detection are the important consideration for constructing such a real-time anomaly detection system. \n \n \nIn this paper, we propose a statistical anomaly detection method based on the consumption patterns of smart grid data. For residential electricity consumption, the daily consumption patterns of a customer usually show quite similar. The proposed algorithm can detect the unusual energy usage from one's history consumption patterns, e.g., the abnormal high usage than the expected (see Fig.~\\ref{fig:anomalycase}). \n\\begin{figure}[htp] \n\\centering \n\\includegraphics[width=0.6\\textwidth]{anomalycase} \n\\caption{Daily pattern of a typical household and anomaly consumption} \n\\label{fig:anomalycase} \n\\end{figure} \n\nTo detect anomalies in time and obtain a better accuracy, we make use of the so-called {\\em Lambda} architecture \\citep{Marz2013}, that can detect anomalies in near real-time, and can efficiently update detection models regularly according to a user-specified time interval. A lambda architecture enables real-time updates through a three-layer structure, including speed layer (or real-time layer), batch layer and serving layer. It is a generic system architecture for obtaining near real-time capability, and its three layers use different technologies in processing data. It is well-suited for constructing an anomaly detection system that requires real-time anomaly detection and efficient model refreshment (we will detail it in the next section). To support big data capability, we choose the Spark Streaming as the speed layer technology for detecting anomalies on a large amount of data streams, Hive as the batch layer technology for computing anomaly detection models, and PostgreSQL as the serving layer for saving the models and detected anomalies; and sending feedbacks to customers. The proposed system can be integrated with smart meters for detecting anomalous energy consumption online. To summarize, we make the following contributions: 1) we propose the statistical-based anomaly detection algorithm based on customers' history consumption patterns; 2) we propose making use of the lambda architecture for the efficiency of the model updating and real-time anomaly detection; 3) we implement the system with a lambda architecture using hybrid technologies; 4) we evaluate our system in a cluster environment using realistic data sets, and show the efficiency and effectiveness of using the lambda architecture in a real-time anomaly detection system. \n \n \nThe rest of this paper is organized as follows. Section 2 discusses the anomaly detection algorithm used in the paper. Section 3 describes the implementation of the lambda detection system. Section 4 evaluates the system. Section 5 surveys the related works. Section 6 concludes the paper and provides the direction for the future works.   \n\\section{Preminaries}\n\n\\subsection{Anomaly Detection Model}\nThe used anomaly detection model is a combination of a short-term energy consumption prediction algorithm, called {\\em periodic auto-regression with eXogenous variables (PARX)} \\citep{omid}, and {\\em Gaussian statistical distribution}. We now first describe the PARX algorithm, which will be used for the prediction based on history consumption patterns. Generally speaking, residential electricity consumption is highly correlated to the weather temperature. In winter, electricity consumption increases since the temperature decrease because of the heating needs. Similarly, in summer, electricity consumption increases when the temperature is higher because of cooling loads. A similar daily consumption pattern may appear repeatedly for a customer, e.g., due to the living habit of the customer. For example, if a customer usually gets up at 7 o'clock, then the consumption pattern will have the morning peak between 7 and 8 o'clock; In the evening, if the customer gets home at 5 o'clock after work, the consumption pattern typically will have the evening peak between 17 and 20 o'clock, due to cooking and washing. \n \n \nThe PARX model, thus, uses a daily period, taking 24 hours of the day as the seasons, \\ie, $t=0...23$,  and uses the previous $p$ days' consumptions at the time $t$ for auto-regression. The PARX model at the $s$-th season and at the $n$-th period is formulated as    \n\n\\vspace{-10pt}\n\\begin{equation}\n\tY_{s,n} = \\sum_{i=1}^{p} \\alpha_{s,i} Y_{s,n-i}  + \\beta_{s,1} XT1 + \\beta_{s,2} XT2 + \\beta_{s,3} XT3 \t+ \\epsilon_s, \\enspace s \\in t\n\\label{eq:parx}\n\\end{equation}\nwhere $Y$ is the data point in the consumption time-series; $p$ is the number of order in the auto-regression; $XT1, XT2$ and $XT3$ are the exogenous variables accounting for the weather temperature, defined in the equations of $(2) - (4)$; $\\alpha$ and $\\beta$ are the coefficients; and $\\epsilon$ is the value of the white noise.   \n{\\small\n\\begin{equation}\nXT1 = \\begin{cases}\nT-20 & \\text{if } T>20 \\\\ \n0 & \\text{otherwise}  \n\\end{cases}\n\\qquad\n\tXT2 = \\begin{cases}\n16-T & \\text{if } T<16 \\\\ \n0 & \\text{otherwise}  \n\\end{cases} \n\\qquad\n\tXT3 = \\begin{cases}\n5-T & \\text{if } T<5 \\\\ \n0 & \\text{otherwise}  \n\\end{cases}\n\\end{equation}\n}\n\nThe variables represent the cooling (temperature above 20 degrees), heating \n(temperature below 16 degrees), and overheating (temperature below 5 degrees), respectively. \nThe anomaly detection algorithm uses unique variate Gaussian distribution described in the following. \nGiven the training data set, $X=\\{x_1, x_2, ..., x_n\\}$ whose data points obey the normal distribution with the mean $\\mu$ and the variance $\\delta^2$, the detection function is defined as \n\\begin{equation} \np(x; \\mu, \\delta)=\\frac{1}{\\delta \\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\delta^2}} \n\\end{equation} \nwhere $\\mu=\\frac{1}{n} \\sum^n_{i=1} x_i$ and $\\delta^2=\\frac{1}{n}\\sum_{i=1}^{n}(x_i-\\mu)^2$. For a new data point, $x$, this function computes its probability density. If the probability is less than a user-defined threshold, \\ie, $p(x)<\\epsilon$, it is classified as an anomaly, otherwise, it is a normal data point. In our model training process, we compute the L1 distance between the actual and predicted consumptions, \\ie, $||Y_t - \\hat{Y}_t||$, where $Y_t$ is the actual hourly consumption at the time $t$, and $\\hat{Y}_t$ is the predicted hourly consumption at the time $t$. The predicted hourly consumption, $\\hat{Y}_t$, is computed using the PARX model in Equation~\\ref{eq:parx}. We find that the L1 distances obey to a log-normal distribution (see Section~\\ref{sec:accuracy}). Therefore, the $x$ in the normal distribution will be the log value of the distance, \\ie, $ln||Y_t - \\hat{Y}_t||$.   \n\n\n\\subsection{Lambda Architecture}\nWe now introduce the Lambda architecture that will be used in our anomaly detection system. As mentioned in Section~\\ref{sec:intr}, the lambda architecture consists of three layers, including speed layer, batch layer and serving layer, illustrated in Fig.~\\ref{fig:lambdaarch}. The speed layer directly ingests data streams from data sources, processes them, and continuously updates the results into the real-time views in the database in the serving layer. The speed layer does not keep any history records, and typically uses main memory based technologies to analyze the incoming data. In contrast, the batch layer runs iteratively and starts from the beginning of the data set once a batch job has finished. When a batch job starts, all the available data in the batch layer storage will be processed. Therefore, the data arriving after the job starts will not be processed until the next job. Since all the data are analyzed in each iteration, each of the new result views will replace its predecessor. As the batch layer does not rely on incremental processing, it is robust to any system failures, which the batch job simply processes all the available data sets in each iteration. The speed and batch usually use different technologies because of their distinct requirements regarding read and write operations. Any query against the data is answered through the serving layer, \\ie, the query processor queries both the views from the speed and the batch layers, and merges them. \n\\begin{figure*}[htp] \n\\vspace{-10pt}\n\\centering \n\\includegraphics[width=0.9\\textwidth]{lambdaarch} \n\\caption{Lambda architecture} \n\\label{fig:lambdaarch} \n\\vspace{-15pt}\n\\end{figure*} \n\nThe lambda architecture itself is only a paradigm. The technologies with which the different layers are implemented are independent of the general idea. The speed layer only deals with new data and compensates for the high latency updates of the batch layer. It can typically leverage stream processing systems, such as Storm, S4, and Spark Streaming, etc. The batch layer needs to be horizontally scalable and supporting random reads, where the technologies like Hadoop with Cascading, Scalding, Pig, and Hive, are suitable. The serving layer requires a system with the ability to perform fast random reads and writes. The system can be a high-performance RDBMS (e.g., PostgreSQL), an in-memory data store (\\eg, Redis, or Memcache), or a high scalable NoSQL system (\\eg, HBase, Cassandra, ElephantDB, MongoDB, or DynamoDB).     \n\n\n\\section{\\uppercase{Implementation}}\n\n\\subsection{System Overview}\nWe now describe the implementation of the anomaly detection system. We choose Spark Streaming, Spark, and PostgreSQL as the speed layer, batch layer and serving layer technology, respectively (see Fig.~\\ref{fig:systemoverview}). The system employs Spark to compute the models for anomaly detection, which reads the data from the Hadoop distributed file system (HDFS) in the batch layer. The batch job runs at a regular time interval, computes and updates the detection models to the table in PostgreSQL database. Spark Streaming is used for processing real-time data streams, e.g., it directly gets data from smart meters, and detects abnormal consumption by the detection algorithm. The detection algorithm always uses the latest models getting from the PostgreSQL database. Spark Stream writes the detected anomalies back to the PostgreSQL database, which will be used for the notification of customers.   \n\\begin{figure}[htp]\n\\vspace{-15pt}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{systemoverview}\n\\vspace{-5pt}\n\\caption{The anomaly detection system}\n\\label{fig:systemoverview}\n\\vspace{-20pt}\n\\end{figure}\n\n\n\\subsection{Training Anomaly Detection Models}\n\nWe employ Spark to train the detection model by running regular batch jobs. All the consumption data from smart meters are written to the append-only HDFS. In each iteration of the batch jobs, Spark uses all the available data in HDFS to compute the detection model. The use of Spark and HDFS supports the computation of the model based on scalable data sets, and since they both are the distributed computing technology, the computation can be finished within a certain time limit, which means that the detection algorithm can use the latest data model for the anomaly detection. Fig.~\\ref{fig:training} illustrates the training process of generating PARX model and Gaussian model using energy consumption and weather temperature time series at the season from 0 to 23. That is, for each season, e.g., $s=0$, we create a new time series with the hourly reading at 0 o'clock of all days, then use the Equation~\\ref{eq:parx} to compute the PARX model (or parameters), and compute the Gaussian model, \\ie, $N(\\mu, \\delta^2)$. Therefore, there are 24 PARX and 24 Gaussian models in total for the hours of the day.   \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{training}\n\\caption{Process of training detection models}\n\\label{fig:training}\n\\end{figure}\n\nAlgorithm~\\ref{alg:training} gives more details about the implementation. This algorithm computes the anomaly detection models with the given training time series collection $\\mathcal{TS}$, weather temperature time series $ts'$, and auto-regression order $p$. Each time series in $\\mathcal{TS}$ represents the hourly energy consumption of a customer. To compute the detection models for each season $s$, we first need to create a new consumption time series and  a new temperature time series (see line 7), then use the two new time series to compute the PARX model (see line 8). According to our analysis in Section~\\ref{sec:accuracy}, the $L1$ distances between predict consumption and actual consumption at season $s$ for all days observes to a log-normal distribution. Therefore, we compute Gaussian statistical model based on the $L1$ distance log values (see line 12-18). The total number of PARX models for all the time series is $||\\mathcal{TS}|| \\times 24$, which is same as the number of the Gaussian models. In the end, all the models are updated to the PostgreSQL database  that will be used for the online anomaly detection in the speed layer. \n\\begin{algorithm}\n\\caption{Training of anomaly detection models}\n{\\scriptsize\n\\begin{algorithmic}[1]\n\\Function {Train}{TimeSeriesCollection $\\mathcal{TS}$, TemperatureTimeSeries $ts'$ Order $p$}\n    \\State $\\mathcal{M} \\gets \\{ \\}$ \\Comment{Initialize the collection of PARX parameters}\n    \\State $\\mathcal{N} \\gets \\{ \\}$ \\Comment{Initialize the collection of the statical model parameters}\n    \\ForAll{$ ts \\in \\mathcal{TS}$ }\n        \\State $id \\gets$ Get the unique identity of $ts$\n        \\ForAll{$ s \\in 0...23$ }\n        \t\\State $ts^c, ts^t \\gets$ Construct a new consumption time series using $ts$, and a new temperature time $ts^t$ using $ts'$ at season $s$\n\t\t    \\State $\\alpha_{1},...,\\alpha_{p}, \\beta_{1},\\beta_{2},\\beta_{3} \\gets$ Compute PARX model using $ts^c$ and $ts^t$\n\t\t    \\State Insert $(id, s, \\alpha_{1},...,\\alpha_{p}, \\beta_{1},\\beta_{2},\\beta_{3})$ into $\\mathcal{M}$\n\t\t     \\State $\\mathcal{L} \\gets \\{ \\}$\n\t\t     \\State $\\mathcal{D} \\gets$ Get the days of $ts$\n\t\t    \\ForAll{$ d \\in \\mathcal{D}$ }\n\t\t        \\State $\\hat{v} \\gets$ Compute the predict reading of the season $s$ using PARX\n\t\t        \\State $v \\gets$ Get the actual hourly reading from $ts$ of the day $d$\n\t\t        \\State $l \\gets$ Compute the ln value of $L_1$ distance of the day $d$, $ln(||\\hat{v} - v||)$\n\t\t        \\State Add $l$ into  $\\mathcal{L}$\n\t\t    \\EndFor\n\t\t    \\State $\\mu, \\beta \\gets$ Compute the mean and standard deviation using the normal distribution statistical model on  $\\mathcal{L}$\n\t\t    \\State Insert $(id, s, \\mu, \\delta )$ into $N$\n\t\t    \\EndFor\n     \\EndFor\t\n    \\Return $\\mathcal{M}, \\mathcal{N}$\n \\EndFunction\n\\end{algorithmic}\n}\n\\label{alg:training}\n\\end{algorithm}\n\nThe implementation is a Spark program. The consumption time series, as well as temperature time series, are read into the distributed main memory as {\\em resilient distributed datasets (RDDs)}, which are fault-tolerant, immutable and partitioned parallel data structures that can be operated in parallel, e.g., by using the operators, including map, reduce, groupByKey, filter, collect, etc \\citep{Zaharia}. To generate the new time series, we use the {\\em groupByKey} operator to aggregate the consumption series by the composite key of meter ID and season (or hours); while use only the season as the key to the temperature time series. Then, we merge the generated time series by the \\texttt{join} operator on the key of the season. The PARX, in fact, can be regarded as a multi-linear regression model, which simply takes the auto-regressors and the exogenous variables as the independent variables. We, then, apply the multiple linear regression function from the Spark machine learning the library, MLib \\citep{Meng}, to compute the coefficients. For all of these operations, the transformation functions are directly applied on RDDs for doing the data processing.    \n\n\\subsection{Real-time Anomaly Detection}\nThe real-time anomaly detection is carried out in the speed layer. Algorithm~\\ref{alg:realtimedetection} describes the anomaly detection process, which is self-explanatory. First, the detection algorithm reads meter readings from all the incoming data streams, and reads the weather temperature and detection models from the PostgreSQL database each hour. For each data stream, the algorithm predicts the reading using the PARX algorithm, with the pre-computed parameters, the previous $p$ day's readings at the current hour, \\ie, the season $s$, and weather temperature (see line 4--7). Then, the algorithm calculates the log value of the $L1$ distance between the predict and the actual readings, then uses it compute the probability using the Gaussian model (see line 8-10). In the end, the algorithm decides whether the current reading is an anomaly or not based on the computed probability value, \\ie, if its value is below the user-defined threshold, $\\epsilon$. If the current reading is classified as an anomaly, it will be written into the database for the customer notification (see line 11--12).\n\\begin{algorithm}[htp]\n\\caption{Real-time anomaly detection}\n{\\scriptsize\n\\begin{algorithmic}[1]\n\\Function {Detect}{CurrentReadingCollection $\\mathcal{V}$, Temperature $t$, PredictModel $\\mathcal{M}$, StaticalModel $\\mathcal{N}$, Threshold $\\epsilon$ }\n    \\State $\\mathcal{R} \\gets \\{ \\}$ \\Comment{Initialize the detection results}\n    \\ForAll{$ v \\in \\mathcal{V}$ }\n        \\State $id \\gets$ Get the unique identity of $v$\n        \\State $s \\gets$ Get the season of $v$\n        \\State $\\alpha_{1},...,\\alpha_{p}, \\beta_{1},\\beta_{2},\\beta_{3} \\gets$ Get the parameters from  $\\mathcal{M}$ by $id$\n        \\State $\\hat{v} \\gets$ Compute the predict reading at $s$ using PARX with the parameters, the $p$ days' readings at $s$, and temperature $t$\n        \\State $x \\gets ln||\\hat{v} - v||$ \\Comment{Compute the ln value of $L_1$ distance at the season $s$}\n        \\State $\\mu, \\delta \\gets$ Get the statical model parameters from $\\mathcal{N}$ by $id$ and $s$\n        \\State $p \\gets$ Compute the probability using the normal distribution function, $\\frac{1}{\\delta \\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\delta^2}}$\n        \\If {$p<\\epsilon$}\n         \\State  Add $(id, s, p, v, \\hat{v})$ into $\\mathcal{R}$\n        \\EndIf\n     \\EndFor\t\n    \\Return $\\mathcal{R}$\n \\EndFunction\n\\end{algorithmic}\n}\n\\label{alg:realtimedetection}\n\\end{algorithm}\n\nWe implement the algorithm to process the real-time data on Spark Streaming. Spark Streaming allows for continuous processing via short interval batches, and its basic data abstraction is called {\\em discretized streams (D-Streams)}, a continuous stream of data \\citep{sparkstreaming}. The data are received in each interval batch, {\\em hourly} in our case, and operations will run upon the data for doing transformations, such as filter unnecessary attribute values, extracting the hour from the timestamp, etc. (see Fig.~\\ref{fig:rddslidewin}). When using PARX for the prediction, we fetch the previous $p$ days' readings of the current hour for auto-regression. For example, in Fig.~\\ref{fig:rddslidewin} we set the order, $p=3$, therefore, the window size is set to 72 hours (\\ie, 3 days)  to keep the past three days' readings at the particular hour within the same window (e.g., the RDDs colored by green). This is done by using the window function, \\texttt{reduceByKeyAndWindow(func, windowLength, slideInterval)}, to aggregate the data with specified key, window length and slide interval (\\eg, meter ID and season as the composite key in this case,  $windowLength=72$ hours and $slideInterval=1$ hour). In the underlying, Spark uses  the data {\\em checkpointing} mechanism to keep the past RDDs in HDFS. At the beginning of each interval batch, the data models are read from the PostgreSQL database in the serving layer, and broadcast to all DStreams. Therefore, the detection program can always use the latest detection models to detect anomalies.   \n\\begin{figure\n\\centering\n\\includegraphics[width=0.85\\textwidth]{rddslidewin}\n\\vspace{-5pt}\n\\caption{Slide windows in the real-time anomaly detection}\n\\label{fig:rddslidewin}\n\\end{figure}\n\n\n\n\n\\section{\\uppercase{Evaluation}}\n\\subsection{Experimental settings}\nIn this section, we will evaluate the effectiveness and the scalability of our anomaly detection system. We conduct the experiments in a cluster with 17 servers.  Five servers\nare used for running the speed layer, while twelve servers are used for the batch layer.\nWe also exploit one of the servers in the speed layer as the serving layer for managing\nthe detection models and sending anomaly detection messages. All the servers have the identical settings, configured with an Intel(R) Core(TM) i7-4770 processor (3.40GHz, 4 Cores, hyper-threading is enabled, two hyper-threads per core), 16GB RAM, and a Seagate Hard driver (1TB, 6 GB\/s, 32 MB Cache and 7200 RPM), running Ubuntu 12.04 LTS with 64bit Linux 3.11.0 kernel. The serving layer uses PostgreSQL 9.4 database with the settings ``shared buffers=4096MB, temp buffers=512MB, work mem=1024MB, checkpoint segments=64'' and default values for the other configuration parameters.\n\nWe have a real-world residential electricity consumption data set (27,300 time series), which will be used for evaluating the accuracy of the anomaly detection. The data are the 2-year time series of hourly resolution. To evaluate the scalability, we use the synthetic data set generated by our data generator with the real-world data as the seed. The size of data tested in the cluster environment is scaled up to one terabyte, corresponding to over twenty million time series. \n\n\\subsection{Anomaly Detection Accuracy}\n\\label{sec:accuracy}\nWe start by evaluating the accuracy of our anomaly detection system using a randomly-selected time series from the real-world data set.\n\nTo provide a basis for comparison, we perform the anomaly detection using a standard boxplot analysis as well. Boxplot is a quick graphic approach for examining data sets, and has been used for decades. A boxplot uses five parameters to describe a numeric data set, including lower fence, lower quartile, media, upper quartile and upper fence (see Fig.~\\ref{fig:boxplot}). According to Fig.~\\ref{fig:boxplot}, a boxplot is constructed by drawing a rectangle between the upper and lower quartiles with a solid line indicating the median. The length of the box is called interquartile range, $IQR$. The sample data points lying outside the fences, $1.5*IQR$, are classified as the outliers, which has been indicated to be acceptable for most situations \\citep{frigge1989}.\n\nTo align boxplot with our detection method, we test the anomalies based on the 24 seasons. There are 17,520 data points in total in the selected time series. Fig.~\\ref{fig:anomalyboxplot} shows the boxplot result where the blue points located on the top of the upper fence represent the anomalies, a total of 1,260 data points. Since the boxplot approach is merely able to detect energy consumption lying unusually far from the main body of the data,  it is difficult to determine which ones are the true anomalies, and to identify the potential reasons for these anomalies because there are too many false positives.\n\n\n\\begin{figure}[htp]\n\t\\vspace{-10pt}\n\\begin{minipage}[b]{0.4\\textwidth}\n\t\\centering\n\t\\includegraphics[width=1.35\\textwidth]{boxplot}\n\t\\vspace{-18pt}\n\t\\caption{Box plot}\n\t\\label{fig:boxplot}\n\\end{minipage}\n\\begin{minipage}[b]{0.75\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.56\\textwidth]{anomalyboxplot}\n\t\t\\vspace{-10pt}\n\t\t\\caption{Anomaly detection using box plot}\n\t\t\\label{fig:anomalyboxplot}\n\t   \\end{minipage}\n\\vspace{-20pt}\n\\end{figure}\n\n\\begin{figure}[t]\n\\vspace{-5pt}\n  \\begin{minipage}[b]{0.33\\textwidth}\n    \\centering\n\t\\includegraphics[width=1.02\\textwidth]{l1distancedist}\n\t\\vspace{-20pt}\n\t\\caption{Log-normal distributions across the L1 distances }\n\t\\label{fig:l1distancedist}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.33\\textwidth}\n    \\centering\n\t\\includegraphics[width=0.9\\textwidth]{detectedanomalies}\n\t\\vspace{-10pt}\n\t\\caption{Anomaly detection using PARX and statistical method}\n\t\\label{fig:detectedanomalies}\n  \\end{minipage}\n   \\begin{minipage}[b]{0.33\\textwidth}\n     \\centering\n\t\\includegraphics[width=0.9\\textwidth]{detectmodelupdate}\n\t\\vspace{-10pt}\n\t\\caption{Impact of detection model update frequency}\n\t\\label{fig:detectmodelupdate}\n  \\end{minipage}\n  \\vspace{-10pt}\n\\end{figure}\n\nWe now use the proposed detection algorithm to analyze the same time series. Fig.~\\ref{fig:l1distancedist} depicts the distribution of the $L1$ distances of a season by the histogram. As shown, the distribution has the shape of a log-normal distribution. We have checked the $L1$ distance distributions for all the 24 seasons, and found that they all have a  similar shape. This is the reason that we choose log-normal distribution in our statistical-based anomaly detection. Besides, we test the anomalies by treating all the days the same, and different, \\ie, discriminating the days into workdays, weekend \\& holidays.  Moreover, we increase the threshold value, $\\epsilon$, from 0.05 to 0.15, and do the test. The results in Fig.~\\ref{fig:detectedanomalies} demonstrate that the detection identifies more anomalies for treating all the days the same than different. The reason is that during weekends and holidays, people tend to stay at home more time, thus use more energy. The consumptions are more likely higher than in the weekdays. For the threshold parameter, its value is for classifying a usual or unusual reading. According to the results, if the value increases, the number of detected anomalies changes significantly. For the real-world deployment of this system, the threshold value can be set by the residents to decide when to receive anomaly alerting messages.\n\n\nWe now evaluate the impact of the model update frequency on the detection accuracy. We use half-year's time series as the initial data set to train the detection models. We design the following three scenarios for model updating: 1) update per day; 2) update per 10 days behind the detection; and 3) without an update. We measure the detected anomalies for the three scenarios by treating all days the same. According to the results shown in Fig.~\\ref{fig:detectmodelupdate}, the frequent updates of the models help to decrease the detected anomalies. It is due to the improvement of the prediction accuracy of the PARX model. Thus, less large $L1$ distances are identified as the anomalies. However, although the update frequency does help to determine the real anomalies, the results do not show a big difference if the models are updated within a certain short-time interval, \\eg, the results of the scenario 1) and 2).\n\nIn the end, we compare our approach with the boxplot, and the result shows that the number of the anomalies reported by the PARX prediction and statistical method can be decreased notably. This increases the chance to determine accurately real anomalies for an energy consumption time series. \n\n\n\n\n\\subsection{System Scalability}\nWe now evaluate the scalability of our anomaly detection system. As our system can scale-out and to efficiently cope with large amounts of data, we vary both the number of executors and the volume of the input data in the following.\n\n{\\bf Scale-out experiment.} Parallel processing is a key feature of the proposed system. To evaluate the scalability of our implementation, we conduct an experiment with a varying number of the executors in Spark. In this experiment, we use a fix-sized synthetic data set with eight million time series of a one-year length (275GB), which were generated by our data generator seeded with the real-world data set. Since we are interested in the real-time and batch capability of our system, we use the full sets of the nodes to test  batch model training and real-time anomaly detection separately. We first test the batch capability by running the training program with the number of executors increased from 8 to 256. We repeat each test for ten times, and record their execution times. The results are depicted by the boxplot shown in Fig.~\\ref{fig:scaleoutbatch}. According to the results, the execution time and the time variance decrease when more executors are added. But, when the number reaches 64, the increasing parallelism does not speed up the batch processing further, which is due to the overhead of the Spark master when managing a large number of executors. We conduct the real-time anomaly detection on Spark Streaming, and likewise, we scale the number of executors from 8 to 256. Since we only study the real-time detection scalability, we use the detection models without updating. Fig.~\\ref{fig:scaleoutstream} shows the results, which indicate that the variance of execution time is larger than the batch model training. It might be due to the variability of real-time batch executions on Spark Streaming when doing the anomaly detection for each hour.\n\n\n\\begin{figure*}[t]\n  \\begin{minipage}[b]{0.33\\textwidth}\n  \\centering\n\t\\includegraphics[width=0.95\\textwidth]{scaleoutbatch}\n\t\\caption{Batch model training}\n\t\\label{fig:scaleoutbatch}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.33\\textwidth}\n    \\centering\n\t\\includegraphics[width=0.95\\textwidth]{scaleoutstream}\n\t\\caption{Real-time detection}\n\t\\label{fig:scaleoutstream}\n\t\\end{minipage}\n  \\begin{minipage}[b]{0.33\\textwidth}\n    \\centering\n\t\\includegraphics[width=1\\textwidth]{sizeup}\n\t\\caption{Size-up experiment}\n\t\\label{fig:sizeup}\n\t\\end{minipage}\n\\end{figure*}\n\n{\\bf Increased data load experiment.}  To evaluate the scalability of the algorithms over large volumes of data, we compare different workloads. According to the above experiments, the optimal number of parallel executors for model training and anomaly detection are 64 and 128, respectively. We use the optimal executor number (the memory of executor is configured to 4GB) in our experiments, but vary the number of time series from 8 to 24 million (the size is from 275GB to 825GB). The processing times of varying data workloads are displayed in Fig.~\\ref{fig:sizeup}. We observe that both of the training and detection processes can scale near linearly with the quantity of the time series. The time of detection, in this case, is the total execution time of handling all the time series of a one-year length, \\eg, it takes less than two minutes to finish eight million time series with the optimized settings. The average time of each real-time batch only takes a few seconds (recall that a batch in Spark Streaming processes the data of each hour). In the real-world deployment, the detection program can be set to run every hour to inspect hourly smart meter readings. According to the results, the anomaly detection system has a very good scalability which can meet the fast-growth of smart meter data.\n\nUnlike the anomaly detection, the training process uses the full set of the data to generate new models each time.  Anomaly detection, instead, is performed for each hour where Spark Streaming runs periodical batch (or impulse) to process the data, which needs more time in overall. The training and detection programs can be deployed either in different clusters or the same cluster. If deployed in the same cluster, it is necessary to allocate computing resources in a reasonable way. For example, since the batch job takes a much longer time, it can be scheduled to run immediately after the anomaly detection job. A scheduling system is, thus, necessary, and this will be for our future work.\n\n\n\\section{\\uppercase{Related Works}}\n{\\bf Anomaly energy consumption detection.} \nAnomaly detection is an important aspect in energy consumption time series management. Chandola et al.  present a survey of different anomaly detection techniques in various application domains including energy \\citep{Chandola2009}. Statistical and data mining are the commonly used techniques for discovering abnormal consumption behaviors \\citep{Janetzko}. Statistical methods are based on modeling data using distributions, and see if the data under test observes to the distributions. Accordingly, the approaches presented in this paper combine PARX and log-normal distribution function to detect anomalies in energy consumption time series. Jakkula and Cook use statistics and clustering to identify outliers in power datasets collected from smart environments \\citep{Jakkula2010}, but they have not considered the impact of the exogenous variables, \\eg, weather temperature,  on the electricity consumption. Linear regression can extract time series features when the dependent variables are well-defined \\citep{Magld2012}. The early experience of identifying outliers in linear regression is through setting a threshold limit, but this yields many false positives for large data sets \\citep{lee1997}. Adnan et al.  combine linear regression with clustering techniques for getting better results \\citep{Adnan2003}. Zhang et al.  \\citep{zhang2011} further  use piecewise linear regression to fit the relation between energy consumption and weather temperature. The results obtained are more favorable than entropy and clustering methods. But, their approach does not take the changes of consumption pattern into account. Brown et al.  use K-nearest neighborhood (KNN) in fast kernel regression to predict electricity consumption \\citep{brown2011}, which requires large datasets. The resulting models are static, thus it is not preferable for online anomaly detection and the situation when consumption pattern is changed. Nadai et al.  combine ARIMA and adaptive artificial neural network (ANN) to detect anomaly consumption \\citep{nadai2015} using a relatively small data set that is from a few buildings. In comparison, we propose the prediction and statistical anomaly models and combine with the lambda architecture for supporting regular model refreshment, and real-time anomaly detections. Besides, the proposed approach can handle scalable data sets, and consumption pattern changing owing to using the PARX model. \n\n\n{\\bf Batch and stream processing on big data.} \nBatch and realtime\/stream processings have attracted much research effort in recent year, with the popularity of Internet of Things (IoT). Liu et al.  make a survey of the existing stream processing systems, and discuss the potential technologies used for lambda architecture \\citep{Liu2014}. The tools \\citep{dawak2011,tldks2011,pvldb2011,ideas2014} Cheng et al.  propose a smart city data platform that supports both batch and real-time data processing \\citep{Cheng}, and they suggest that anomaly detection should be implemented as the chief component of any platform for processing sensor data. Different to proposing the generic lambda architecture  \\citep{Marz2013}, Preuveneers et al.  \\citep{Preuveneers} and Gao \\citep{Gao} present the big data architectures for processing domain-specific big data, including health care, context-aware user authentication and social media.  Schneider et al.  study batch data and streaming data anomaly detection, respectively \\citep{Schneider2016}. The used detection model, however, is static, and the use case is different to ours which employs the batch job to update the models iteratively while use the real-time job to detect the anomalies online in data streams.  \n\n\\begin{comment}\nBatch and stream processing on big data The data explosion of the Internet of Things (IoT) is often linked with the Big Data paradigm. MapReduce [10] \u2013 and its Hadoop [48] implementation \u2013 is a software framework and programming model that allows\ndevelopers to write programs that process massive amounts of unstructured data in parallel across a distributed cluster of computers. The shortcomings and drawbacks of batch-oriented data processing have been widely recognized as many applications are in need of real-time [5,44] and in-stream processing capabilities [9]. This concept got a lot of traction with\nvarious distributed event stream processing (ESP) engines emerging. Yahoo's S4 [29] and Twitter's Storm project [26,47] were among the first to attract a lot of attention. Also Google acknowledged the limitations of MapReduce, with its MillWheel [2] framework and programming model dedicated to fault-tolerant stream processing at Internet scale. Spark [49] is another\nstate-of-practice software solution for large-scale data processing. It runs up to 100 times faster in memory than Hadoop MapReduce and supports scalable fault-tolerant streaming applications. Spark Streaming [50] builds upon the Spark's foundations to build big data application that act on data in real time. Apache Samza2 \u2013 originally developed at LinkedIn \u2013 is\nanother popular open source frameworks for scalable stream processing.\n\\end{comment}\n\n{\\bf The use of lambda architecture.}\nLambda architecture has attracted a growing interest due to its mix capabilities to process both real-time and batch data. Sequeira et al.  use lambda architecture in an industrial EMS solution with cloud computing capabilities \\citep{Sequeira2014}. Kro\\ss et al.  develop on-demand stream processing within the lambda architecture to optimize the usage of computing resources in clusters \\citep{Brunnert}. Martnez-Prieto et al.  adapts the lambda architecture in semantic data processing \\citep{Martnez2015}; Liu et al.  applies it to smart grid complex event processing (CEP) \\citep{Liu2015}; Villari et al.  proposes AllJoyn Lambda, the platform for managing embedded devices of smart homes \\citep{Villari}; and Hasani et al.  use it for real-time big data analytics \\citep{Hasani}. Besides, the works \\citep{Casado2015,Liu2014} both give an extensive review of the technologies for the lambda architecture. There are various use cases of the lambda architecture. In contrast, we focus on a particular use case, \\ie, energy consumption anomaly detection as has been mentioned in future extensions of \\citep{Kiran} work. More specifically, we use it in the model update and the real-time anomaly detection using the models, which is significant to the large deployment of smart meters and sensors of IoT today.\n\n\n\\section{\\uppercase{Conclusions and Future Work}}\n\\label{sec:conclusion}\nAnalyzing and detecting anomalies is an important task for live energy consumption data while the improvement of detection accuracy and scalability is quite challenging. In this paper, we applied the novel lambda architecture technique to an anomaly detection system, which supports iterative batch updates of the detection models, and real-time anomaly detection on scalable data streams. We have proposed the detection algorithm for finding the anomalies based on one's history consumption pattern via the supervised learning and statistical algorithm. Furthermore, the system supports alerting service for customers by setting a personalized threshold value for conspicuous energy consumption. We have evaluated the accuracy of the anomaly detection algorithm using a real-world data set, and the scalability using a large synthetic data set. The results have validated the effectiveness and the efficiency of the proposed system with the lambda architecture. \n\n\nFor the future work, we will implement a scheduling system that can coordinate the running of the batch and real-time jobs within the same cluster. We intend to explore the ways to detect a greater range of anomalies, such as missing values, negative energy consumption and device errors. Besides, we plan to support additional types of data, such as gas, heating and water data, and to implement the corresponding detection algorithm.   \n\n \n\n\\section*{Acknowledgements}\n\\noindent This work is part of the CITIES (NO. 1035-00027B) research project funded by Innovation Found Denmark.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStatistical topic models are generative unsupervised models that describe the content of documents in large textual collections. Prior research has investigated the application of topic models such as Latent Dirichlet Allocation (LDA) \\cite{blei2003latent} in a variety of domains ranging from image analysis to political science. \nMost of the work on topic models assumes exchangeability between words and treats documents in a bag-of-words fashion. As a result, the words' grouping in coherent text segments, such as sentences or phrases, is lost. \n\nHowever, the inner structure of documents is generally useful, when identifying topics. For instance, one would expect that in each sentence, after standard pre-processing steps such as stop-word removal, only a very limited number of latent topics would appear. Thus, we argue that coherent text segments should pose ``constraints'' on the amount of topics that appear inside those segments.  \n\nIn this paper, we propose \\textit{sentenceLDA} (\\textit{senLDA}), whose purpose is to incorporate part of the text structure in the topic model. Motivated by the argument that coherent text spans should be produced by only a handful of topics, we propose to modify the generative process of LDA. Hence, we argue that the latent topics of short text spans should be consistent across the units of those spans. In our approach, such text spans can vary from paragraphs to sentences and phrases depending on the task's purpose. Also, note that in the extreme case where words are the coherent text segments, the standard LDA model becomes a special case of \\textit{senLDA}. \n\nIn the remainder of the paper we present the \\textit{senLDA }  and we derive its collapsed Gibbs sampler in Section 2, we illustrate its advantages by comparing it with  LDA on intrinsic (\\textit{in vitro}) and extrinsic (\\textit{ex vivo}) evaluation experiments using collections of Wikipedia and PubMed articles in Section 3, and we conclude in Section 4.\n\n\\section{The proposed model}\nA statistical topic model represents the words in a collection of $D$ documents\nas mixtures of $K$ ``topics'', which\nare multinomials over a vocabulary of size $V$. In the case of LDA, for each document\n$d_i$ a multinomial over topics is sampled from a Dirichlet prior with parameters $\\boldsymbol{\\alpha}$. \nThe probability $p(w|z=k)$ of a term $w$, given the topic $k$, is represented by\n$\\phi_{k,t}$. We refer to the complete $K\\times V$ matrix of word-topic \nprobabilities as $\\Phi$. \nThe multinomial parameters $\\phi_k$\nare again drawn from a Dirichlet prior parametrized by $\\boldsymbol{\\beta}$. Each observed term $w$ in the collection is drawn from a multinomial for the topic\nrepresented by a discrete hidden indicator variable $z_i$. For simplicity in the mathematical development and notation, we assume symmetric Dirichlet priors but the extension to the asymmetric case is straightforward. Hence, the values of $\\alpha$ and $\\beta$ are model hyper-parameters.\n \n\\input{senLDAfig.tex}\n\nWe extend LDA by adding an extra plate denoting the coherent text segments of a document. In the rest, without loss of generality we use sentences as coherent segments. A finer level of granularity can be achieved though, by analysing the structure of sentences and using phrases as such segments. The graphical representation of the \\textit{senLDA } model is shown in Figure \\ref{fig:senlda} and the generative process of a document collection using \\textit{senLDA } is described in Algorithm \\ref{algo:senlda}. \nFor inference, we use a collapsed Gibbs sampling method \\cite{griffiths2004finding}. \nWe now derive the Gibbs sampler equations by estimating the hidden topic variables. \n\n\nIn \\textit{senLDA } the joint distribution can be factored:\n\\begin{equation}\np(w,z|\\alpha,\\beta) = p(w |z,\\beta)p(z|\\alpha)\n\\label{eq:factors}\n\\end{equation}\nbecause the first term is independent of $\\alpha$ and the second from $\\beta$. \nAfter standard manipulations as in the paradigm of \\cite{heinrich2005parameter} one arrives at: \n\\begin{equation}\np(\\vec{z}, \\vec{w}|\\alpha,\\beta) = \\prod_{z=1}^{K}\\frac{\\Delta(\\vec{n}_z+\\beta)}{\\Delta(\\beta)} \\prod_{m=1}^D \\frac{\\Delta(\\vec{n}_m+\\alpha)}{\\Delta(\\alpha)}\\label{eq:joint}\n\\end{equation}\n\\noindent where $\\Delta(\\vec{x})=Beta(x_1,\\ldots,x_m)=\\frac{\\prod_{k=1}^{dim \\vec{x}} \\Gamma(x_k)}{\\Gamma(\\sum_{k=1}^{dim \\vec{x}} x_k)}$ is a multidimensional extension of the beta function used for notation convenience, and $\\vec{n}_m$, $\\vec{n}_z$ refer to the occurrences of topics with documents and topics with terms respectively. To calculate the full conditional\nwe take into account the structure of the document $d$ and the fact that  $\\vec{w_d}=\\{\\vec{w}_{d\\neg{s}}, \\vec{w}_{\\neg{s}}\\}$, $\\vec{z}=\\{\\vec{z}_{d\\neg{s}}, \\vec{z}_{\\neg{s}}\\}$. The subscript $s$ in $\\vec{w}_s,\\vec{z}_s$ denotes the words and the topic respectively of sentence $s$. For the full conditional  of topic $k$ we have:\n\\begin{equation}\n\\begin{split}\np(z_s=k|\\vec{z}_{\\neg{s}}, \\vec{w}) = \\frac{p(\\vec{w},\\vec{z})}{p(\\vec{w}, \\vec{z}_{\\neg{s}})} = \\frac{p(\\vec{w}|\\vec{z})}{p(\\vec{w}_{\\neg{s}}|\\vec{z}_{\\neg{s}}) p(w_s)}\\frac{p(\\vec{z})}{p(\\vec{z}_{\\neg{s}})}=\\\\\n= \\frac{p(\\vec{w},\\vec{z})}{p(\\vec{w}_{\\neg{s}}, \\vec{z}_{\\neg{s}})} \\propto\n \\frac{\\Delta(\\vec{n}_z+\\beta)}{\\Delta(\\vec{n}_{z,\\neg{s}}+\\beta)} \\frac{\\Delta(\\vec{n}_m+\\alpha)}{\\Delta(\\vec{n}_{m,\\neg{s}}+\\alpha)} \n\n\\end{split}\n\\label{eq:fullConditional}\n\\end{equation}\n\n\\noindent For the first term of equation Eq. \\eqref{eq:fullConditional} we have:\n\\begin{multline}\n  \\frac{\\Delta(\\vec{n}_z+\\beta)}{\\Delta(\\vec{n}_{z,\\neg{s}}+\\beta)} =\n  \\frac{ \\frac{\\prod_{w\\in s} \\Gamma(\\vec{n}_z+\\beta)}{\\Gamma(\\sum_{w\\in s}(\\vec{n}_z+\\beta))}}\n  {\\frac{\\prod_{w\\in s} \\Gamma(\\vec{n}_{z,\\neg{s}}+\\beta)}{\\Gamma(\\sum_{w\\in s}(\\vec{n}_{z,\\neg{s}}+\\beta))}}= \\\\ =\\prod_{w\\in s} (\\frac{ \\Gamma(\\vec{n}_z+\\beta)}{\\Gamma(\\vec{n}_{z,\\neg{s}}+\\beta)}) \\frac{\\Gamma(\\sum_{w\\in s}(n_{z,\\neg{s}}+\\beta))}{\\Gamma(\\sum_{w\\in s}(n_z+\\beta))} = \\\\\n  =   \\underbrace{\\frac{\\overbrace{\\prod_{w\\in s} (n_{k,\\neg{s}}^{(w)}+\\beta)\\cdots (n_{k,\\neg{s}}^{(w)}+\\beta+(n_{k,s}^{(w)}-1))  }^{\\mbox{A}}}{(\\sum_{w\\in V} (n_{k,\\neg{s}}^{(w)}+\\beta))\\cdots(\\sum_{w\\in V} n_{k,\\neg{s}}^{(w)}+\\beta+(N_{k,s}^{(w)}-1))} }_{\\mbox{B}}\n\\label{eq:inferenceDetails}\n\\end{multline}\nHere, for the generation of A and B  we used the recursive property of the $\\Gamma$ function: $\\Gamma(x+m)= (x+m-1)(x+m-2)\\cdots(x+1)x\\Gamma(x)$; $w$ is a term that can occur many times in a sentence and $n_{k,s}^{(w)}$ denotes $w$'s frequency in sentence $s$ given that the sentence $s$ belongs to topic $k$; $N_{k,s}^{(w)}$ denotes how many words of sentence $s$ belong to topic $t$. \n\n\\RestyleAlgo{boxruled}\n\\begin{algorithm}[t]\\small\n\\DontPrintSemicolon\n \\For{ \\upshape document $d \\in[1,\\ldots,D]$}{\n  sample mixture of topics $\\theta_m \\sim$ Dirichlet(a)\\;\n  sample sentence number $S_d \\sim  Poisson (\\xi)$  \\;\n   \/\/Sentence plate\\;\n  \\For{ \\upshape sentence $s\\in[1,S_d]$}{\n  sample number of words  $W_{s} \\sim  Poisson (\\xi_d)$ \\;\n  sample topic $z_{d,s} \\sim Multinomial(\\theta_m)$\\;\n  \/\/Word plate in each language\\;\n  \\For{ \\upshape words $w\\in [1, W_{d,s}]$ in sentence $s$ }{\n  sample term for $w \\sim Multinomial(\\phi_{z_{d,s}}) $\\;\n  }\n  } \n }\n\\caption{Text collection generation with \\textit{senLDA}}\\label{algo:senlda}\n\\end{algorithm}\n\nThe development of the second factor in the final step of Eq. \\eqref{eq:fullConditional} is similar to the LDA calculations with the difference that the counts of topics per document are calculated given the allocation of sentences to topics and not the allocation of words to topics. This yields:\n\\begin{multline}\n p(z_s=k|\\vec{z}_{\\neg{s}}, \\vec{w}) = (n_{m,\\neg{s}}^{(k)} + \\alpha) \\times \\\\ \\times \\frac{\\prod_{w\\in s} (n_{k,\\neg{s}}^{(w)}+\\beta)\\cdots (n_{k,\\neg{s}}^{(w)}+\\beta+(n_{k,s}^{(w)}-1))}{(\\sum_{w\\in V} (n_{k,\\neg{s}}^{(w)}+\\beta))\\cdots(\\sum_{w\\in V} n_{k,\\neg{s}}^{(w)}+\\beta+(N_{k,s}^{(w)}-1))}  \n\\label{eq:finalMonolingual}\n \\end{multline}\nwhere $n_{m,\\neg{s}}^{(w)}$ denotes the number of times that topic $k$ has been observed with a sentence from document $d$, excluding the sentence currently sampled. \nNote that Eq. \\eqref{eq:finalMonolingual} reduces to the standard  LDA collapsed Gibbs sampling inference equations if the coherent text spans are reduced to words.\n\n\nThe idea of integrating the sentence limits in the LDA model has been previously investigated. For instance, in \\cite{wang2009multi} in the context of summarization the authors combine the unigram language model with topic models over sentences so that the latent topics are represented by sentences instead of terms. In \\cite{chen2010adaptation} the notion of \\textit{sentence topics} is introduced and they are sampled from separate topic distributions and co-exist with the word topics. Also, Boyd et al. \\cite{boyd2009syntactic} propose an adaptation of topic models to the text structure obtained by the parsing tree of a document. Our method resembles these works in that it integrates the notion of sentences to extend LDA. In our case though, we directly extend LDA maintaining the association of words to topics, we retain its simplicity without adding extra hyper-parameters thus allowing a fast, gibbs sampling inference, and we do not require any language-dependent tools such as parsers.\n\n\\section{Empirical results}\n\n\nWe conduct experiments to verify the applicability and evaluate the performance of \\textit{senLDA }  compared to LDA. The process is divided into two steps: (i) the training phase, where the topic models are trained to learn the their parameters, and (ii) the inference phase that is for new, unseen documents their topic distributions are estimated. We use the Gibbs sampling inference approach given by Eq. \\eqref{eq:finalMonolingual}. The hyper-parameters $\\alpha$ and $\\beta$ are set to $\\frac{1}{K}$, with $K$ being the number of topics. Table \\ref{table:trainData} shows the  datasets we used. They come from the publicly available collections of Wikipedia \\cite{partalas15lshtc} and PubMed \\cite{tsatsaronis2015overview}. The first four datasets (WikiTrain* and PubMedTrain*) were used for learning the topic model parameters; they differ in their respective size. Also, the vocabulary of the PubMed datasets is significantly larger due to the medical terms that appear. During preprocessing we only applied lower-casing, stop-word removal and lemmatization using the WordNet Lemmatizer.\\footnote{The code and the data are publicly available  at \\url{https:\/\/github.com\/balikasg\/topicModelling\/}}  The rest of the document collections of Table \\ref{table:trainData} are used for classification purposes and are discussed later in the section. \n\n\\begin{table}[t]\n\\small\\centering\n \\begin{tabular}{lcccc}\n  \\toprule\n  &Documents & $|V|$ & Classes & Timing (sec) \\\\\\midrule\n  WikiTrain1 & 10,000 & 46,051 & - &182|271 \\\\\n  WikiTrain2 & 30,000 & 65,820 & - &332|434 \\\\\n  PubMedTrain1 &10,000& 55,115 & - &304|433 \\\\\n  PubMedTrain2 & 60,000& 150,440 & - & 1830|2799 \\\\\n  Wiki37& 2,459 & 23,559& 37 & -\\\\\n  Wiki46& 3,657 & 27,914& 46 & -\\\\\n  PubMed25 & 7,248 &40,173 & 25 & -\\\\\n  PubMed50 & 9,035 &47,199 & 50 & -\\\\\n\\bottomrul\n \\end{tabular}\n\\caption{Description of the data used after pre-processing. ``Timing'' refers to the 25 first training iterations with the left (resp. right) values corresponding to \\textit{senLDA } (resp. LDA).}\\label{table:trainData}\n\\end{table}\n\n\\noindent\\textbf{Intrinsic evaluation} Topic model evaluation has been the subject of intense research. For intrinsic evaluation we report here perplexity \\cite{azzopardi2003investigating}, which is probably the dominant measure for topic models evaluation in the bibliography. The perplexity of $d$ held out documents given the model parameters $\\vec{\\vartheta}$ is defined as the reciprocal geometric mean of the token likelihoods of those data, given the parameters of the model:\n\\begin{equation}\n p(w_{\\text{heldOut}}) = \\exp-\\frac{\\sum_{i=1}^d \\sum_{j=1}^{w_i} \\log p(w_{i,j}|\\vec{\\vartheta})}{\\sum_{i=1}^d \\sum_{j=1}^{w_i} 1}  \n\\end{equation}\nNote that \\textit{senLDA }  samples per sentence and thus results in less flexibility at the word level where perplexity is calculated. Even though, the comparison between \\textit{senLDA } and LDA, at word level using perplexity, gives insights in the relative merits of the the proposed model. \n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=0.47\\textwidth]{perplexity_both.pdf}\n \\caption{The ratio of perplexities of \\textit{senLDA } and LDA calculated on Wiki37 and PubMed25.  }\\label{fig:perplexity}\n\\end{figure}\n\nFigure \\ref{fig:perplexity} depicts the ratio of the perplexity values between \\textit{senLDA } and LDA. We set $K=125$ after grid searching $K\\in \\{25, 75, 125, 175\\}$ for perplexity with 5-fold cross-validation on the training data. Values higher (resp. lower) than one signify that \\textit{senLDA } achieves lower (resp. higher) perplexity than LDA. The figure demonstrates that in the first iterations before convergence of both models, \\textit{senLDA } performs better. What is more, \\textit{senLDA } converges after only around 30 iterations, whereas LDA converges after 160 iterations on Wikipedia and 200 iterations on the PubMed  datasets respectively. We define convergence as the situation where the model's perplexity does not any more decrease over training iterations. The shaded area in the figure highlights the period while \\textit{senLDA } performs better. It is to be noted, that although competitive, \\textit{senLDA }  does not outperform  LDA given unlimited time resources. However, that was expected since for \\textit{senLDA } the training instances are sentences, thus the model's flexibility is restricted when evaluated against a word-based \nmeasure. \n\n\\begin{figure*}[ht]\n  \\centering\n \\includegraphics[width=0.90\\textwidth]{classification_30k.pdf}\n \\caption{Classification performance on PubMed and Wikipedia text collections using $F_1$ measure.}\\label{fig:classfication}\n\\end{figure*}\n\nAn important difference between the models however, lies in the way they converge. From Figure \\ref{fig:perplexity} it is clear that \\textit{senLDA } converges faster. We highlight this by providing exact timings for the first 25 iterations of the models (column ``Timing'' of Table \\ref{table:trainData}) on a machine using an Intel Xeon CPU E5-2643 v3 at 3.40GHz. For both models we use our own Python implementations with the same speed optimisations.  Using ``WikiTrain2'' and 125 topics, for 25 iterations the \\textit{senLDA } needs 332 secs, whereas LDA needs 434 sec., an improvement of 30\\%. Furthermore, comparing the convergence, \\textit{senLDA } needs 332 secs (25 iterations) whereas LDA needs more than 2770 secs (more than 160 iterations) making \\textit{senLDA } more than 8 times faster. Similarly for the ``PubMedTrain2'' dataset which is more complex due to its larger vocabulary size, \\textit{senLDA } converges around 12 times (an order of magnitude) faster.\nNote that \\textit{senLDA}'s fast convergence is a strong advantage and can be highly appreciated in different application scenarios where unlimited time resources are not available.  \n\n\n\\noindent\\textbf{Extrinsic evaluation} Previous studies have shown that perplexity does not always agree with human evaluations of topic  models \\cite{azzopardi2003investigating} and it is recommended to evaluate topic models on real tasks. To better support  our development for \\textit{senLDA } applicability we also evaluate it using text classification as the evaluation task. \n For text classification, each document is represented by its topic distribution, which is the vectorial input to Support Vector Machines (SVMs). The classification collections are split on train\/test (75\\%\/25\\%) parts. The SVM regularization hyper-parameter $\\lambda$ is selected from $\\lambda\\in[10^{-4},\\ldots,10^4]$ using 5-fold cross-validation on the training part of the classification data.\nThe PubMed testsets are multilabel, that is each instance is associated with several classes, 1.4 in average in the sets of Table \\ref{table:trainData}. For the multilabel problem with the SVMs we used a binary relevance approach.    \nTo assess the classification performance, we report the $F_1$ evaluation measure, which is the harmonic mean of precision and recall.\n\n\n\nThe classification performance on $F_1$ measure for the different classification datasets is shown in Figure \\ref{fig:classfication}. \nFirst note that in the majority of the classification scenarios, \\textit{senLDA }  outperforms LDA. In most cases, the performance difference increases when the larger train sets (``WikiTrain2'' and ``PubMedTrain2'') are used. For instance, in the second line of figures with the PubMed classification experiments, increasing the topic models' training data benefits both LDA and \\textit{senLDA }, but \\textit{senLDA } still performs better. \nMore importantly though and in consistence with the perplexity experiments, the advantage of \\textit{senLDA } remains: the faster \\textit{senLDA } convergence benefits the classification performance. The \\textit{senLDA } curves are steeper in the first training iterations and stabilize after roughly 30 iterations when the model converges. We believe that assigning the latent topics to coherent groups of words such as sentences results in document representations of finer level. In this sense,  spans larger than single words can capture and express the document's content more efficiently for discriminative tasks like classification. \n\n\nTo investigate the correlation of topic model representations learned on different levels of text, we report the classification  performance using as document representations the concatenation of a document's topic distributions output by LDA and \\textit{senLDA }. For instance, the concatenated vectorial representation of a document when $K=125$ for each model is a vector of 250 dimensions. The resulting concatenated representations are denoted by ``senLDA+'' in Figure \\ref{fig:classfication}. As it can be seen, ``senLDA+'' performs better compared to both LDA and \\textit{senLDA }. Its performance combines the advantages of both models: during the first iterations it is as steep as the \\textit{senLDA } representations and in the later iterations benefits by the LDA convergence to outperform the simple \\textit{senLDA } representation. Hence,  the concatenation of the two distributions creates a richer representation where the two models contribute complementary information that achieves the best classification performance. Achieving the optimal performance using those representations suggests that the relaxation of the independence assumptions between the text structural units can be beneficial; this is also among the contributions of this work.\n\n\n\\section{Conclusion}\nWe proposed \\textit{senLDA}, an extension of LDA where topics are sampled per coherent text spans. This resulted in very fast convergence and good classification and perplexity performance. LDA and \\textit{senLDA } differ in that the second assumes a very strong dependence of the latent topics between the words of sentences, whereas the first assumes independence between the words of documents in general. In our future research, our goal is to investigate this dependence and further adapt the sampling process of topic models to cope with the rich text structure.\n\n\n\n\\section{Acknowledgements} \nThis work is partially supported by the CIFRE N 28\/2015.\n\n\\bibliographystyle{abbrv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\\ \n\n\\vspace{-1em}\n\n\n\nLet $G$ be a complex connected semisimple algebraic group (e.g.\\ $\\mathrm{SL}_n$ or $\\mathrm{SO}_n$), $B$ a Borel subgroup of $G$, and $U$ the unipotent radical of $B$. For example, in the case $G=\\mathrm{SL}_n$, we can take $U$ to be the collection of all upper triangular matrices with all diagonal entries equal to $1$.\n\nThe homogeneous space $G\/U$ is called the ``basic affine space\". While $G\/B$ is projective, the basic affine space $G\/U$ is a quasi-affine variety. It turns out that many interesting problems in representation theory are related to the basic affine space. In particular, the algebra $\\mathcal{D}(G\/U)$ of algebraic differential operators on $G\/U$ is well-studied, for example in  \\cite{Gelfand}\\cite{GK}\\cite{LS}\\cite{BBP}.\n\nIn this paper, we study the cotangent bundle $T^*(G\/U)$, of which the coordinate ring $\\mathbb{C}[T^*(G\/U)]$ is the quasi-classical counterpart of $\\mathcal{D}(G\/U)$. From a result by Ginzburg and Riche \\cite{GR}, the coordinate ring $\\mathbb{C}[T^*(G\/U)]$ is finitely generated, and the affine closure of the basic affine space is defined as $$\\overline{T^*(G\/U)}\\coloneqq \\Spec \\mathbb{C}[T^*(G\/U)].$$\n\nFollowing Beauville \\cite{B}, a normal variety $X$ has\nsymplectic singularities if its regular locus carries a symplectic $2$-form whose pull-back\nalong any resolution $\\widetilde{X}\\rightarrow X$ extends to a holomorphic $2$-form on $\\widetilde{X}$.\nThere are many interesting and important examples of symplectic singularities. These include finite quotient singularities \\cite{B}, normal closures of nilpotent coadjoint orbits \\cite{Pan}, Nakajima quiver varieties \\cite{BS}, and more recently Coulomb branches associated to simple quivers \\cite{Wee}.\n\nSymplectic singularities play an important role in representation theory, and they have a lot of interesting properties. For instance, in the conical case (affine symplectic singularities with a good $\\mathbb{C}^*$-action), there is a universal family of deformations \\cite{Nam}\\cite{Nam2}\\cite{KV}\nand filtered quantizations \\cite{BK}\\cite{BLPW1}\\cite{Los}. Thanks to these two properties, one can generalize results in Lie theory to conical symplectic singularities. For example, there is a category $\\mathscr{O}$ for a conical symplectic resolution \\cite{BLPW2}. And in this general setting, the classical BGG category $\\mathscr{O}$ \\cite{BGG} can be viewed as the category $\\mathscr{O}$ associated with the Springer resolution \\cite{Springer}. Hence A. Okounkov once said, ``Symplectic singularities are the Lie algebras of the 21st century\".\n\n\nIn \\cite{GK}, Ginzburg and Kazhdan conjectured that the affine closure of the basic affine space also has symplectic singularities.\nIn the case $G=\\mathrm{SL}_n$, we have the following result proved in section \\ref{dq}.\n\\begin{thm}\\label{main}\n    The affine closure $\\overline{T^*(\\mathrm{SL}_n\/U)}$ has symplectic singularities.\n\\end{thm}\n\nIn the Lie algebra $\\mathfrak{so}_8$, there is a unique nilpotent adjoint orbit $\\mathcal{O}_{\\textrm{min}}\\subset\\mathfrak{so}_8$ of minimal (positive) dimension $10$. The closure of the minimal orbit is \\mbox{$\\overline{\\mathcal{O}}_\\textrm{min}=\\mathcal{O}_{\\textrm{min}}\\cup\\{0\\}$}. In section \\ref{kostant}, we discuss Kostant's theorem on the highest weight variety and its application to the minimal nilpotent orbits of type $D$.\nIn section \\ref{sl3}, we show that there is an isomorphism of affine varieties\n    \\begin{equation}\\label{1.3}\n        \\overline{T^*(\\mathrm{SL}_3\/U)}\\rightarrow\\overline{\\mathcal{O}}_\\textrm{min}.\n    \\end{equation}\nIn section \\ref{symplectic}, we show that this map (\\ref{1.3}), when restricted on smooth points, is a symplectic isomorphism.\n\nIn \\cite{GK}, Ginzburg and Kazhdan constructed an action on $\\overline{T^*(G\/U)}$ by the Weyl group $W$ of $G$, called the ``Gelfand--Graev action\". So in the case $G=\\mathrm{SL}_3$ we have an $S_3$-action on $\\overline{T^*(\\SL_3\/U)}$. On the other hand, the Lie algebra $\\mathfrak{so}_8$ has an $S_3$-symmetry called the triality action \\cite{Cartan}\\cite{MW}, and the restriction of the triality action gives an $S_3$-action on $\\overline{\\mathcal{O}}_\\textrm{min}$. In section \\ref{triality}, we give a new interpretation of this triality action. In section \\ref{ggaction}, we show that the isomorphism (\\ref{1.3}) is $S_3$-equivariant. So we have proved\n\\begin{thm}\nThere is an $S_3$-equivariant isomorphism\n    \\begin{equation}\\label{2.7}\n        \\overline{T^*(\\mathrm{SL}_3\/U)}\\rightarrow\\overline{\\mathcal{O}}_\\textrm{min}\n    \\end{equation} of affine Poisson varieties.\n\\end{thm}\n\n\n\n\n\\vspace{2em}\n\\textbf{Acknowledgments.}\\ The author is very grateful to his advisor, Victor Ginzburg, for all the encouragement, suggestions, and discussions. He would like to thank David Kazhdan, for the helpful correspondence containing statements related to the affine closure $\\overline{T^*(\\SL_3\/U)}$, the minimal nilpotent orbit $\\mathcal{O}_{\\textrm{min}}$ in $\\mathfrak{so}_8$ and the triality action. He would also like to thank Sam Evens, Yu Li, and Minh-Tam Trinh for their useful suggestions and comments.\n\n\\vspace{3em}\n\\section{The affine closure $\\overline{T^*(SL_n\/U)}$ has symplectic singularities.}\\label{dq}\n\nA quiver $Q$ is a finite directed graph consisting of \na vertex set $I$ and an edge set $E$.\nWrite $Q^\\textrm{op}=(I,\\overline{E})$\nfor the opposite quiver obtained from $Q$ by reversing the orientation of edges.\nThe\ndouble quiver of $Q$ is defined by $\\overline Q=(I,E\\sqcup \\overline{E})$. \nA representation of $Q$ assigns a vector space $V(i)$ to each vertex $i\\in I$ and a linear map $V(e):V(i)\\rightarrow V(j)$ to each arrow $e\\in E$ whose source and target are $i$ and $j$ respectively.\n\nLet $Q$ be the following Dynkin quiver of type $\\mathrm{A}_{n}$.\n$$\n    \\begin{tikzcd}\n\\bullet \\arrow[r]\n\t& \\bullet\n\t\\arrow[r] \n\t& \\cdots\n\t\\arrow[r] \n\t& \\bullet\n\t\\arrow[r] \n\t& \\bullet\n\\end{tikzcd}\n$$\nLet $V=\\bigoplus_{k=1}^{n-1}\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1})$, so each element of $V$ defines a representation of the quiver $Q$.\nThe cotangent space $T^*V$ is identified with $$\\bigoplus_{k=1}^{n-1}\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1})\\oplus\\Hom(\\mathbb{C}^{k+1},\\mathbb{C}^{k})$$ via the  trace pairing\n$\n    \\sum_{k=1}^{n-1}\\mathrm{Tr} (\\beta_k\\circ\\alpha_k),\n$ for $\\alpha_k\\in\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1})$ and $ \\beta_k\\in\\Hom(\\mathbb{C}^{k+1},\\mathbb{C}^{k})$.\nSo each element of $T^*V$ gives a representation of the double quiver $\\overline{Q}$  \n\\begin{equation}\\label{double quiver}\n    \\begin{tikzcd}\n {\\mathbb{C}} \\arrow[r,shift left=.7ex,\"\\alpha_1\"]\n\t& {\\mathbb{C}^2} \\arrow[l,shift left=.7ex,\"\\beta_1\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_2\"] \n\t& {\\cdots} \\arrow[l,shift left=.7ex,\"\\beta_2\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_{n-2}\"] \n\t& {\\mathbb{C}^{n-1}} \\arrow[l,shift left=.7ex,\"\\beta_{n-2}\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_{n-1}\"] \n\t& \\mathbb{C}^n \\arrow[l,shift left=.7ex,\"\\beta_{n-1}\"] \n\\end{tikzcd}.\n\\end{equation}\nThis construction was applied in \\cite{KP} to show the normality of the nilpotent cone of $\\mathfrak{gl}_n$. Similar constructions can also be found in \\cite{KS}\\cite{CB1}\\cite{CB2}.\n\n\nThere is an action of $H\\coloneqq\\prod_{k=2}^{n-1}\\mathrm{SL}_i(\\mathbb{C})$ on $V$ by $$(g_2,g_3,\\cdots,g_{n-1}).(\\alpha_1,\\alpha_2,\\cdots,\\alpha_{n-1})=(g_{2}\\circ\\alpha_1,g_{3}\\circ\\alpha_2\\circ g_{2}^{-1},\\cdots,g_{n}\\circ\\alpha_k\\circ g_{n-1}^{-1}).$$ This $H$-action on $V$\ninduces a Hamiltonian $H$-action on $T^*V$, of which the moment map is\n$$\n    \\mu_H(\\alpha,\\beta)(X)=\\sum_{k=2}^{n-1}\\mathrm{Tr}\\left((\\alpha_{k-1}\\beta_{k-1}-\\beta_k\\alpha_k)X_k\\right),\n$$\nwhere $(\\alpha,\\beta)=\\bigoplus_{k=1}^{n-1}(\\alpha_k,\\beta_k)\\in T^*V$, and $X=(X_2,X_3,\\cdots,X_{n-1})\\in\\mathfrak{sl}_2\\times\\mathfrak{sl}_3\\times\\cdots\\times\\mathfrak{sl}_{n-1}$.\n\n\n\\begin{defn}\n    We denote the zero fiber of the moment map by\n    $$\n        N\\coloneqq\\mu_H^{-1}(0)\\subset T^*V\n    $$\n    So $N$ is the subvariety consists of all the $(\\alpha,\\beta)$ in  $T^*V$ such that for all $k\\in\\{1,2,3,\\cdots,n-1\\}$\n\\begin{equation}\\label{N}\n    \t\\beta_k\\alpha_k-\\alpha_{k-1}\\beta_{k-1}=\\lambda_k\\,\\textrm{Id}_{\\mathbb{C}^k},\n\\end{equation}\nfor some $\\lambda_k\\in\\mathbb{C}.$\n\\end{defn}\n\n\n\\begin{notn}\nThroughout this paper, we use the expression $(\\alpha,\n\\beta)$ to denote an element $$\\bigoplus_{k=1}^{n-1}(\\alpha_k,\\beta_k)\\in\\bigoplus_{k=1}^{n-1}\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1})\\oplus\\Hom(\\mathbb{C}^{k+1},\\mathbb{C}^{k})=T^*V.$$\n\\end{notn}\n\nWe recall the following theorem.\n\\begin{thm}[\\textrm{Theorem 7.18} in \\cite{DKS}]\nThe affine closure $\\overline{T^*(\\mathrm{SL}_n\/U)}$ is isomorphic to the categorical quotient $N\\sslash H$ as affine varieties.\n\\end{thm}\n\n\n\\begin{lem}\n\tLet $0=m_0\\leq m_1\\leq m_2\\leq\\cdots\\leq m_{n-1}\\leq m_n=n$.\n\tSuppose $m_k\\leq k$ for all $k$. Then\n\t$$\n\t\\sum_{k=1}^{n-1}m_k(m_{k+1}-m_k)\\leq\\sum_{k=1}^{n-1}k,\n\t$$\n\tand the equality holds if and only if $m_k=k$ for all $k$.\n\\end{lem}\n\\begin{proof}\n\tSince $m_0=0$, we show that \n\t\\begin{align*}\n\t\t\\sum_{k=0}^{n-1}m_k(m_{k+1}-m_k)\n\t\t=\\sum_{k=0}^{n-1}\\sum_{j=m_k+1}^{m_{k+1}}m_k\n\t\t\\leq\\sum_{k=0}^{n-1}\\sum_{j=m_k+1}^{m_{k+1}}(j-1)\n\t\t=\\sum_{k=1}^{n-1}k.\n\t\\end{align*}\n\tIn fact, the LHS is the dimension of some (partial) flag variety of $\\mathrm{GL}_n$, so it obtains maximum if and only if in the case of complete flag variety, which has dimension RHS.\n\\end{proof}\n\n\\pagebreak\n\\begin{lem}\\label{codim}\nThe singular locus of $\\overline{T^*(\\SL_n\/U)}$ has codimension at least $4$.\n\\end{lem}\n\\begin{proof}\n\tFor each $\\underline{m}=(m_1,m_2,m_3,\\cdots,m_{n-1})$ satisfying the condition in the previous lemma, we define $H(\\underline{m})\\coloneqq\\prod_{k=1}^{n-1}\\mathrm{SL}_{m_k}(\\mathbb{C})$,\n\t$\\tilde{H}(\\underline{m})\\coloneqq\\prod_{k=1}^{n-1}\\mathrm{GL}_{m_k}(\\mathbb{C})$, $V(\\underline{m})=\\bigoplus_{k=1}^{n-1}\\Hom(\\mathbb{C}^{m_k},\\mathbb{C}^{m_{k+1}})$.\n\tThen by Theorem 6.13 in \\cite{DKS} (page 30), the affine variety $X$ can be written as disjoint union \n\t   $$\n\t           X=\\bigsqcup_{S,\\delta}Q_{(S,\\delta)}\n\t   $$\n\twhere each $Q_{(S,\\delta)}$ is a smooth hyperk\\\"ahler manifold that is a locally closed subset of $X$.\n\tLet $x$ be a singular point of $\\overline{T^*(\\SL_n\/U)}$. Then $x$ lies in some strata $Q_{(S,\\delta)}$ with the same dimension as the Hamiltonian reduction of $T^*V(\\underline{m})$ by a subgroup $H_S\\leq\\tilde{H}(\\underline{m})$, where $\\underline{m}\\neq(1,2,3,\\cdots,n-1)$ and $H_S$ is defined after (6.1) in \\cite{DKS} (page 26), from which we also know $H(\\underline{m})$ is a proper subgroup of $H_S$.\n\tSo \n\t\\begin{align*}\n\t\t\\dim Q_{(S,\\delta)}& =\\dim(T^*V(\\underline{m})\/\\!\/\\!\/ H_S)\\\\\n\t\t& = 2(\\dim V(\\underline{m})-\\dim H_S)\\\\\n\t\t& \\leq 2(\\dim V(\\underline{m})-\\dim H(\\underline{m})-1)\\\\\n\t\t& =2\\left(\\sum_{k=1}^{n-1}m_km_{k+1}-(m^2_k-1)\\right)-2\\\\\n\t\t& =2\\left(n-2+\\sum_{k=1}^{n-1}m_k(m_{k+1}-m_k)\\right)\n\t\\end{align*}\n\tSince $\\underline{m}\\neq(1,2,3,\\cdots,n-1)$, by the previous lemma, we know $\\sum_{k=1}^{n-1}m_k(m_{k+1}-m_k)\\leq\\left(\\sum_{k=1}^{n-1}k\\right)-1$.\n\tThus\n\t$$\n\t\\dim Q_{(S,\\delta)}\\leq2\\left((n-3)+\\sum_{k=1}^{n-1}k\\right)\n\t=2\\left(-1+\\sum_{k=1}^{n}k\\right)-4=\\dim(\\overline{T^*(\\SL_n\/U)})-4.\n\t$$\n\\end{proof}\n\n\\begin{defn}\\label{Nsurj}\n    We define the surjective part (resp.\\ principal part) of $N$ as follows.\n    \\begin{align*}\n        N_\\textrm{surj}& =\\{(\\alpha,\\beta)\\in N\\,|\\, \\textrm{all the $\\beta_k$ are surjective}\\},\\\\\n        N_\\textrm{pr}& =\\{(\\alpha,\\beta)\\in N\\,|\\, \\textrm{the $H$-orbit of $(\\alpha,\\beta)$ is closed and the stabilizer }H_{(\\alpha,\\beta)}=1\\}.\n    \\end{align*}\n    Then we have\n    $\n        N_\\textrm{surj}\\subset N_\\textrm{pr}.\n    $\n\\end{defn}\n\n\\begin{prop}\\label{symp}\n\tThe smooth locus of $\\overline{T^*(\\SL_n\/U)}$ admits a holomorphic symplectic form.\n\\end{prop}\n\\begin{proof}\n\tLet $X=\\overline{T^*(\\SL_n\/U)}$. Let $\\pi:N\\rightarrow N\\sslash H=X$ be the categorical quotient map. Let $X_\\textrm{sm}$ be the smooth locus of $X$.\n\tBy proposition 5.7 in \\cite{Dre}, we have $\\pi(N_\\textrm{pr})\\subset X_\\textrm{sm} \\subset X$. From the proof of \\textrm{Lemma 7.17} in \\cite{DKS} (page 38), we know $N\\setminus N_\\textrm{surj}$ has codimension at least 2, so $$\\mathrm{codim}(N\\setminus N_\\textrm{pr})\\geq2.$$ By the upper semicontinuity of the dimensions of the fibers of $\\pi$ we have $$\\mathrm{codim}(X_\\textrm{sm}\\setminus \\pi(N_\\textrm{pr}))\\geq2.$$  From the Marsden-Weinstein theorem, there is a holomorphic symplectic form $\\omega_0$ on $\\pi(N_\\textrm{pr})$. Then by Hartog's lemma, we can extend $\\omega_0$ to a closed holomorphic 2-form $\\omega$ on $X_\\textrm{sm}$. By taking the top wedge of $\\omega$, we have its points of degeneracy has codimension $1$, so $\\omega$ has to be non-degenerated on $X_\\textrm{sm}$, hence symplectic. \n\\end{proof}\n\n\\pagebreak\n\\begin{thm}\\label{main2}\n\tThe affine closure $\\overline{T^*(\\mathrm{SL}_n\/U)}$ has symplectic singularities.\n\\end{thm}\n\\begin{proof}\n   Applying Theorem 18.4 in \\cite{gros}, we know that $X\\coloneqq\\overline{T^*(\\SL_n\/U)}$ is a normal variety. Let $\\omega$ be the holomorphic symplectic form on the smooth locus of $X$ as in Proposition \\ref{symp}. Let $$\\nu:\\widetilde{X}\\rightarrow X$$ be any resolution of singularity. By Lemma \\ref{codim} and the main theorem in \\cite{Flenner}, the pull back map $$\\nu^*:\\Omega^2_\\textrm{hol}(X)\\rightarrow\\Omega^2_\\textrm{hol}(\\widetilde{X})$$ is a bijection. So $\\nu^*\\omega$ extends holomorphically everywhere on $\\widetilde{X}$.\n\\end{proof}\n\\begin{rem}\n    Let $P$ be any parabolic subgroup of $\\mathrm{SL}_n$.\n    Let $[P,P]$ denote the commutator subgroup of $P$.\n    Then I believe one can use similar method to prove that the affinization $\\overline{T^*(\\mathrm{SL}_n\/[P,P])}$ also has symplectic singularities, which generalizes Theorem \\ref{main2}.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{3em}\n\\section{The case of $n=3$ and minimal nilpotent orbits.}\n\n\\subsection{The minimal nilpotent orbit of $\\mathfrak{so}_{2m}$}\\label{kostant}\nLet $\\mathcal{O}_{\\textrm{min}}$ be the minimal nilpotent orbit of a semisimple Lie algebra $\\mathfrak{g}$.\nLet $\\mathfrak{g} = \\mathfrak{n} \\oplus \\mathfrak{h} \\oplus \\mathfrak{n}^{-}$ be a triangular decomposition of $\\mathfrak{g},$ such that $\\mathfrak{n}=\\mathrm{Lie}(U)$.\nWrite $\\Delta = \\Delta^{+} \\sqcup \\Delta^{-}$\nfor the root system of $(\\mathfrak{g}, \\mathfrak{h})$ so that the roots in $\\Delta^+$ (resp.\\ in $\\Delta^-$) correspond to $\\mathfrak{n}$ (resp.\\ $\\mathfrak{n}^-$).\nLet $\\psi\\in\\Delta^{+}$ be the highest root of $\\mathfrak{g}$. Let $v_\\psi$ be a nonzero root vector corresponding to $\\psi$.\n\\begin{prop}[\\textrm{Theorem 4.3.3} in \\cite{CM}] The minimal nilpotent orbit $\\mathcal{O}_{\\textrm{min}}$ is the adjoint orbit of $v_\\psi$.\\end{prop}\n\nLet $\\mathcal{I}\\subset\\mathbb{C}[\\mathfrak{g}]$ be the ideal of polynomials which vanish identically on $\\mathcal{O}_{\\textrm{min}}$.\nFor each dominant weight $\\lambda\\in\\mathfrak{h}^*$, define $V_\\lambda$ as an irreducible $\\mathfrak{g}$-representation with highest weight $\\lambda$. One has the following result\n\\begin{thm}[Kostant's Theorem (\\textrm{Theorem III.2.1} in \\cite{Garfinkle})]\nDecompose the $\\mathfrak{g}$-representation\n$$\\mathrm{Sym}^2(\\mathfrak{g})^*=V_{2\\psi}^*\\oplus C.$$\nThe ideal $\\mathcal{I}$ is generated by $C$ in $\\mathrm{Sym}(\\mathfrak{g})^*=\\mathbb{C}[\\mathfrak{g}]$.\n\\end{thm}\n\n\n\nLet $m\\geq4$. We consider the case $\\mathfrak{g}=\\mathfrak{so}_{2m}$.\nLet $e_1,e_2,\\cdots,e_{2m}$ be the natural basis of $\\mathbb{C}^{2m}$.\nDefine an Euclidean inner product $(\\ ,\\ )$ on $\\mathbb{C}^{2m}$ by\n    \\begin{equation}\\label{inner}\n        (e_i,e_j)=\\delta_{i,2m+1-j}.\n    \\end{equation}\nFor $v_1\\wedge v_2\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$, we define\n\\begin{align}\\label{varphi}\n\t\\qquad \\varphi_{v_1\\wedge v_2}:\\ \\mathbb{C}^{2m}\\ &\\longrightarrow \\ \\mathbb{C}^{2m}\\\\\n\tu\\ \\ &\\longmapsto\\ \\, \\left(v_1,u\\right)v_2-\n\t\\left(v_2,u\\right)v_1.\\nonumber\n\\end{align}\nExtend by linearity, we get the following isomorphism of $\\mathfrak{so}_{2m}$-representations:\n\\begin{equation}\\label{Lso}\n    \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}=\\mathfrak{so}_{2m}. \n\\end{equation}\n\n\n\\begin{defns}\nWe say a subspace $W\\subset \\mathbb{C}^{2m}$ is isotropic if $(W,W)=0$. An element $f\\in\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})$ is isotropic if its image is isotropic.\nAn element $\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ is isotropic if for all $v_1^*, v_2^*\\in (\\mathbb{C}^{2m})^*$,\n\t$$\n\t    (\\iota_{v_1^*}\\alpha, \\iota_{v_2^*}\\alpha)=0.\n\t$$\nWe say $\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ is decomposable if there exist some $v_1,v_2\\in\\mathbb{C}^{2m}$ such that $\\alpha=v_1\\wedge v_2$. \n\\end{defns}\n\\begin{rem}\n    By a theorem of Pl\\\"ucker we know that $\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ is decomposable if and only if\n$$\\alpha\\wedge\\alpha=0\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^4\\mathbb{C}^{2m}.$$\n\\end{rem}\n\\begin{lem}\\label{iso}\n\tLet $v_1\\wedge v_2\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ \n\tbe a nonzero decomposable element. \n\tThen \n\t$$\n\t\\mathrm{span} (v_1,v_2) \\textrm{ is isotropic }\n\t\\iff v_1\\wedge v_2 \\textrm{ is isotropic.}\n\t$$\n\\end{lem}\n\\begin{proof}\n\t($\\Longrightarrow$)\\ Clear.\\\\\n\t($\\Longleftarrow$)\\ It suffices to show that \n\t$$\n\t(v_1,v_1)=(v_1,v_2)=(v_2,v_2)=0.\n\t$$\n\tSince $v_1\\wedge v_2\\neq0$, we have $v_1$ and $v_2$ are linearly independent. So there exist $v_1^*,v_2^*\\in(\\mathbb{C}^{2m})^*$ such that\n\t$$\n\tv_1^*(v_1)=v_2^*(v_2)=1,\\textrm{ and } \n\tv_1^*(v_2)=v_2^*(v_1)=0.\n\t$$\n\tThen\n\t\\vspace{-1em}\n\t$$\n\tv_1=-\\iota_{v_2^*}(v_1\\wedge v_2)\\textrm{, }\n\tv_2=\\iota_{v_1^*}(v_1\\wedge v_2),\n\t$$\n\tand our conclusion follows from the definition.\n\\end{proof}\n\nLet $\\overline{\\mathcal{O}}_\\textrm{min}$ be the  closure of the minimal nilpotent orbit $\\mathcal{O}_{\\textrm{min}}\\subset\\mathfrak{so}_{2m}$.\n\\begin{prop}\\label{ominiso} Under the identification (\\ref{Lso}),\n\t\\begin{equation}\\label{Omin}\n\t    \\overline{\\mathcal{O}}_\\textrm{min}=\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\textrm{ is decomposable and isotropic}\\}\n\t\\end{equation}\n\\end{prop}\n\\begin{proof}\n\tSince $\\overline{\\mathcal{O}}_\\textrm{min}=\\mathcal{O}_{\\textrm{min}}\\cup\\{0\\}$, it suffices to show that\n\t$$\n\t\\mathcal{O}_{\\textrm{min}}=\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\textrm{ is a nonzero isotropic decomposable element}\\}.\n\t$$\n\tSince both decomposable and isotropic properties are invariant under the $\\mathrm{SO}_{2m}$ action, and $e_1\\wedge e_2$ is isotropic decomposable, so all elements in the minimal orbit are isotropic decomposable. But $\\mathrm{SO}_{2m}$ acts transitively on the isotropic planes in $\\mathbb{C}^{2m}$, and there is a scaling  $\\mathbb{C}^*$-action on $\\mathcal{O}_{\\textrm{min}}$, so $\\mathrm{SO}_{2m}$ acts transitively on the nonzero isotropic decomposable elements.\n\\end{proof}\n\n\n\\begin{defns}\n    Define the following two maps:\n\t\\begin{align*}\n\t\t\\Phi:(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^4\\mathbb{C}^{2m})^* & \\longrightarrow\\ \\mathbb{C}[\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}]\\\\\n\t\t\\varphi\\quad\\ \\ & \\longmapsto\\  [\\alpha\\mapsto\\varphi(\\alpha\\wedge\\alpha)],\\\\[0.5em]\n        \\Psi:\\mathrm{Sym}^2(\\mathbb{C}^{2m})^* & \\longrightarrow\\mathbb{C}[\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}]\\\\\n\t    v_1^*\\cdot v_2^* & \\longmapsto[\\alpha\\mapsto(\\iota_{v_1^*}\\alpha,\\iota_{v_2^*}\\alpha)].\n\t\\end{align*}\n\tLet $V_\\textrm{dec}$ (resp.\\ $V_\\textrm{iso}$) denote the image of $\\Phi$ (resp.\\ $\\Psi$).\n\tNote that $V_\\textrm{dec}$ (resp.\\ $V_\\textrm{iso}$) generates the ideal $\\mathcal{I}_\\textrm{dec}$ (resp.\\ $\\mathcal{I}_\\textrm{iso}$) of polynomial functions on $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$ vanishing on decomposable (resp.\\ isotropic) elements of $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}$.\n\\end{defns}\n\n\n\\begin{prop}\nBoth $\\Phi$ and $\\Psi$ are injective, and\n\t$$\n\t    \\mathrm{Sym}^2(\\mathfrak{so}_{2m})^*=V^*_{2\\psi}\\oplus V_\\textrm{dec}\\oplus V_\\textrm{iso}\n\t$$\nas $\\mathfrak{so}_{2m}$-representions.\n\\end{prop}\n\\begin{proof}\nFirst we show the following elements are linearly independent in $\\mathbb{C}[\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}]$:\n$$\n    \\{\\Phi(e_i^*\\wedge e_j^*\\wedge e_k^*\\wedge e_l^*)\\suchthat1\\leq i<j<k<l\\leq 2m\\}\\cup\\{\\Psi(e_r^*\\cdot e_s^*)\\suchthat1\\leq r\\leq s\\leq 2m\\}.\n$$\nSuppose there exist some $A_{ijkl},B_{rs}\\in\\mathbb{C}$ such that\n\\begin{equation}\\label{sum}\n    \\sum_{1\\leq i<j<k<l\\leq 2m}\\!\\!A_{ijkl}\\ \\Phi(e_i^*\\wedge e_j^*\\wedge e_k^*\\wedge e_l^*)\\ +\\sum_{1\\leq r\\leq s\\leq 2m}\\!\\!B_{rs}\\,\\Psi(e_r^*\\cdot e_s^*)=0.\n\\end{equation}\nLet $1\\leq t\\leq 2m$. Evaluating the LHS of (\\ref{sum}) at $e_{t}\\wedge e_{2m+1-t}$, we have $B_{t,2m+1-t}=0$. So\n\\begin{equation}\\label{sum1}\n    \\sum_{1\\leq i<j<k<l\\leq 2m}\\!\\!A_{ijkl}\\ \\Phi(e_i^*\\wedge e_j^*\\wedge e_k^*\\wedge e_l^*)\\ +\\sum_{s\\neq 2m+1-r}\\!\\!B_{rs}\\,\\Psi(e_r^*\\cdot e_s^*)=0.\n\\end{equation}\nLet $1\\leq r\\leq s\\leq 2m$ with $s\\neq 2m+1-r$.\nEvaluating the LHS of (\\ref{sum1}) at $e_{r}\\wedge (e_s+e_{2m+1-r})$, we have $B_{rs}=0$. \nSo\n\\begin{equation}\\label{sum2}\n    \\sum_{1\\leq i<j<k<l\\leq 2m}\\!\\!A_{ijkl}\\ \\Phi(e_i^*\\wedge e_j^*\\wedge e_k^*\\wedge e_l^*)=0.\n\\end{equation}\nLet $1\\leq i<j<k<l\\leq 2m$, evaluating the LHS of (\\ref{sum2}) at $(e_i\\wedge e_j+e_k\\wedge e_l)$, we have $A_{ijkl}=0$.\nSo both $\\Phi$ and $\\Psi$ are injective, and \n$$\n    V_\\textrm{dec}\\cap V_\\textrm{iso}=\\{0\\}.\n$$\nSince the highest weight vector ${(e_{1}^*}\\wedge {e_{2}^*})^2\\in \\mathrm{Sym}^{2}(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^{2}(\\mathbb{C}^{2m})^{*})$ does not belong to the $\\mathfrak{so}_8$-representation $V_\\textrm{dec}\\oplus V_\\textrm{iso}$. By the irrducibility of $V_{2\\psi}^*$ we have\n$$\n    V_{2\\psi}^*\\cap(V_\\textrm{dec}\\oplus V_\\textrm{iso})=\\{0\\}.\n$$\nBy the injectivity of $\\Phi$ and $\\Psi$ we have\n$$\n        \\dim V_\\textrm{dec} =\\binom{2m}{4},\\  \n        \\dim V_\\textrm{iso} =\\binom{2m+1}{2}.\n$$\nBy Weyl's dimension formula\n    $$\n        \\dim V_{2\\psi}^*=\\frac{m}{3}(4m^3-7m-3).\n    $$\nSo, we find\n    \\begin{align*}\n    \\dim V_{2\\psi}^*+\\dim V_\\textrm{dec}+\\dim V_\\textrm{iso}&=\\frac{m}{3}(4m^3-7m-3)+\\binom{2m}{4}+\\binom{2m+1}{2}\\\\\n        &=2 m^4 - 2 m^3 + \\frac{3 m^2}{2} - \\frac{m}{2}\n        =\\binom{\\binom{2m}{2}+1}{2}\\\\\n        &=\\dim\\mathrm{Sym}^2(\\mathfrak{so}(2m)).\n    \\end{align*}\nTherefore, we obtain\n\t$$\n\t    \\mathrm{Sym}^2(\\mathfrak{so}_{2m})=V_{2\\psi}^*\\oplus V_\\textrm{dec}\\oplus V_\\textrm{iso}.\n\t$$\n\\end{proof}\n\nThe following Hamiltonian reduction construction of the closure of the minimal nilpotent orbit in $\\mathfrak{so}_{2m}$ is known. We repeat this construction to set up the notations.\nFix the natural basis $e_1,e_2\\in\\mathbb{C}^2$, and define a symplectic bilinear form $\\omega_1$ on $\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})$ by\n\\begin{equation}\\label{o1}\n    \\omega_1(f_1,f_2)=(f_1(e_1),f_2(e_2))-(f_1(e_2),f_2(e_1)).\n\\end{equation}\n\nThe action of $\\mathrm{SL}_2$ on  $\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})$ by precomposing is Hamiltonian with moment map\n\\begin{align*}\n    & \\mu_{\\mathrm{SL}_2}:\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m}) \\longrightarrow \\mathfrak{sl}_2^*\\cong\\mathfrak{sl}_2,\\\\\n    & \\mu_{\\mathrm{SL}_2}(f) =\n    \\renewcommand*{\\arraystretch}{1.5}\n    \\begin{pmatrix*}\n    (f(e_1),f(e_2)) && \\hspace{-1em}(f(e_1),f(e_1))\\\\\n    (f(e_2),f(e_2)) && \\hspace{-1em}-(f(e_1),f(e_2))\n    \\end{pmatrix*}.\n\\end{align*}\n\n\nSo the zero fiber of the $\\mathrm{SL}_2$-moment map on \\mbox{$\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})$ is}\n    \\begin{equation}\n        N_1\\coloneqq\\mu_{\\mathrm{SL}_2}^{-1}(0)=\\{f\\in\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})\\,|\\, f \\textrm{ is isotropic with respect to }(\\ ,\\ )\\}.\n    \\end{equation}\n\n\\pagebreak\nFor the reader's convenience we include the proof of the following result:\n\\begin{prop}\\label{ominhamil} The Hamiltonian reduction $N_1\\sslash\\mathrm{SL}_2$ is isomorphic to the closure $\\overline{\\mathcal{O}}_\\textrm{min}$ of the minimal orbit in $\\mathfrak{so}_{2m}$ as affine varieties.\n\\end{prop}\n\\begin{proof}\nRecall from Proposition \\ref{ominiso}, $$\\overline{\\mathcal{O}}_\\textrm{min}=\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\textrm{ is decomposable and isotropic}\\}.$$\nDenote\n\t\\vspace{-1em}\n\t\\begin{align*}\n\t\t& \\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}\\coloneqq\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\mathrm{\\ is\\ decomposable}\\},\\\\\n\t\t& \\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_{\\textrm{dec},\\,\\textrm{iso}}\\coloneqq\\{\\alpha\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\,|\\, \\alpha\\mathrm{\\ is\\ decomposable\\ and\\ isotropic}\\}.\n\t\\end{align*}\n\t\n\t\n\tBy the fundamental theorem of invariant theory for $\\mathrm{SL}_2$ (\\textrm{Theorem 2} in page 387 of \\cite{Procesi}), we have $$\\mathbb{C}[(\\mathbb{C}^2)^*\\otimes\\mathbb{C}^{2m}]^{\\mathrm{SL}_2}=\\mathbb{C}\\left[\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}\\right].$$ \n\tSince $\\mathrm{SL}_2$ is reductive and $N_1\\subset\\Hom(\\mathbb{C}^2,\\mathbb{C}^{2m})=(\\mathbb{C}^2)^*\\otimes\\mathbb{C}^{2m}$ is an $\\mathrm{SL}_2$-invariant sub-variety, the restriction from $(\\mathbb{C}^2)^*\\otimes\\mathbb{C}^{2m}$ to $N_1$ gives a surjective map\n\t\\begin{align}\\label{surj}\n\t\t\\mathbb{C}\\left[\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}\\right]\\quad& \\longrightarrow\\qquad\\qquad\\mathbb{C}[N_1]^{\\mathrm{SL}_2}\\\\\n\t\tf\\qquad \\qquad& \\longmapsto \\quad\\left(e_1^*\\otimes v_1+e_2^*\\otimes v_2\\mapsto f(v_1\\wedge v_2)\\right)\\nonumber\n\t\\end{align}\nBy Lemma \\ref{iso} we have the kernel of (\\ref{surj}) is precisely the ideal $\\mathcal{I}_\\textrm{iso}$ of functions on $\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}$ vanishing identically on the isotropic elements. Hence\n$$\n    \\mathbb{C}[N_1]^{\\mathrm{SL}_2}=\\mathbb{C}\\left[\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec}\\right]\/\\mathcal{I}_\\textrm{iso}=\\mathbb{C}\\left[\\left(\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^{2m}\\right)_\\textrm{dec,\\,iso}\\right].\n$$\n\\end{proof}\t\n\n\n\\vspace{3em}\n\\subsection{The Lie algebra $\\mathfrak{so}_8$ as an $\\mathfrak{sl}_3$-representation}\nFix the following set of simple roots for the root system of type $D_4$.\n\\begin{align*}\n\t\\alpha_1& =(1,-1,0,0)\\\\\n\t\\alpha_2& =(0,1,-1,0)\\\\\n\t\\alpha_3& =(0,0,1,-1)\\\\\n\t\\alpha_4& =(0,0,1,1)\n\\end{align*}\nSo that the Dynkin Diagram $D_4$ is labeled as follows.\n    $$\n     \\dynkin[text style\/.style={scale=1},label,label macro\/.code={\\alpha_{\\drlap{#1}}},scale=1.2,edge\nlength=0.9cm]D4\n    $$\nChoose a Chevalley basis of $\\mathfrak{so}_8$ as in the \\hyperref[appendix]{Appendix}  $$\\{X_\\alpha\\}_{\\alpha\\in\\Delta^+},\\{Y_{-\\alpha}\\}_{\\alpha\\in\\Delta^+},\\{H_{\\alpha_i}\\}_{i=1,2,3,4}.$$\nChoose simple roots $\\alpha=(1,-1,0),\\ \\beta=(0,1,-1)$ of $\\mathfrak{sl}_3$. There exists a unique embedding of Lie algebras\n$$\n    \\varphi_1:\\mathfrak{sl}_3\\longhookrightarrow\\mathfrak{so}_8.\n$$\nwhich restricts to an embedding $\\varphi_1|_{\\mathfrak{h}_{\\mathfrak{sl}_3}}:\\mathfrak{h}_{\\mathfrak{sl}_3}\\hookrightarrow\\mathfrak{h}_{\\mathfrak{so}_8}$ such that\n$$\n\\varphi_1^*(\\alpha_2)=\\alpha,\\textrm{ and } \\varphi_1^*(\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4)=\\beta.\n$$\nSo\n$\\mathfrak{so}_8$ becomes an $\\mathfrak{sl}_3$-representation with respect to the adjoint action. \n\n\\begin{defn}\\label{F}\n    Define an isomorphism $\\eta:\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*\\longrightarrow\\mathbb{C}^8$ by the formula\n    \\begin{equation}\\label{eta}\n          \\eta(z_1,z_2,z_3,a,b,w_1,w_2,w_3)=z_1e_2+z_2e_3-z_3e_8+ae_4\n        +be_5-w_3e_1+w_2e_6+w_1e_7\n    \\end{equation}\n\\end{defn}\n\n\\begin{prop}\n    The map $\\eta$ induces an isomorphism of $\\mathfrak{sl}_3$-representations\n    $$\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta:\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)\\longrightarrow\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^8)=\\mathfrak{so}_8.$$  \n\\end{prop}\n\\begin{proof}\n    First, let $V_0\\coloneqq\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*$, and $<\\,,\\,>$ denote the inner product on $V_0$ given by the natural pairing in between $\\mathbb{C}^3\\oplus\\mathbb{C}$ and $\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*$. Then\n    $$\n        \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)=\\mathfrak{so}(V_0,<\\,,\\,>).\n    $$\n    Since $\\eta$ is an isometry between $(V_0,<\\,,\\,>)$ and $(\\mathbb{C}^8,(\\,,\\,))$, the map $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta$ is a Lie algebra isomorphism between $\\mathfrak{so}(V_0,<\\,,\\,>)$ and $\\mathfrak{so}_8$.\n    We identify $$\\mathfrak{sl}_3=\\{A\\in\\mathbb{C}^3\\otimes(\\mathbb{C}^3)^*\\,|\\, \\mathrm{tr}(A)=0\\}\\subset\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*).$$\n    Then $\\mathfrak{sl}_3$ becomes an Lie subalgebra of $\\mathfrak{so}(V_0,<\\,,\\,>)$, and the $\\mathfrak{sl}_3$-representation structure on $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)$ is given by the adjoint action.\n    By an explicit calculation, we have the restriction of $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta$ to $\\mathfrak{sl}_3$ equals to the embedding $\\varphi_1$. For example, for positive root vectors,\n    \\begin{align*}\n        \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta\\,(e_1\\wedge e^*_2)& =e_2\\wedge e_6= X_{\\alpha_2},\\\\\n        \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta\\,(e_2\\wedge e^*_3)& =e_3\\wedge (-e_1)= X_{\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4},\\\\\n        \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\eta\\,(e_1\\wedge e^*_3)& =e_2\\wedge (-e_1)= X_{\\alpha_1+2\\alpha_2+\\alpha_3+\\alpha_4}.\n    \\end{align*}\n    Since the $\\mathfrak{sl}_3$-representation structure on $\\mathfrak{so}_8$ is given by the Lie algebra embedding $\\varphi_1$, our statement follows. \n\\end{proof}\n\nDecompose $\\mathfrak{so}_8=\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)$ into irreducible $\\mathfrak{sl}_3$-representations, \n\\begin{equation}\\label{decomp}\n    \\mathfrak{so}_8=\\mathfrak{sl}_3\\oplus\\mathbb{C}_{\\textrm{trace}}\\oplus\\mathbb{C}\\oplus\\mathbb{C}^3\\oplus\\mathbb{C}^3\\oplus\\mathbb{C}^3\\oplus(\\mathbb{C}^3)^*\\oplus(\\mathbb{C}^3)^*\\oplus(\\mathbb{C}^3)^*.\n\\end{equation}\n\n\n\\begin{defns}\\label{decomp2}\nLet $\\mathfrak{h}=\\{(c_1,c_2,c_3)\\in\\mathbb{C}^3\\,|\\,c_1+c_2+c_3=0\\}$. Define $\\varphi_2:\\mathfrak{h}\\rightarrow\\mathfrak{so}_8$ by \n\\begin{align*}\n\t\\varphi_2(c_1,c_2,c_3)\\coloneqq\\ & c_1H_{\\alpha_1}+c_2H_{\\alpha_2}+c_3H_{\\alpha_3}\\\\\n\t=\\ & c_1(H_1-H_2)+c_2(H_3-H_4)+c_3(H_3+H_4)\\\\\n\t=\\ & -c_1(H_2+H_3-H_1)+(c_3-c_2)H_4\\in\\mathbb{C}_{\\textrm{trace}}\\oplus\\mathbb{C},\n\\end{align*}\nwhere $H_i\\coloneqq e_i\\wedge e_{9-i}$ for each $i$.\nSo the image of $\\varphi_2$ is precisely the sub-representation $\\mathbb{C}_{\\textrm{trace}}\\oplus\\mathbb{C}\\subset\\mathfrak{so}_8.$\n\n\n\nLet $V_1,V_2,V_3$ (resp.\\ $V^*_1,V^*_2,V^*_3$) be the three copies of $\\mathbb{C}^3$ (resp.\\ $(\\mathbb{C}^3)^*$) in the decomposition (\\ref{decomp}) whose highest weight vectors are root vectors for $\\mathfrak{so}_8$ corresponding to the roots\n$\\alpha_1+\\alpha_2, \\alpha_2+\\alpha_3,\\alpha_2+\\alpha_4$ (resp.\\ $\\alpha_2+\\alpha_3+\\alpha_4, \\alpha_1+\\alpha_2+\\alpha_4, \\alpha_1+\\alpha_2+\\alpha_3$). Identify\n $V_1,V_2,V_3$ (\\textrm{resp.\\ }$V^*_1,V^*_2,V^*_3$) via the unique $\\mathfrak{sl}_3$-equivariant linear isomorphisms which map the above choice of highest weight vectors to each other.\n Explicitly,\n\\begin{align*}\n& V_1=\\mathbb{C}\\langle -X_{\\alpha_1+\\alpha_2},X_{\\alpha_1},Y_{-\\alpha_2-\\alpha_3-\\alpha_4}\\rangle=\\mathbb{C}^3,\\\\\n& V_2=\\mathbb{C}\\langle X_{\\alpha_2+\\alpha_3},X_{\\alpha_3},Y_{-\\alpha_1-\\alpha_2-\\alpha_4}\\rangle=\\mathbb{C}^3,\\\\\n& V_3=\\mathbb{C}\\langle X_{\\alpha_2+\\alpha_4},X_{\\alpha_4},Y_{-\\alpha_1-\\alpha_2-\\alpha_3}\\rangle=\\mathbb{C}^3,\\\\\n& V_1^*=\\mathbb{C}\\langle X_{\\alpha_2+\\alpha_3+\\alpha_4},Y_{-\\alpha_1},-Y_{-\\alpha_1-\\alpha_2}\\rangle=(\\mathbb{C}^3)^*,\\\\\n& V_2^*=\\mathbb{C}\\langle X_{\\alpha_1+\\alpha_2+\\alpha_4},Y_{-\\alpha_3},Y_{-\\alpha_2-\\alpha_3}\\rangle=(\\mathbb{C}^3)^*,\\\\\n& V_3^*=\\mathbb{C}\\langle X_{\\alpha_1+\\alpha_2+\\alpha_3},Y_{-\\alpha_4},Y_{-\\alpha_2-\\alpha_4}\\rangle=(\\mathbb{C}^3)^*.\n\\end{align*}\nSo we have an identification\n\\begin{equation}\\label{phi}\n       \\varphi: \\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus V_1\\oplus V_2\\oplus V_3\\oplus V^*_1\\oplus V^*_2\\oplus V^*_3\\longrightarrow\\mathfrak{so}_8\n\\end{equation}\n\\end{defns}\n\n\n\\begin{rem}\nFix the standard basis $e_1,e_2,e_3$ of $\\mathbb{C}^3$, and identify\n$$\n\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^3\\mathbb{C}^3=\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^3(\\mathbb{C}^3)^*=\\mathbb{C},\\ \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2\\mathbb{C}^3=(\\mathbb{C}^3)^*,\\ \\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^3)^*=\\mathbb{C}^3.\n$$\nFor any given $A\\in\\End(\\mathbb{C}^3)$ and $v\\in\\mathbb{C}^3$ we define $A\\wedge v\\in\\mathrm{Sym}^2(\\mathbb{C}^3)^*$ by\n$$\n    (A\\wedge v)(w_1,w_2)=(Aw_1)\\wedge v\\wedge w_2+(Aw_2)\\wedge v\\wedge w_1.\n$$\nSimilarly we can define $A\\wedge v^*\\in\\mathrm{Sym}^2\\mathbb{C}^3$ for $v^*\\in(\\mathbb{C}^3)^*$.\n\nLet $(M,{c},{u},{u}^*)$ be an element in $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^3)^*)^3$. Then \n$\\varphi(M,{c},{u},{u}^*)\\in\\overline{\\mathcal{O}}_\\textrm{min}$ if and only if:\n\\begin{align*}\n\t& u_i^*(u_j)=0\\textrm{ for all distinct $i,j\\in\\{1,2,3\\}$},\\\\\n\t& u^*_1(u_1)=(c_1-c_2)(c_1-c_3),\\textrm{ and its cyclic permutations},\\\\\n\t& u_1\\wedge u_2 = (c_1-c_2)u_3^*,\\textrm{ and its cyclic permutations},\\\\\n\t& u_1^*\\wedge u_2^* = (c_1-c_2)u_3,\\textrm{ and its cyclic permutations},\\\\\n\t& (M-c_3\\,\\mathrm{Id}_{\\mathbb{C}^3})\\wedge u_3+u_1^*\\cdot u_2^*=0, \\textrm{ and its cyclic permutations},\\\\\n\t& (M-c_3\\,\\mathrm{Id}_{\\mathbb{C}^3})\\wedge u_3^*+u_1\\cdot u_2=0, \\textrm{ and its cyclic permutations},\\\\\n\t& (M-c_3\\,\\mathrm{Id}_{\\mathbb{C}^3})^2+u_1\\otimes u_1^*+u_2\\otimes u_2^*-u_3\\otimes u_3^*+u_3^*(u_3)\\,\\mathrm{Id}_{\\mathbb{C}^3}=0, \\textrm{ and its cyclic permutations}.\n\\end{align*}\n\\end{rem}\n\n\n\\vspace{3em}\n\\subsection{Isomorphism between $\\overline{T^*(\\mathrm{SL}_3\/U)}$ and $\\overline{\\mathcal{O}}_\\textrm{min}\\subset\\mathfrak{so}_8$}\\label{sl3}\n\nIn the special case $n=3$, the construction of $N$ defined in section 2 consist of elements in\n\\begin{equation*}\n    \\begin{tikzcd}\n\t\\mathbb{C} \\arrow[r,shift left=.7ex,\"\\alpha_1\"]   \n\t& {\\mathbb{C}^2} \\arrow[l,shift left=.7ex,\"\\beta_1\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_2\"] \n\t& \\mathbb{C}^3 \\arrow[l,shift left=.7ex,\"\\beta_2\"] \n\\end{tikzcd}\n\\vspace{-1em}\n\\end{equation*}\nsuch that\n$$(\\beta_2 \\alpha_2-\\alpha_1 \\beta_1)=\\lambda_2\\,\\mathrm{Id}_{\\mathbb{C}^2}\\text{ for some }\\lambda_2\\in\\mathbb{C}$$\nWe denote the scalar $\\beta_1\\alpha_1$ by $\\lambda_1$.\nBy \\cite{DKS} again, we have the affine closure $\\overline{T^*(\\mathrm{SL}_3\/U)}$ is identified with $N\\sslash \\mathrm{SL}_2$ as affine varieties.\nIn the special case $m=4$, Proposition \\ref{ominhamil} says that $\\overline{\\mathcal{O}}_\\textrm{min}=N_1\\sslash\\mathrm{SL}_2$ as affine varieties, where\n$$\n    N_1=\\{f\\in\\Hom(\\mathbb{C}^2,\\mathbb{C}^{8})\\,|\\, f \\textrm{ is isotropic with respect to }(\\ ,\\ )\\}.\n$$\n\n\n\tWe identify $\\mathbb{C}^8=(\\mathbb{C}^3\\oplus\\mathbb{C})\\oplus(\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*)=\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*$ via $\\eta$ in (\\ref{eta}). Then $\\eta$ is an isometry with respect to the inner product on $\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*$ given by the natural pairing of $\\mathbb{C}^4$ and $(\\mathbb{C}^4)^*$ and the inner product $(\\,,\\,)$ on $\\mathbb{C}^8$ defined by taking $m=4$ in (\\ref{inner}).\n\\begin{defn}\n    We define the map\n\\begin{align*}\n\tF:\n\tT^*V& \\longrightarrow \\Hom(\\mathbb{C}^2,\\mathbb{C}^8)=\\Hom(\\mathbb{C}^2,\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)\\\\\n\t(\\alpha,\\beta) &\\longmapsto\\left(v\\mapsto (\\alpha_2\\oplus-\\beta_1)(v)\\oplus\\frac{(\\beta_2\\oplus\\alpha_1)\\wedge v}{e_1\\wedge e_2}\\right)\n\\end{align*}\n\\end{defn}\n\n\\begin{prop}\\label{Fthm}\n\tThe map $F$ is a $\\mathrm{SL}_2$-equivariant symplectic isomorphism between $T^*V$ and $(\\Hom(\\mathbb{C}^2,\\mathbb{C}^8),\\omega_1)$. \n\\end{prop}\n\\begin{proof}\n\tConsider the following $\\mathrm{SL}_2$-equivariant symplectic isomorphism.\n\tFirst, we have a symplectic isomorphism $$F_0:T^*V\\longrightarrow T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4)$$ which maps $(\\alpha,\\beta)$ to the following element in $\\Hom(\\mathbb{C}^2,\\mathbb{C}^4)\\oplus\\Hom(\\mathbb{C}^4,\\mathbb{C}^2)$:\n\t$$\n\t\\begin{tikzcd}[row sep=large, column sep = large]\n\t\t\\mathbb{C}^2 \\arrow[r,shift left=.7ex,\"\\alpha_2\\oplus-\\beta_1\"]   \n\t\t& {(\\mathbb{C}^3\\oplus\\mathbb{C})=\\mathbb{C}^4} \\arrow[l,shift left=.7ex,\"\\beta_2\\oplus\\alpha_1\"] \n\t\\end{tikzcd}.\n\t$$\n\t\n\tNext, with respect to the basis $e_1,e_2\\in\\mathbb{C}^2$ we decompose\n\t\\begin{align*}\n\t\t& T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4))=T^*(\\mathbb{C}^4\\oplus\\mathbb{C}^4)=(\\mathbb{C}^4\\oplus\\mathbb{C}^4)\\oplus((\\mathbb{C}^4)^*\\oplus(\\mathbb{C}^4)^*).\\\\\n\t\t& \\Hom(\\mathbb{C}^2,\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)=(\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)\\oplus(\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*).\n\t\\end{align*}\n\tWe define the map\n\t\\begin{align*}\n\t\tF_1:\\qquad T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4))\\ &\\longrightarrow\\ \\Hom(\\mathbb{C}^2,\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)\\\\\n\t\t(v_1\\oplus v_2)\\oplus(v_1^*\\oplus v_2^*)\\ &\\longmapsto\\ (v_1\\oplus (-v_2^*))\\oplus(v_2\\oplus v_1^*).\n\t\\end{align*}\n\tAnd we check $F_1$ is a symplectic isomorphism:\n\t\\begin{align*}\n\t\t&\\ \\ \\ F_1^*\\omega_1\\big((v_1\\oplus v_2)\\oplus(v_1^*\\oplus v_2^*),(w_1\\oplus w_2)\\oplus(w_1^*\\oplus w_2^*)\\big)\\\\\n\t\t& = \\omega_1\\big((v_1\\oplus (-v_2^*))\\oplus(v_2\\oplus v_1^*),(w_1\\oplus (-w_2^*))\\oplus(w_2\\oplus w_1^*)\\big)\\\\\n\t\t& =\\big(\\!-\\!v_2^*(w_2)+w_1^*(v_1)\\big)-\\big(v_1^*(w_1)-w_2^*(v_2)\\big)\\\\\n\t\t& =(w_1^*\\oplus w_2^*)(v_1\\oplus v_2)-(v_1^*\\oplus v_2^*)(w_1\\oplus w_2),\n\t\\end{align*}\n\twhich is the natural symplectic form on $T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4))$.\n\tNotice that all three spaces $$T^*V,\\quad T^*(\\Hom(\\mathbb{C}^2,\\mathbb{C}^4)),\\quad \\Hom(\\mathbb{C}^2,\\mathbb{C}^4\\oplus(\\mathbb{C}^4)^*)$$ have natural Hamiltonian $\\mathrm{SL}_2$-actions. Since both $F_0$ and $F_1$ are $\\mathrm{SL}_2$-equivariant, $F$ is also an $\\mathrm{SL}_2$ equivariant symplectic isomorphism as\n\t$$\n\t\tF=F_1\\circ F_0.\n\t$$\n\\end{proof}\n\\begin{thm}\\label{TO}\n\tThe map $F$ induces an isomorphism between $\\overline{T^*(\\mathrm{SL}_3\/U)}$ and $\\overline{\\mathcal{O}}_\\textrm{min}$ as affine varieties.\n\\end{thm}\n\\begin{proof}\nThis is a direct corollary of Propositions \\ref{Fthm} and \\ref{ominhamil}.\n\\end{proof}\n\\begin{rem}\n    Let $P$ be the parabolic subgroup of $\\mathrm{SL}_n$ corresponding to the standard partial flag \n$$\\mathbb{C}^1 \\subset \\mathbb{C}^2 \\subset \\mathbb{C}^n.$$\nLet $[P,P]$ denote the commutator subgroup of $P$.\nThen I expect that one can use similar method to prove that the affinization $\\overline{T^*(\\mathrm{SL}_n\/[P,P])}$ is isomorphic to the closure of the minimal orbit in $so(2n+2)$.\n\\end{rem}\n\n\n\n\n\n\n\\vspace{3em}\n\\subsection{Symplectic form on the smooth locus}\\label{symplectic}\nLet $\\mathcal{O}_{\\textrm{nilp}}$ be a nilpotent adjoint orbit in $\\mathfrak{g}$.\nUsing Killing form $\\kappa$ we identify $\\mathcal{O}_{\\textrm{nilp}}$ with a nilpotent coadjoint orbit in $\\mathfrak{g}^*$. For any $Y\\in\\mathfrak{g}$, we denote $\\xi_Y$ to be the vector field on $\\mathcal{O}_{\\textrm{nilp}}$ generated by the infinitesimal action by $Y$.\nAnd the Kirillov-Kostant-Souriau symplectic strucutre $\\omega^\\textrm{KKS}$ on $\\mathcal{O}_{\\textrm{nilp}}$ is given by the formula:\n$$\n    \\omega^\\textrm{KKS}_{X}(\\xi_Y,\\xi_Z)=\\kappa(X,[Y,Z]).\n$$\nSince $\\mathcal{O}_{\\textrm{nilp}}$ is a nilpotent orbit, there is a $\\mathbb{C}^*$-action on it such that $\\omega^\\textrm{KKS}=\\mathcal{L}_{\\textsl{Eu}}\\omega^\\textrm{KKS},$ where $\\textsl{Eu}$ is the Euler vector field on $\\mathcal{O}_{\\textrm{nilp}}$. Define $$\\lambda^\\textrm{KKS}\\coloneqq\\iota_{\\textsl{Eu}}\\omega^\\textrm{KKS}.$$\nSo we have $$d\\lambda^\\textrm{KKS}=d\\iota_{\\textsl{Eu}}\\omega^\\textrm{KKS}=\\mathcal{L}_{\\textsl{Eu}}\\omega^\\textrm{KKS}=\\omega^\\textrm{KKS}.$$\nLet us recall the following known result.\n\\begin{prop}\n    Let $X\\in\\mathcal{O}_{\\textrm{nilp}}$ and $Y\\in\\mathfrak{g}$.  Then\n    $$\\lambda^\\textrm{KKS}_X(\\xi_Y)=\\kappa(X,Y).$$\n\\end{prop}\n\\begin{proof}\nSince $X$ is nilpotent, we have $Y\\in\\mathfrak{g}_X$ if and only if $\\kappa(X,Y)=0$. So the RHS of the formula only depends on $\\xi_Y$ (i.e. independent of the choice of $Y$). Let $E\\in\\mathcal{O}_{\\textrm{nilp}}$. Fix some $\\mathfrak{sl}_2$-triple $(E,H,F)$ in $\\mathfrak{g}$. Then we have\n$$\n    (\\xi_H)_E=2\\,\\textsl{Eu}_{E}\\in T_E\\mathcal{O}_{\\textrm{nilp}}.\n$$\nSince $\\lambda^\\textrm{KKS}$ is invariant under the $G$-action, it suffices to prove the statement for $X=E$. Let $Y\\in\\mathfrak{g}$. We compute\n$$\n    \\lambda^\\textrm{KKS}_E(\\xi_Y)=\\iota_{\\textsl{Eu}}\\omega_E^\\textrm{KKS}(\\xi_Y)=\\frac{1}{2}\\omega_E^\\textrm{KKS}(\\xi_Y,\\xi_H)=\\frac{1}{2}\\kappa(E,[Y,H])=\\frac{1}{2}\\kappa([H,E],Y)=\\kappa(E,Y).\n$$\n\\end{proof}\n\n\\begin{thm}\n\tThe symplectic form on $\\overline{T^*(\\mathrm{SL}_3\/U)}_\\textrm{sm}$ given in Proposition \\ref{symp} coincides (up to a scalar multiple) with $\\omega^\\textrm{KKS}$ on $\\mathcal{O}_{\\textrm{min}}\\subset\\mathfrak{so}_8$.\n\\end{thm}\n\\begin{proof}\n\tLet $v_1\\wedge v_2\\in \\mathcal{O}_{\\textrm{min}}\\subset\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2 {\\mathbb{C}^8}$.\n\tRecall the element $\\varphi_{v_1\\wedge v_2}\\in \\mathfrak{so}_8$ is defined in (\\ref{varphi}) by the formula\n\t$$\n\t\\varphi_{v_1\\wedge v_2}(u)=(v_1,u)v_2-(v_2,u)v_1.\n\t$$\n\tObserve that the tangent space $T_{v_1\\wedge v_2}\\mathcal{O}_{\\textrm{min}}$ is spanned by vectors given by infinitesimal actions of some $w_1\\wedge w_2\\in\\mathfrak{so}_8$. Let \n\t$$\n\t\\xi_{w_1\\wedge w_2}=[w_1\\wedge w_2,v_1\\wedge v_2]\n\t$$\n\tbe such a tangent vector.\n\tTo calculate $\\xi_{w_1\\wedge w_2}$ we first we compute\n\t\\begin{align*}\n\t\t(\\varphi_{v_1\\wedge v_2}\\circ \\varphi_{w_1\\wedge w_2})(u)\\ \\ =\\ \\ \\ &[(w_2,u)(v_2,w_1)-\\,(w_1,u)(v_2,w_2)] v_1\\\\\n\t\t+&[(w_1,u)(v_1,w_2)-\\,(w_2,u)(v_1,w_1)] v_2,\\\\\n\t\t(\\varphi_{w_1\\wedge w_2}\\circ \\varphi_{v_1\\wedge v_2})(u)\\ \\ =\\ \\ \\ &[(v_2,u)(w_2,v_1)-\\,(v_1,u)(w_2,v_2)] w_1\\\\\n\t\t+&[(v_1,u)(w_1,v_2)-\\,(v_2,u)(w_1,v_1)] w_2.\n\t\\end{align*}\n\tThen we compute the Lie bracket $[\\varphi_{v_1\\wedge v_2}, \\varphi_{w_1\\wedge w_2}]$ and get\n\t\\begin{align*}\n\t\t\\xi_{w_1\\wedge w_2}& =(v_2,w_2)\\, w_1\\wedge v_1-(v_1,w_2)\\, w_1\\wedge v_2-(v_2,w_1)\\, w_2\\wedge v_1+(v_1,w_1)\\, w_2\\wedge v_2\\\\\n\t\t& =\\varphi_{w_1\\wedge w_2}(v_1)\\wedge v_2+ v_1\\wedge \\varphi_{w_1\\wedge w_2}(v_2).\n\t\\end{align*}\n\tThe value of the one form $\\lambda^\\textrm{KKS}$ on $\\mathcal{O}_{\\textrm{min}}$ at $v_1\\wedge v_2$ is\n\t\\begin{align*}\n\t\t\\lambda^\\textrm{KKS}_{v_1\\wedge v_2}(\\xi_{w_1\\wedge w_2})=\\mathrm{Tr}(\\varphi_{v_1\\wedge v_2}\\circ \\varphi_{w_1\\wedge w_2})=2((v_2,w_1)(v_1,w_2)-(v_1,w_1)(v_2,w_2)).\n\t\\end{align*}\n\tRecall \\textrm{Proposition\\ \\ref{Fthm}}, the symplectic form on $\\overline{T^*(\\mathrm{SL}_3\/U)}_\\textrm{sm}$ given as in \\textrm{Proposition \\ref{symp}} coincides with the pull-back of the symplectic form on $(N_1\\sslash \\mathrm{SL}_2)_\\textrm{sm}$ by the induced isomorphism $$F:\\left(\\overline{T^*(\\SL_3\/U)}\\right)_\\textrm{sm}\\rightarrow \\mathcal{O}_{\\textrm{min}}.$$\n\tSo it suffices to show that the symplectic form on $(N_1\\sslash \\mathrm{SL}_2)_\\textrm{sm}$ coming from the Hamiltonian reduction coincides (up to a scalar multiple) with $\\omega^\\textrm{KKS}$. It suffices to check for one forms.\n\tWe know on $\\Hom(\\mathbb{C}^2,\\mathbb{C}^8)=\\mathbb{C}^8\\oplus\\mathbb{C}^8$, the one form\n\t$$\\lambda'_{v_1\\oplus v_2}(x_1\\oplus x_2)=(v_1,x_2)-(v_2,x_1)$$\n\tis $\\mathrm{SL}_2$-invariant. \n\tRecall from (\\ref{o1}), we have $\\omega_1=(1\/2)d\\lambda'$.\n\tLift the tangent vector $\\xi_{w_1\\wedge w_2}$ to $(x_1\\oplus x_2)\\in T_{v_1\\oplus v_2}(\\mathbb{C}^8\\oplus\\mathbb{C}^8)$, where\n\t\\begin{align*}\n\t\tx_1& =\\varphi_{w_1\\wedge w_2}(v_1)=(w_1,v_1)w_2-(w_2,v_1)w_1,\\\\\n\t\tx_2& =\\varphi_{w_1\\wedge w_2}(v_2)=(w_1,v_2)w_2-(w_2,v_2)w_1.\n\t\\end{align*}\n\tSo we have $$\\lambda'_{v_1\\oplus v_2}(x_1\\oplus x_2)=2((w_1,v_2)(v_1,w_2)-(w_2,v_2)(v_1,w_1))=\\lambda^\\textrm{KKS}_{v_1\\wedge v_2}(\\xi_{w_1\\wedge w_2}).$$\n\\end{proof}\n\n\n\n\n\n\n\n\\vspace{3em}\n\\section{The Gelfand-Graev action on $\\overline{T^*(\\mathrm{SL}_3\/U)}$.}\n\n\n\n\\subsection{The triality action on $\\mathfrak{so}_8$}\\label{triality}\n\nThe triality action was first discovered by E.\\ Cartan in his 1925 paper \\cite{Cartan} in which he constructed a $S_3$-action on $\\mathfrak{so}_8$ preserving the Lie bracket lifting the $S_3$-symmetry of the Dynkin Diagram $D_4$:\n    $$\n     \\dynkin[text style\/.style={scale=1},label,label macro\/.code={\\alpha_{\\drlap{#1}}},scale=1.2,edge\nlength=0.9cm]D4\n    $$\nThe action of an order three element of the triality action is constructed from the following matrix\n$$\n\\begingroup\n\\renewcommand*{\\arraystretch}{1.5}\n\\begin{pmatrix*}[r]\n    -\\frac{1}{2} && -\\frac{1}{2} && -\\frac{1}{2} && -\\frac{1}{2}\\\\\n    \\frac{1}{2} && \\frac{1}{2} && -\\frac{1}{2} && -\\frac{1}{2}\\\\\n    \\frac{1}{2} && -\\frac{1}{2} && \\frac{1}{2} && -\\frac{1}{2}\\\\\n    \\frac{1}{2} && -\\frac{1}{2} && -\\frac{1}{2} && \\frac{1}{2}\n\\end{pmatrix*}.\n\\endgroup\n$$\n\n\nGiven the Chevalley basis of $\\mathfrak{so}_8$ as in the \\hyperref[appendix]{Appendix}, each automorphism of the Dynkin diagram does lift uniquely to a Lie algebra automorphism (c.f. Theorem 2.108 in \\cite{Knapp}, and Lemma 2.6 in \\cite{EL}). But a priori it is not clear that the these lifts gives an $S_3\\hookrightarrow\\mathrm{Aut}(\\mathfrak{so}_8)$. For example, not every lift of the cyclic permutation of the simple roots $\\alpha_1,\\alpha_3,\\alpha_4$ has order three in $\\mathrm{Aut}(\\mathfrak{so}_8)$.\nIn the work of \\cite{MW}, a lifted triality action on the Lie algebra $\\mathfrak{so}(4,4)$ is constructed, but the authors proved their result by an explicit calculation in coordinates by computer.\n\nIn this section, a new interpretation of the triality action on $\\mathfrak{so}_8=\\mathfrak{so}(4,4)_\\mathbb{C}$ is given by applying the decomposation of $\\mathfrak{so}_8$ into irreducible $\\mathfrak{sl}_3$-representations as in (\\ref{decomp}).\nRecall the notations and definitions in Definition \\ref{decomp2}, we have an isomorphism (\\ref{phi}) of $\\mathfrak{sl}_3$-representations\n$$\n       \\varphi: \\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus V_1\\oplus V_2\\oplus V_3\\oplus V^*_1\\oplus V^*_2\\oplus V^*_3\\longrightarrow\\mathfrak{so}_8.\n$$\n\n\\begin{defn}\n\nDefine an $S_3$-action on $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus V_1\\oplus V_2\\oplus V_3\\oplus V^*_1\\oplus V^*_2\\oplus V^*_3$ by fixing $\\mathfrak{sl}_3$, acting on $\\mathfrak{h}$ as the Weyl group $S_3$-action, and permuting subscripts of $V_i$ and $V^*_i$. Then under the identification $\\varphi$, we have an $S_3$-action $\\mathsf{act}$ on $\\mathfrak{so}_8$.\n\\end{defn}\n\n\\begin{thm}\n    The $S_3$-action $\\mathsf{act}$ gives an embedding $S_3\\hookrightarrow \\mathrm{Aut}(\\mathfrak{so}_8)$ which coincides with the triality action.\n\\end{thm}\n\n\\begin{proof}\n\t\n\tLet $(M,{c},{u},{u}^*)$ be an element in $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^3)^*)^3$. Express each components in terms of the Chevalley basis, we see the $S_3$-action $\\mathsf{act}$ does permutes the root vectors accordingly (i.e. fixing $\\alpha_2$, and permutes $\\alpha_1,\\alpha_3,\\alpha_4$). So it suffices to show that this action preserves the Lie bracket of $\\mathfrak{so}_8$.\n\t\nThere is a linear action of $(M,{c},{u},{u}^*)$ on elements $(v,a,b,v^*)\\in\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*$ coming from the linear action of $\\mathfrak{so}_8$ on $\\mathbb{C}^8$ given by\n\t$$\n\t(M,c,u,u^*).\\begin{pmatrix}\n\t\tv\\\\\n\t\ta\\\\\n\t\tb\\\\\n\t\t\\ v^*\n\t\\end{pmatrix}=\n\t\\begin{pmatrix}\n\t\tM v+c_1v+au_2+bu_3+v^*\\wedge u_1^*\\\\\n\t\tu_2^*(v)+(c_3-c_2)a-v^*(u_3)\\\\\n\t\tu_3^*(v)-(c_3-c_2)b-v^*(u_2)\\\\\n\t\t-v^* M-c_1v^*-au_3^*-bu_2^*+v\\wedge u_1\n\t\\end{pmatrix}.\n\t$$\n\t\n\tWe calculate the Lie bracket structure on $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^3)^*)^3$.\n\tLet $A=(M_A,{c}_A,{u}_A,{u}^*_A)$ and $B=(M_B,c_B,u_B,u^*_B)$ be two elements in $\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^*)^3)^3$. Then the Lie bracket can be written as $[A,B]=(M_C,c_C,u_C,u^*_C)$ where\n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\begin{align*}\n\t\t& M_C=[M_A,M_B]+\\sum_{i=1}^{3}(u_{A,i}\\otimes u^*_{B,i}-u_{B,i}\\otimes u^*_{A,i})+\\frac{1}{3}\\sum_{i=1}^{3}(u^*_{A,i}(u_{B,i})-u^*_{B,i}(u_{A,i}))\\,\\mathrm{Id}_{\\mathbb{C}^3}\\\\\n\t\t& c_C=\n\t\t\\begin{pmatrix}\n\t\t\t-2\/3 && 1\/3 && 1\/3\\\\\n\t\t\t1\/3 && -2\/3 && 1\/3\\\\\n\t\t\t1\/3 && 1\/3 && -2\/3\n\t\t\\end{pmatrix}\n\t\t\\begin{pmatrix}\n\t\t\tu^*_{A,1}(u_{B,1})-u^*_{B,1}(u_{A,1}) \\\\\n\t\t\tu^*_{A,2}(u_{B,2})-u^*_{B,2}(u_{A,2}) \\\\\n\t\t\tu^*_{A,3}(u_{B,3})-u^*_{B,3}(u_{A,3})\n\t\t\\end{pmatrix}\\\\\n\t\t& u_C=\t\t\\begin{pmatrix}\n\t\t\tM_A\\,u_{A,1}-M_B\\,u_{B,1}+u^*_{A,2}\\wedge u^*_{B,3}-u^*_{B,2}\\wedge u^*_{A,3}+2c_{A,1}\\,u_{B,1}-2c_{B,1}\\,u_{A,1}\\\\\n\t\t\tM_A\\,u_{A,2}-M_B\\,u_{B,2}+u^*_{A,3}\\wedge u^*_{B,1}-u^*_{B,3}\\wedge u^*_{A,1}+2c_{A,2}\\,u_{B,2}-2c_{B,2}\\,u_{A,2}\\\\\n\t\t\tM_A\\,u_{A,3}-M_B\\,u_{B,3}+u^*_{A,1}\\wedge u^*_{B,2}-u^*_{B,1}\\wedge u^*_{A,2}+2c_{A,3}\\,u_{B,3}-2c_{B,3}\\,u_{A,3}\\\\\n\t\t\\end{pmatrix}\\\\\n\t\t& u^*_C=\t\t-\\begin{pmatrix}\n\t\t\tu^*_{A,1}M_A-u^*_{B,1}M_A+u_{A,2}\\wedge u_{B,3}-u_{B,2}\\wedge u_{A,3}+2c_{A,1}\\,u^*_{B,1}-2c_{B,1}\\,u^*_{A,1}\\\\\n\t\t\tu^*_{A,2}M_A-u^*_{B,2}M_A+u_{A,3}\\wedge u_{B,1}-u_{B,3}\\wedge u_{A,1}+2c_{A,2}\\,u^*_{B,2}-2c_{B,2}\\,u^*_{A,2}\\\\\n\t\t\tu^*_{A,3}M_A-u^*_{B,3}M_A+u_{A,1}\\wedge u_{B,2}-u_{B,1}\\wedge u_{A,2}+2c_{A,3}\\,u^*_{B,3}-2c_{B,3}\\,u^*_{A,3}\\\\\n\t\t\\end{pmatrix}\n\t\\end{align*}\n\tThen we see that the Lie bracket is manifestly equivariant under the lifted triality action. \n\\end{proof}\n\n\\begin{rem}\n\tThe Killing form can also be written in this fashion.\n\t$$\n\t\\frac{1}{2}\\,\\mathrm{Tr}(A B)=\\mathrm{Tr}(M_A M_B)+\\sum_{i=1}^{3}(u^*_{A,i}(u_{B,i})+u^*_{B,i}(u_{A,i}))+\\sum_{i=1}^{3}2\\,c_{A,i}\\,c_{B,i}\n\t$$\n\t\n\tThen the Cartan three form can be expressed as\n\t\\begin{align*}\n\t\t\\frac{1}{2}\\,\\mathrm{Tr}(A\\,[B,C])& =\\mathrm{Tr}(M_A\\,[M_B,M_C])+\\sum_{\\sigma\\in S_3}\\left(u^*_{A,\\sigma(1)}\\wedge u^*_{B,\\sigma(2)}\\wedge u^*_{C,\\sigma(3)}-u_{A,\\sigma(1)}\\wedge u_{B,\\sigma(2)}\\wedge u_{C,\\sigma(3)}\\right)\\\\\n\t\t&\\quad +\\sum_{\\textrm{Cyclic } A,B,C}\\left(2c_A\\cdot\\left(u^*_C(u_B)-u^*_B(u_C)\\right)+\\sum_{i=1}^{3}\\,\\left(u^*_{C,i}M_Au_{B,i}-u^*_{B,i}M_Au_{C,i}\\right)\\right).\n\t\\end{align*}\n\\end{rem}\n\n\n\n\n\n\\vspace{3em} \n\\subsection{The Gelfand-Graev action on $\\overline{T^*(\\SL_3\/U)}$}\\label{ggaction}\n\nRecall that $N$ is defined as elements in the following double quiver representation.\n\\[\n\\begin{tikzcd}\n\t0 \\arrow[r,shift left=.7ex,\"\\alpha_0\"]   \n\t& {\\mathbb{C}} \\arrow[r,shift left=.7ex,\"\\alpha_1\"]\n\t\\arrow[l,shift left=.7ex,\"\\beta_0\"] \n\t& {\\mathbb{C}^2} \\arrow[l,shift left=.7ex,\"\\beta_1\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_2\"] \n\t& {\\cdots} \\arrow[l,shift left=.7ex,\"\\beta_2\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_{n-2}\"] \n\t& {\\mathbb{C}^{n-1}} \\arrow[l,shift left=.7ex,\"\\beta_{n-2}\"] \n\t\\arrow[r,shift left=.7ex,\"\\alpha_{n-1}\"] \n\t& \\mathbb{C}^n \\arrow[l,shift left=.7ex,\"\\beta_{n-1}\"]\n\\end{tikzcd},\n\\]\nsuch that $\\forall k\\in\\{1,2,3,\\cdots,n-1\\}$\n\\begin{align*}\n\t\\beta_k\\alpha_k-\\alpha_{k-1}\\beta_{k-1}=\\lambda_k\\,\\textrm{Id}_{\\mathbb{C}^k}\n\\end{align*}\nfor some $\\lambda_k\\in\\mathbb{C}.$\n\n\n\\begin{defn}[Following \\cite{Wang} with slightly different convention]\\ \n\t\t\n\tLet $B=(\\alpha,\\beta)\\in N$.\n\tLet $k\\in\\{1,2,3,\\cdots,n-1\\}$. Define\n\t\\begin{align*}\n\t\t& \\mathrm{out}_k(B) \\coloneqq\\alpha_k\\oplus\\beta_{k-1}\\in\\Hom(\\mathbb{C}^k,\\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1}),\\\\\n\t\t& \\mathrm{in}_k(B) \\coloneqq\\beta_k\\oplus(-\\alpha_{k-1})\\in\\Hom(\\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1},\\mathbb{C}^k).\n\t\\end{align*}\nThen\n$$\n\t\\mathrm{in}_k(B)\\,\\mathrm{out}_k(B)=\\lambda_k\\,\\textrm{Id}_{\\mathbb{C}^k}.\n$$\n\tWe also define $Z_k$ to be a subvariety of $N\\times N$ consisting of pairs $$(B,B')=((\\alpha,\\beta),(\\alpha',\\beta')),$$ such that\\\\\n(1)\\quad $\\forall j\\notin\\{k-1,k\\},\\ \\alpha_j=\\alpha'_j,\\textrm{ and }\\beta_j=\\beta'_j.$\\\\\n(2)\\quad The following short sequence is exact\n$$\n\\begin{tikzcd}[column sep=large]\n\t0\\arrow[r]& \\mathbb{C}^k\\arrow[r,\"\\mathrm{out}_k(B')\"]& \\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1}\\arrow[r,\"\\mathrm{in}_k(B)\"]& \\mathbb{C}^k\\arrow[r] & 0\n\\end{tikzcd}\n$$\nMoreover, for each $j$ we fix some volume form $\\mathsf{vol}_j\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^j\\mathbb{C}^j$, and require\n$$\n\t\\mathrm{out}_k(B')(\\mathsf{vol}_k)\\wedge\\mathrm{in}_k(B)^{-1}(\\mathsf{vol}_k)=\\mathsf{vol}_{k-1}\\wedge \\mathsf{vol}_{k+1}\\in\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^{2k}(\\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1}).\n$$\n(Since the short sequence is exact, we may choose any representative of $\\mathrm{in}_k(B)^{-1}(\\mathsf{vol}_k)$.)\\\\\n(3)\n$$\n\t\\mathrm{out}_k(B')\\,\\mathrm{in}_k(B')=\\mathrm{out}_k(B)\\,\\mathrm{in}_k(B)-\\lambda_k\\,\\textrm{Id}_{\\mathbb{C}^{k+1}\\oplus\\mathbb{C}^{k-1}}.\n$$\n\\end{defn}\n\n\\begin{defn}\n\t  As in Proposition-Definition 3.6.2 in \\cite{Wang}, for any simple reflection $s_k\\in W=S_n$, one constructs an automorphism $S_k$ on $N\\sslash H$, such that for all $(B,B')\\in Z_k$,\n\t  $$\n\t  \t\tS_k(\\pi(B))=\\pi(B').\n\t  $$\n\t  (Remark: for a given $B$, with $\\mathrm{in}(B)$ surjective, the choice of $B'$ such that $(B,B')\\in Z_k$ is unique up to an $\\mathrm{SL}_k$ action, see Lemma 3.5.3 in \\cite{Wang}, hence $\\pi(B')$ is well defined.\n\t  This property also uniquely determines $S_k$. Then in that paper, the author verified all the Coxeter relations, hence constructed a Weyl group action on $N\\sslash H$.)\n\\end{defn}\n\n\\begin{thm}[Theorem 4.7.1 in \\cite{Wang}]\n\tThe Weyl group action on $N\\sslash H$ we described above coincides with the Gelfand-Graev action on $\\overline{T^*(\\SL_3\/U)}$ constructed in \\cite{GK}, Theorem 5.2.7.\n\\end{thm}\n\n\\begin{prop}\n\tIn the case of $n=3$, and under the identification $$\\overline{T^*(\\SL_3\/U)}=\\overline{\\mathcal{O}}_\\textrm{min}\\subset\\mathfrak{so}_8,$$ the Gelfand-Graev action coincides with the triality $S_3$-action $\\mathsf{act}$ on $\\mathfrak{so}_8$ restricted on $\\overline{\\mathcal{O}}_\\textrm{min}$.\n\\end{prop}\n\n\\begin{proof}\n\tSince Lie algebra automorphism preserves the minimal orbit, so the triality action can be restricted to an $S_3$ action on $\\overline{\\mathcal{O}}_\\textrm{min}$. It suffices to check for two of the simple transpositions $s_1=(23),s_2=(13)\\in S_3$, the triality action satifies the condition that for each $k\\in\\{1,2\\}$, and $(B,B')\\in Z_k$, we have\n\t$$\n\t\tS_k(\\pi(B))=\\pi(B').\n\t$$\n\tSince the Gelfand-Graev action is uniquely determined by its restriction on the regular semisimple open part, we take $(M,{c},{u},{u}^*)$ to be an element in $\\mathcal{O}_{\\textrm{min}}\\subset\\mathfrak{sl}_3\\oplus\\mathfrak{h}\\oplus(\\mathbb{C}^3)^3\\oplus((\\mathbb{C}^3)^*)^3$ with distinct $c_1, c_2, c_3$.\n\tThen the corresponding isotropic decomposable element in $\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2 {\\left(\\mathbb{C}^3\\oplus\\mathbb{C}\\oplus\\mathbb{C}^*\\oplus(\\mathbb{C}^3)^*\\right)}$ is\n\t$$\n\t\n\t\t\\left(u_3\\oplus 0 \\oplus -(c_3-c_2)\\oplus -u_2^*\\right)\\wedge\\left(\\frac{u_2}{c_3-c_2}\\oplus 1 \\oplus 0 \\oplus -\\frac{u_3^*}{c_3-c_2}\\right).\n\t$$\n\tSo with this choice of representative, we have a lift $B=(\\alpha,\\beta)\\in N$\n$$\n\t\\alpha_1=\n\t\\begin{pmatrix}\n\t\t0\\\\\n\t\tc_3-c_2\n\t\\end{pmatrix},\\quad\n\t\\beta_1=\n\t\\begin{pmatrix}\n\t\t0 & 1\n\t\\end{pmatrix},\\quad\n\t\\alpha_2=\\hspace{-1.5em}\n\t\\vcenter{\\begin{tikzpicture}\\hspace{-1.5em}\n\t\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.2em,minimum height=2.2em)]\n\t\t\t{\n\t\t\t\t\\   & \\    \\\\\n\t\t\t\tu_3 & \\!\\!\\!\\displaystyle\\frac{u_2}{c_3\\!-\\!c_2} \\!\\!\\!  \\\\\n\t\t\t\t\\   & \\     \\\\\n\t\t\t} ;\n\t\t\t\\draw (m-1-1.north west) rectangle (m-3-2.south east);\n\t\t\t\\draw (m-1-1.north east) -- (m-3-1.south east);\t\n\t\\end{tikzpicture}}\\hspace{-36em},\\quad\n\t\\beta_2=\n\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.6em,minimum height=2.75em)]\n\t\t{\n\t\t\t\\quad\\ \\displaystyle\\ \\ -\\frac{u_3^*}{c_3-c_2} \\qquad \\\\\n\t\t\t\\qquad\\ \\ \\ \\ u_2^* \\qquad\\ \\ \\ \\\\\n\t\t} ;\n\t\t\\draw (m-2-1.south west) rectangle (m-1-1.north east);\n\t\t\\draw (m-2-1.north west) -- (m-1-1.south east);\t\n\\end{tikzpicture}}\n\\hspace{-36em}\\qquad.\n$$\n\nFirst, for $k=1$, we have\n\t$$\n\t\\mathrm{out}_1(B)=\n\t\t\\begin{pmatrix}\n\t\t\t0\\\\\n\t\t\tc_3-c_2\n\t\t\\end{pmatrix},\\quad\n\t\\mathrm{in}_1(B)=\n\t\t\\begin{pmatrix}\n\t\t\t0 & 1\n\t\t\\end{pmatrix},\\quad\n\t\\lambda_1=c_3-c_2.\n\t$$\n\tApply the action by $s_1=(23)$, and rewrite the bivector so that the first defining property of $Z_1$ holds.\n\t\\begin{align*}\n\t\t&\\ \\ \\ \\left(u_2\\oplus 0 \\oplus -(c_2-c_3)\\oplus -u_3^*\\right)\\wedge\\left(\\frac{u_3}{c_2-c_3}\\oplus 1 \\oplus 0 \\oplus -\\frac{u_2^*}{c_2-c_3}\\right)\\\\\n\t\t& = \\left(u_3\\oplus (c_2-c_3) \\oplus 0\\oplus -u_2^*\\right)\\wedge\\left(\\frac{u_2}{c_3-c_2}\\oplus 0 \\oplus 1 \\oplus -\\frac{u_3^*}{c_1-c_2}\\right).\n\t\\end{align*}\nSo we have\n$$\n\t\\quad\\ \\mathrm{out}_1(B')=\n\t\\begin{pmatrix}\n\t\t1\\\\\n\t\t0\n\t\\end{pmatrix},\\quad\n\t\\mathrm{in}_1(B')=\\begin{pmatrix}\n\t\t\\,c_2\\!-\\!c_3 & 0\\,\n\t\\end{pmatrix}.\n$$\nThen $(B,B')\\in Z_1$. Hence the triality action of $s_1$ is the same as the Gelfand-Graev action by $S_1$.\n\nNext, for $k=2$, we have\n$$\n\\mathrm{out}_2(B)=\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.2em,minimum height=2em)]\n\t\t{\n\t\t\t\\   & \\    \\\\\n\t\t\tu_3 & \\!\\!\\!\\displaystyle\\frac{u_2}{c_3\\!-\\!c_2} \\!\\!\\!  \\\\\n\t\t\t\\   & \\     \\\\\n\t\t\t0 & \\!\\!\\!1  \\!\\!\\! \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-4-1.north west) -- (m-4-2.north east);\n\t\t\\draw (m-1-1.north east) -- (m-4-1.south east);\t\n\\end{tikzpicture}}\t\n\\hspace{-36em},\\quad\\mathrm{in}_2(B)=\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.6em,minimum height=2.75em)]\n\t\t{\n\t\t\t\\quad\\ \\  \\displaystyle-\\frac{u_3^*}{c_3-c_2} \\qquad  & 0\\\\\n\t\t\t\\qquad\\qquad u_2^* \\ \\ \\ \\qquad\\ \\ \\ & c_2\\!-\\!c_3\\\\\n\t\t} ;\n\t\t\\draw (m-2-1.south west) rectangle (m-1-2.north east);\n\t\t\\draw (m-1-2.north west) -- (m-2-2.south west);\t\n\t\t\\draw (m-2-1.north west) -- (m-2-2.north east);\t\n\\end{tikzpicture}}\n\\hspace{-29.5em},\\quad\\lambda_2=c_1-c_3\\,.\n$$\nApplying the triality action by element $s_2=(13)\\in S_3$, we get $B''$ such that\n$$\n\\mathrm{out}_2(B'')=\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.2em,minimum height=2em)]\n\t\t{\n\t\t\t\\   & \\    \\\\\n\t\t\tu_1 & \\!\\!\\!\\displaystyle\\frac{u_2}{c_1\\!-\\!c_2} \\!\\!\\!  \\\\\n\t\t\t\\   & \\     \\\\\n\t\t\t0 & \\!\\!\\!1  \\!\\!\\! \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-4-1.north west) -- (m-4-2.north east);\n\t\t\\draw (m-1-1.north east) -- (m-4-1.south east);\t\n\\end{tikzpicture}}\t\n\\hspace{-36 em},\\qquad\\mathrm{in}_2(B'')=\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.6em,minimum height=2.75em)]\n\t\t{\n\t\t\t\\quad\\ \\  \\displaystyle-\\frac{u_1^*}{c_1-c_2} \\qquad  & 0\\\\\n\t\t\t\\qquad\\qquad u_2^* \\ \\ \\ \\qquad\\ \\ \\ & c_2\\!-\\!c_1\\\\\n\t\t} ;\n\t\t\\draw (m-2-1.south west) rectangle (m-1-2.north east);\n\t\t\\draw (m-1-2.north west) -- (m-2-2.south west);\t\n\t\t\\draw (m-2-1.north west) -- (m-2-2.north east);\t\n\\end{tikzpicture}}\n\\hspace{-29.5em}.\n$$\nThen we can check\n$$\n\\mathrm{out}_2(B)\\,\\mathrm{in}_2(B) =\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.3em,minimum height=2.5em)]\n\t\t{\n\t\t\t\\ \\hspace{8em}\\ & \\    \\\\\n\t\t\t\\displaystyle\\frac{u_2\\wedge u^*_2-u_3\\wedge u_3^*}{c_3-c_2} & u_2   \\\\\n\t\t\t\\ \\hspace{8em}\\ & \\     \\\\\n\t\t\tu^*_2 & c_2\\!-\\!c_3 \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-3-1.south west) -- (m-3-2.south east);\n\t\t\\draw (m-1-2.north west) -- (m-4-2.south west);\t\n\\end{tikzpicture}}\n\\hspace{-30em}=\\hspace{-1.2em}\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.3em,minimum height=2.5em)]\n\t\t{\n\t\t\t\\ \\hspace{7em}\\ & \\    \\\\\n\t\t\t-M+c_1\\mathrm{Id}_{\\mathbb{C}^3} & u_2   \\\\\n\t\t\t\\ \\hspace{7em}\\ & \\     \\\\\n\t\t\tu^*_2 & c_2\\!-\\!c_3 \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-3-1.south west) -- (m-3-2.south east);\n\t\t\\draw (m-1-2.north west) -- (m-4-2.south west);\t\n\\end{tikzpicture}}\\hspace{-30.5em},\n$$\n\n$$\n\\mathrm{out}_2(B'')\\,\\mathrm{in}_2(B'') =\\hspace{-1.5em}\n\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.3em,minimum height=2.5em)]\n\t\t{\n\t\t\t\\ \\hspace{8em}\\ & \\    \\\\\n\t\t\t\\displaystyle\\frac{u_2\\wedge u^*_2-u_1\\wedge u_1^*}{c_1-c_2} & u_2   \\\\\n\t\t\t\\ \\hspace{8em}\\ & \\     \\\\\n\t\t\tu^*_2 & c_2\\!-\\!c_1 \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-3-1.south west) -- (m-3-2.south east);\n\t\t\\draw (m-1-2.north west) -- (m-4-2.south west);\t\n\\end{tikzpicture}}\n\\hspace{-30em}=\\hspace{-1.2em}\\vcenter{\\begin{tikzpicture}\n\t\t\\matrix (m)[matrix of math nodes, nodes in empty cells, left delimiter={(},right delimiter={)},minimum width=3.3em,minimum height=2.5em)]\n\t\t{\n\t\t\t\\ \\hspace{7em}\\ & \\    \\\\\n\t\t\t-M+c_3\\mathrm{Id}_{\\mathbb{C}^3} & u_2   \\\\\n\t\t\t\\ \\hspace{7em}\\ & \\     \\\\\n\t\t\tu^*_2 & c_2\\!-\\!c_1 \\\\\n\t\t} ;\n\t\t\\draw (m-1-1.north west) rectangle (m-4-2.south east);\n\t\t\\draw (m-3-1.south west) -- (m-3-2.south east);\n\t\t\\draw (m-1-2.north west) -- (m-4-2.south west);\t\n\\end{tikzpicture}}\\hspace{-30.5em}.\n$$\n\n\nSo \n$$\n\\mathrm{out}(B'')\\,\\mathrm{in}(B'')=\\mathrm{out}(B)\\,\\mathrm{in}(B)-\\lambda_2\\,\\mathrm{Id}_{\\mathbb{C}^4}\n$$\nSo $(B,B'')\\in Z_2$.\nThis shows that the Gelfand-Graev action coincides with the triality action.\n\\end{proof}\n\n\n\n\n\\vspace{3em}\n\\section{A $\\mathbb{C}^*$-action on $\\overline{T^*(\\mathrm{SL}_n\/U)}$.}\\label{TDS}\n\nLet $\\mathfrak{b}$ (resp.\\ $\\mathfrak{n}$) be the Lie algebra of $B$ (resp.\\ $U$). Then $\\mathfrak{b}$ is the orthogonal complement of $\\mathfrak{n}$ with respect to the Killing Form, and the cotangent bundle $T^*(\\mathrm{SL}_n\/U)$ can be identified with\n$$\n\\mathrm{SL}_n\\times_U \\mathfrak{b}.\n$$\n\\begin{defn}[Definition 4.3 in \\cite{Wang}]\\label{Xi}\nLet $(\\alpha,\\beta)\\in N_\\textrm{surj}$ as in Definition \\ref{Nsurj}.\nFor each $k$, we choose some matrices of special forms,\n$\n    \\beta^0_k=(\\textbf{0}_{k,1}\\,|\\, \\mathrm{Id}_{\\mathbb{C}^k} )\\in\\Hom(\\mathbb{C}^{k+1},\\mathbb{C}^k).\n$\nChoose some\n$$\n    (g_1,g_2,g_3,\\cdots,g_{n-1},g_n)\\in\\mathrm{SL}_1\\times\\mathrm{SL}_2\\times\\cdots\\times\\mathrm{SL}_{n-1}\\times\\mathrm{SL}_n,\n$$\nsuch that for each $k\\in\\{1,2,3,4,\\cdots,n-1\\}$,\n$$\n    g_{k}\\beta'_kg_{k+1}^{-1}=\\beta^0_k.\n$$\nLet $\\Gamma:\\mathfrak{gl}_n\\rightarrow\\mathfrak{sl}_n$ denote the projection $\\Gamma(X)=X-(1\/n)\\mathrm{Tr}(X)\\mathrm{Id}_{\\mathbb{C}^n}$.\nThen as in equation (4.5) in \\cite{Wang}, we define the following map:\n\\begin{align}\n    \\Xi:N_\\textrm{surj}\/H & \\longrightarrow\\mathrm{SL}_n\\times_U \\mathfrak{b}\\\\\n    [(\\alpha,\\beta)]& \\longmapsto [g_n^{-1},\\Gamma(g_n\\alpha_{n-1}\\beta_{n-1}g_n^{-1})]\\nonumber\n\\end{align}\nThen the map $\\Xi$ is well defined and extends to an isomorphism\n$$\n    \\Xi:N\\sslash H\\rightarrow\\overline{T^*(\\SL_n\/U)}.\n$$\n\\end{defn}\n\nFor each $k\\geq2$, fix the following principal $\\mathfrak{sl}_2$-triple in $\\mathfrak{sl}_k$\n$$\n        e_k={\\begin{pmatrix}\n        0 & 1 \\\\\n        \\ & 0 & \\!1 \\\\\n        \\ & \\ & \\!\\ddots & 1 \\\\\n        \\ & \\ & \\ & 0 & 1 \\\\  \n        \\ & \\ & \\ & \\ & 0 \\\\ \n    \\end{pmatrix}},\\ \\ \n   h_k={\\begin{pmatrix}\n        {k\\!-\\!1} & \\hspace{-1em}0 \\\\\n         \\hspace{-.5em}0 & \\hspace{-1em}{k\\!-\\!3}\\\\\n        \\ & \\ & \\hspace{-1em}\\ddots\\\\\n        \\ & \\ & \\ & \\hspace{-2em}{-(k\\!-\\!3)} & \\hspace{-1em}0 \\\\  \n        \\ & \\ & \\ & \\hspace{-2em}0 & \\hspace{-2em}{-(k\\!-\\!1)} \\\\ \n    \\end{pmatrix}},\\ \\ \n  f_k={\\begin{pmatrix}\n        0 &\\\\\n        1(k\\!-\\!1) & \\hspace{-1.5em}0 \\\\\n        \\ & \\hspace{-2em}2(k\\!-\\!2) \\\\\n        \\ & \\ &\\hspace{-2.5em}\\ddots & \\hspace{-1em}0 & \\\\  \n        \\ & \\ & \\ & \\hspace{-2.5em}{(k\\!-\\!1)1} & 0 \\\\ \n    \\end{pmatrix}}.\n$$\n\n\\begin{defn}\n    Let $\\gamma:\\mathbb{C}^*\\rightarrow \\mathrm{SL}_n$ be the following one-parameter subgroup generated by the element $h_n$ of our fixed\n$\\mathfrak{sl}_2$-triple,\n$$\n    \\gamma(z)\\coloneqq z^{h_n}={\\begin{pmatrix}\n        z^{n\\!-\\!1} & 0 \\\\\n         0 & z^{n\\!-\\!3}\\\\\n        \\ & \\ & \\ddots\\\\\n        \\ & \\ & \\ & z^{-(n\\!-\\!3)} & 0 \\\\  \n        \\ & \\ & \\ & 0 & z^{-(n\\!-\\!1)}\n    \\end{pmatrix}}.\n$$\nLet $z\\in\\mathbb{C}^*, [g,X]\\in \\mathrm{SL}_n\\times_U\\mathfrak{b}$. Define a $\\mathbb{C}^*$-action on $T^*(\\mathrm{SL}_n\/U)$ by\n\\begin{equation}\\label{Cstar}\n        z.[g,X]=[g.\\gamma(z),z^2\\, \\mathrm{Ad}_{\\gamma(z^{-1})}X].\n\\end{equation}\n\\end{defn}\n\n\\begin{prop}\n    In terms of the construction of $\\overline{T^*(\\SL_n\/U)}$ in section \\ref{dq}, this $\\mathbb{C}^*$-action (\\ref{Cstar}) is given by \n    \\begin{equation}\n        z.\\bigoplus_{k}(\\alpha_k,\\beta_k)\\coloneqq\\bigoplus_{k}(z\\alpha_k,z\\beta_k)\n    \\end{equation}\n\\end{prop}\n\\begin{proof}\nLet $[g,X]\\in \\mathrm{SL}_n\\times_U\\mathfrak{b}$. Take a representative\n$$\n    (g,X)\\in\\mathrm{SL}_n\\times\\mathfrak{b}.\n$$\nWe are going to construct an $(\\alpha,\\beta)\\coloneqq\\bigoplus_{k=1}^{n-1}(\\alpha_k,\\beta_k)\\in N$ such that\n$$\\Xi(\\alpha,\\beta)=[g,X].$$ \nLet $\\iota:\\mathbb{C}^{n-1}\\hookrightarrow\\mathbb{C}^n$ be the embedding of $\\mathbb{C}^{n-1}$ into the last $(n-1)$ coordinates of $\\mathbb{C}^n$, i.e.\n$$\n    \\iota(z_1,z_2,\\cdots,z_{n-1})=(0,z_1,z_2,\\cdots,z_{n-1})\n$$\nDefine\n$$\n    \\alpha_{n-1}=g\\circ(X-x_{11}\\mathrm{Id}_{\\mathbb{C}^n})\\circ\\iota,\n$$\nwhere $x_{11}$ is the first row and first column entry of the matrix $X$.\nDefine\n$$\n    \\beta_{n-1}=\\beta^0_{n-1}\\circ g^{-1},\n$$\nand for each $k\\in\\{1,2,3,\\cdots,n-2\\}$, define\n$$\n    \\beta_k=\\beta^0_k.\n$$\nAnd all the linear maps $\\alpha_1,\\alpha_2,\\cdots,\\alpha_{n-2}$ are uniquely determined by $\\alpha_{n-1},\\beta_1,\\beta_2,\\cdots,\\beta_{n-1}$ and the condition (\\ref{N}). We check\n\\begin{align*}\n    \\Xi(\\alpha,\\beta)& =[g,\\Gamma(g^{-1}\\alpha_{n-1}\\beta_{n-1}g)]\\\\\n    & =[g,\\Gamma[g^{-1}g(X-x_{11}\\mathrm{Id}_{\\mathbb{C}^n})\\circ\\iota\\circ \\beta^0_{n-1}g^{-1}g]\\\\\n        & =[g,\\Gamma((X-x_{11}\\mathrm{Id}_{\\mathbb{C}^n})\\circ\\mathrm{diag}(0,1,1,\\cdots,1))]\\\\\n        & =[g,\\Gamma(X-x_{11}\\mathrm{Id}_{\\mathbb{C}^n})]=[g,X].\n\\end{align*}\n\nLet $z\\in\\mathbb{C}^*$, and define\n$$\n    (\\alpha',\\beta')=\\bigoplus_{k}(\\alpha'_k,\\beta'_k)\\coloneqq\\bigoplus_{k}(z\\alpha_k,z\\beta_k).\n$$\nTake\n$$\n    (g_1,g_2,g_3,\\cdots,g_{n-1},g_n)\\coloneqq(1,z^{-h_2},z^{-h_3},\\cdots,z^{-h_{n-1}},\\gamma(z)^{-1}g^{-1})\\in\\mathrm{SL}_1\\times\\mathrm{SL}_2\\times\\cdots\\times\\mathrm{SL}_{n-1}\\times\\mathrm{SL}_n.\n$$\nThen for each $k\\in\\{1,2,3,4,\\cdots,n-1\\}$\n$$\n    g_{k}\\beta'_kg_{k+1}^{-1}=\\beta^0_k.\n$$\nSo by Definition \\ref{Xi} again,\n$$\n        \\Xi(\\alpha',\\beta')=[g_n^{-1},\\Gamma(g_n\\alpha'_{n-1}\\beta'_{n-1}g_n^{-1})]=[g_n^{-1},z^2\\mathrm{Ad}_{\\gamma(z^{-1})}\\Gamma(g^{-1}\\alpha_{n-1}\\beta_{n-1}g)]=[g.\\gamma(z),z^2\\,\\mathrm{Ad}_{\\gamma(z^{-1})}X]\n$$\n\n\\end{proof}\nIn the special case of $n=3$, applying the identification $\\overline{T^*(\\SL_3\/U)}=\\overline{\\mathcal{O}}_\\textrm{min}$ in \\textrm{Corollary \\ref{TO}}, we have\n\\begin{cor}\n    The above $\\mathbb{C}^*$-action (\\ref{Cstar}) on $\\overline{T^*(\\SL_3\/U)}$ corresponds to the action of $z^2$ for the natural \\mbox{$\\mathbb{C}^*$-action on $\\overline{\\mathcal{O}}_\\textrm{min}$.}\n\\end{cor}\n\n\n\n\\section{Appendix: a Chevalley basis of $\\mathfrak{so}_8={\\scalebox{0.8}{\\raisebox{0.4ex}{$\\bigwedge$}}^2(\\mathbb{C}^8)}$.}\\label{appendix}\n\n\n\\begin{align*}\n\t& X_{\\alpha_1}=e_1\\wedge e_7 \n\t&& Y_{-\\alpha_1}=e_2\\wedge e_8\\\\\n\t& X_{\\alpha_2}=e_2\\wedge e_6\n\t&& Y_{-\\alpha_2}=e_3\\wedge e_7\\\\\n\t& X_{\\alpha_3}=e_3\\wedge e_5\n\t&& Y_{-\\alpha_3}=e_4\\wedge e_6\\\\\n\t& X_{\\alpha_4}=e_3\\wedge e_4\n\t&& Y_{-\\alpha_4}=e_5\\wedge e_6\\\\\n\t& X_{\\alpha_1+\\alpha_2}=e_1\\wedge e_6 \n\t&& Y_{-\\alpha_1-\\alpha_2}=e_3\\wedge e_8\\\\\n\t& X_{\\alpha_2+\\alpha_3}=e_2\\wedge e_5 \n\t&& Y_{-\\alpha_2-\\alpha_3}=e_4\\wedge e_7\\\\\n\t& X_{\\alpha_2+\\alpha_4}=e_2\\wedge e_4\n\t&& Y_{-\\alpha_2-\\alpha_4}=e_5\\wedge e_7\\\\\n\t& X_{\\alpha_2+\\alpha_3+\\alpha_4}=e_2\\wedge e_3\\hspace{-3em}\n\t&& Y_{-\\alpha_2-\\alpha_3-\\alpha_4}=e_6\\wedge e_7\\\\\n\t& X_{\\alpha_1+\\alpha_2+\\alpha_4}=e_1\\wedge e_4\\hspace{-3em} \n\t&& Y_{-\\alpha_1-\\alpha_2-\\alpha_4}=e_5\\wedge e_8\\\\\n\t& X_{\\alpha_1+\\alpha_2+\\alpha_3}=e_1\\wedge e_5\\hspace{-3em} \n\t&& Y_{-\\alpha_1-\\alpha_2-\\alpha_3}=e_4\\wedge e_8\\\\\n\t& X_{\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4}=e_1\\wedge e_3\\hspace{-3em}\n\t&& Y_{-\\alpha_1-\\alpha_2-\\alpha_3-\\alpha_4}=e_6\\wedge e_8\\\\\n\t& X_{\\alpha_1+2\\alpha_2+\\alpha_3+\\alpha_4}=e_1\\wedge e_2\\hspace{-3em}\n\t&& Y_{-\\alpha_1-2\\alpha_2-\\alpha_3-\\alpha_4}=e_7\\wedge e_8\\\\\n\t& H_{\\alpha_1}=e_1\\wedge e_8-e_2\\wedge e_7\\hspace{-3em}\n\t&& H_{\\alpha_3}=e_3\\wedge e_6-e_4\\wedge e_5\\\\\n    &  H_{\\alpha_2}=e_2\\wedge e_7-e_3\\wedge e_6\\hspace{-3em}\n    && H_{\\alpha_4}=e_3\\wedge e_6+e_4\\wedge e_5\n\\end{align*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{2em}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nA covering of a topological space $X$ is said to be \\emph{categorical} if every set in the covering is open and contractible in $X$.\nThat is, the inclusion map of each set into $X$ is nullhomotopic.\nThe \\emph{Lusternik-Schnirelmann category} (or simply \\emph{category}) $\\cat(X)$ of $X$ is the smallest integer $k$\nsuch that $X$ admits a categorical covering by $k+1$ open sets $\\{U_0,\\ldots,U_k\\}$. \n\nLusternik-Schnirelmann category and related invariants have been computed for polyhedral products of the form $(X,\\ast)^K$ \nfor certain nice spaces $X$ in~\\cite{MR2457428,2015arXiv150107474G}, \nthough the methods there do not extend to moment-angle complexes $\\mathcal Z_K=(D^2,\\ensuremath{\\partial} D^2)^K$ for several reasons.\nFor example, the contractibility of the disk $D^2$ does not give nice lower bounds in terms of the dimension of $K$. \nIndeed, there is a fairly large literature that is focused on determining those $K$ for which the moment-angle complex $\\mathcal Z_K$ has category $1$ \n(c.f.~\\cite{2014arXiv1409.4462B,2015arXiv151005283B,MR3461047,MR3084441,MR2321037,2014arXiv1412.4866I,2013arXiv1306.6221I,BerglundRational}).\n\nWe will mostly be interested in moment-angle complexes $\\mathcal Z_K$ over triangulated spheres $K$.\nThese are known as \\emph{moment-angle manifolds} since it is here that $\\mathcal Z_K$ takes the form of a topological manifold. \nMoment-angle complexes of this form have generated a lot of interest due to their connections to quasitoric manifolds in toric topology \nand intersection of quadrics in complex geometry, amongst other things.\nTheir topology and cohomology is, however, very intricate, with many questions remaining open even for low dimensional $K$ \n(see for example~\\cite{MR3363157,MR2330154,MR3073929,MR1104531,MR531977,MR2285318,2015arXiv150904302L,2015arXiv151007778L}). \nNone-the-less, we will characterize those triangulations of $d$-spheres where $d\\leq 2$ for which $\\mathcal Z_K$ has a given category,\nas well as certain higher dimensional spheres built up via join, connected sum, and vertex doubling operations.\n\nOur motivation for doing this is a combinatorial and algebraic characterization of Golod complexes $K$ and $co$-$H$-space \n(category $\\leq 1$) moment-angle complexes $\\mathcal Z_K$ given in~\\cite{MR3461047} in the case of flag complexes $K$.\nThe authors there showed that both of these concepts are equivalent, and moreover, that they both coincide with chordality \nof the $1$-skeleton of $K$ and the triviality of the multiplication on $\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$\nfor $R=\\mb Z$ or $R$ any field, where $R[K]$ is the Stanley-Reisner ring of $K$ over $R$.\nAn interesting algebraic consequence of this from the perspective of commutative algebra was that triviality of the multiplication on \n$\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$ implies that all higher Massey products are also trivial, at the very least when $K$ is flag.\n\\footnote{A theorem in~\\cite{MR2344344} claims this is true for all $K$, but a recent paper~\\cite{2015arXiv151104883K} provides a simple counter-example.}  \nThis depended on the general fact that the cohomology ring of a space of category $\\leq 1$ has trivial multiplication and Massey products vanish~\\cite{MR0231375,MR0356037}.\nIt is natural to ask what the corresponding statement is for spaces with larger category, \nmore so, if the characterization for Golod flag complexes in~\\cite{MR3461047} can be generalized in this sense. \nAn answer to the first question was given by Rudyak in~\\cite{MR1644063}, which inspires the following.\n\\begin{definition}\nA simplicial complex $K$ on vertex set $[n]$ is $m$-\\emph{Golod} over $R$ if\n\\begin{itemize}\n\\item[(1)] $\\Nill(\\mathrm{Tor}_{R[v_1,\\ldots,v_n]}(R[K],R))\\leq m+1$;\n\\item[(2)] Massey products $\\vbr{v_1,\\ldots,v_k}$ vanish in $\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$ \nwhenever $v_i=a_1\\cdots a_{m_i}$ and $v_j=b_1\\cdots b_{m_j}$, and $m_i+m_j>m$ for some odd $i$ and even $j$ and\n$a_s,b_t\\in\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{proposition}\nIf $\\cat(\\mathcal Z_K)\\leq m$, then $K$ is $m$-Golod.~$\\hfill\\square$ \n\\end{proposition}\n\nHere the \\emph{nilpotency} $\\Nill A$ of a graded algebra $A$ is the smallest integer $k$ such that all length $k$ products in the \npositive degree part $A^+$ vanish. \nNotice that $K$ is $(m+1)$-Golod whenever it is $m$-Golod, and $1$-Golodness of $K$ coincides with the classical notion of Golodness~\\cite{MR0138667},\nnamely, that all products and (higher) Massey products are trivial in $\\mathrm{Tor}^+_{R[v_1,\\ldots,v_n]}(R[K],R)$. \nAll of this can be restated equivalently in terms of the cohomology of $\\mathcal Z_K$ due to an isomorphism of graded commutative algebras\n$H^*(\\mathcal Z_K;R)\\cong\\mathrm{Tor}_{R[v_1,\\ldots,v_n]}(R[K],R)$ when $R$ is a field or $\\ensuremath{\\mathbb{Z}}$~\\cite{MR2117435}.\nWe will consider (co)homology with integer coefficients. \n\n\\begin{theorem}\n\\label{TM0}\nIf $K$ is $k$-neighbourly, $\\cat(\\mathcal Z_K)\\leq \\frac{1+\\dim K}{k}$ and $K$ is $\\paren{\\frac{1+\\dim K}{k}}$-Golod.~$\\hfill\\square$\n\\end{theorem}\n\n\n\\begin{theorem}\n\\label{TM1}\nIf $K$ is any $d$-sphere for $d\\leq 2$, or (under a few conditions) built up as a connected sum of joins of such spheres,\nthen the following are equivalent: (a) $\\cat(\\mathcal Z_K)\\leq k$; (b) $K$ is $k$-Golod; \n(c) length $k+1$ cup products of positive degree elements in $H^*(\\mathcal Z_K)$ vanish;\n(d) there does not exist a spherical filtration of full subcomplexes of $K$ of length more than $k$. Moreover, $k\\leq d+1$.~$\\hfill\\square$\n\\end{theorem}\n\nFor instance, $k$ is precisely $3$ when $K$ is the boundary of the dual of a \\emph{fullerene} $P$. \n\n\\begin{theorem}\nFor fullerences $P$, $\\cat(\\z{\\ensuremath{\\partial} P^\\ast})=3$ and $\\ensuremath{\\partial} P^\\ast$ is $3$-Golod.\nIn particular, all Massey products consisting of decomposable elements in $H^+(\\z{\\ensuremath{\\partial} P^\\ast})$ must vanish.~$\\hfill\\square$\n\\end{theorem}\n\nApplying \\emph{vertex doubling operations}, the range of spheres in Theorem~\\ref{TM1} can be extended using the following.\n\n\n\\begin{theorem}\n\\label{TM2}\nIf $K(J)$ is the simplicial wedge of $K$ for some integer sequence $J=(j_1,\\ldots,j_n)$, then $\\cat(\\z{K(J)})\\leq \\cat(\\mathcal Z_K)$.~$\\hfill\\square$\n\\end{theorem}\n\nMore general bounds for $\\cat(\\mathcal Z_K)$ will be given along the way,\nfor example, when $K$ is formed by gluing simplicial complexes along common full subcomplexes.  \nWe remark that many of the results in this paper extend to polyhedral products of the form $(Cone(X),X)^K$ in place of $\\mathcal Z_K=(D^2,S^1)^K$.\n\n\n\\section{Preliminary}\n\nRecall the following concepts from~\\cite{MR516214,MR1990857,MR0004108,MR0229240}. \nThe \\emph{geometric category} $\\gcat(X)$ of a space $X$ is the smallest integer $k$ such that $X$ admits\na categorical covering $\\{U_0,\\ldots,U_k\\}$ of $X$ with each $U_i$ contractible (in itself),\nand the \\emph{category} $\\cat(f)$ of a map $f\\colon\\seqm{X}{}{Y}$ is the smallest $k$ such that $X$ admits\nan open covering $\\{V_0,\\ldots,V_k\\}$ such that $f$ restricts to a nullhomotopic map on each $V_i$.\nIt is easy to see that $\\cat(X)=\\cat(\\ensuremath{\\mathbbm{1}}\\colon\\seqm{X}{}{X})$, $\\cat(f)\\leq\\min\\{\\cat(X),\\cat(Y)\\}$, $\\cat(h\\circ h')\\leq \\cat(h')$.\nFor path-connected paracompact spaces,\n$$\n\\cat(f\\times g)\\leq \\cat(f)+\\cat(g),\n$$ \nwhich follows from the also well-known fact that $\\cat(X\\times Y)\\leq \\cat(X)+\\cat(Y)$ together with the preceding inequalities.\nUnlike $\\cat()$, $\\gcat()$ is not a homotopy invariant,\nthough one can obtain a homotopy invariant from $\\gcat()$ by defining the \\emph{strong category}\n$$\n\\Cat(X)=\\min\\cset{\\gcat(Y)}{Y\\simeq X}.\n$$\nIn fact, the strong category satisfies $\\Cat(X)-1\\leq \\cat(X)\\leq\\Cat(X)\\leq \\gcat(X)$.\nWe shall let $\\cuplen(X)=\\Nill H^*(X)-1$ denote the length of the longest non-zero cup product of positive degree elements in $H^*(X)$. \nThe main use of this is the classical lower bound \n$$\n\\cuplen(X)\\leq \\cat(X). \n$$\n\n\n\n\\subsection{Some General Bounds}\n\nWe begin by giving upper bounds for the Lusternik-Schnirelmann category over some general spaces.\n\n\n\\begin{lemma}\n\\label{LOpen}\nLet $A$ be a subcomplex of $X$ and $S$ an open subset of $A$. \nThen $S$ is a deformation retract of an open subset $U$ of $X$ such that $U\\cap A=S$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\mc I_j$ be an index set for the $j$-cells $e_\\alpha^j$ of $X-A$, $\\Phi_\\alpha\\colon\\seqm{D^j}{}{X}$ its characteristic map,\nand $\\phi_\\alpha\\colon\\ensuremath{\\partial} D^j\\incladdr{}{D^j}\\mapaddr{\\Phi_\\alpha}{X}$ its attaching map. \nGiven a subset $B\\subseteq X$, let $V_{\\alpha,B}$ be the image of \n$\\phi_\\alpha^{-1}(B)\\times[0,\\frac{1}{2})\\subseteq D^j\\cong (\\ensuremath{\\partial} D^j\\times [0,1])\/(\\ensuremath{\\partial} D^j\\times\\{1\\})$ under $\\Phi_\\alpha$.\nNotice $V_{\\alpha,B}$ deformation retracts onto a subspace of $\\phi_\\alpha(\\ensuremath{\\partial} D^j)\\cap B$,\nand if $B\\cap e_\\alpha^j=\\emptyset$, $B\\cup V_{\\alpha,B}$ deformation retracts onto $B$.\n\nConstruct $R_{i+1}\\subseteq X$ such that $R_i\\subseteq R_{i+1}$, $R_i$ is a deformation retract of $R_{i+1}$, \nand $R_i\\cap X^{\\vbr{i}}$ is open in the $i$-skeleton $X^{\\vbr{i}}$, \nby letting $R_0=S$ and $R_{i+1}=R_i\\cup \\bigcup_{\\alpha\\in\\mc I_{i+1}} V_{\\alpha,S}$.\nThen $U=\\bigcup_{i\\geq 0} R_i$ is open in $X$, deformation retracts onto $S$, and $U\\cap A=S$. \n\n\\end{proof}\n\n\n\n\\begin{lemma}\n\\label{LFiltration}\n\nGiven a filtration $X_0\\subseteq\\cdots\\subseteq X_m=X$ of subcomplexes of a $CW$-complex $X$,\nsuppose $X_{i+1}-X_i$ is contractible in $X$ for each $i$. Then $\\cat(X)\\leq \\cat(\\incl{X_0}{}{X})+m\\leq \\cat(X_0)+m$.\n\n\\end{lemma}\n\n\\begin{proof}\n\nLet $k=\\cat(\\incl{X_0}{}{X})$, and $\\{U_0,\\ldots, U_k\\}$ be a categorical cover of the inclusion \\incl{X_0}{}{X}.\nNote $V_i=X_{i+1}-X_i$ is open in $X_{i+1}$ since the subcomplex $X_i$ is closed in $X_{i+1}$. \nThen iterating Lemma~\\ref{LOpen}, we have open subsets $\\bar U_i$ and $\\bar V_i$ that deformation retract onto\neach $U_i$ and $V_i$ respectively. Since the $U_i$'s and $V_i$'s cover $X$ and are contractible in $X$,\nso do the $\\bar U_i$'s and $\\bar V_i$'s, thus they form a categorical cover of $X$.\n\n\\end{proof}\n\n\n\nFor any spaces $X$ and $Y$, and a fixed basepoint $\\ast\\in X$, \nwe let $X\\rtimes Y=(X\\times Y)\/(\\ast\\times Y)$ denote the \\emph{right half-smash} of $X$ and $Y$,\nand $Y\\ltimes X=(Y\\times X)\/(Y\\times\\ast)$ the \\emph{left half-smash}.\n\n\\begin{lemma}\n\\label{LHalfSmash}\nIf $X$ and $Y$ are $CW$-complexes and $X$ is path-connected, then $\\cat(X\\rtimes Y)=\\cat(X)$. \n\\end{lemma}\n\n\\begin{proof}\n\nLet  $\\tilde X$ be given by attaching the interval $[0,1]$ to $X$ \nby identifying $0\\in [0,1]$ with the basepoint $\\ast\\in X$, and fix $1\\in\\tilde X$ to be the basepoint.\nGiven $k=\\cat(\\tilde X)$ and $\\{U_0,\\ldots,U_k\\}$ a categorical cover of $\\tilde X$,\ntake the open cover $\\{U_0\\rtimes Y,\\ldots, U_k\\rtimes Y\\}$ of $\\tilde X\\rtimes Y=(\\tilde X\\times Y)\/(1\\times Y)$\n(here $U_i\\rtimes Y=U_i\\times Y$ if $1\\nin U_i$).\nNotice that the contractions of each $U_i$ in $\\tilde X$ can be taken so that $1$ remains fixed if $1\\in U_i$.\nIf $U_i$ contracts to a point $b_i$ in $\\tilde X$, $U_i\\rtimes Y$ deforms onto $\\{b_i\\}\\rtimes Y$ in $\\tilde X\\rtimes Y$,\nwhich in turn contracts to the basepoint in $\\tilde X\\rtimes Y$ by homotoping the coordinate $b_i$ to $1$.\nTherefore $\\cat(\\tilde X\\rtimes Y)\\leq k$, and we have $\\cat(X\\rtimes Y)\\leq k$ since $X\\simeq\\tilde X$ \nand $X\\rtimes Y\\simeq \\tilde X\\rtimes Y$. Moreover, $\\cat(X\\rtimes Y)\\geq k$ since $X$ is a retract of $X\\rtimes Y$.\n\n\\end{proof}\n\n\nLet $\\mc S$ be $m$ copies of the interval $[0,1]$ glued together at the endpoints $1$ in some order.\nGiven a collection of maps \\seqm{X}{f_i}{Y_i} for $i=1,\\ldots,m$, the \\emph{homotopy pushout} $P$ of the maps $f_i$ \nis the $m$-fold mapping cylinder \n$$\nP=\\paren{Y_1\\coprod\\cdots\\coprod Y_m\\coprod (X\\times\\mc S)}\/\\sim\n$$ \nunder the identification $(x,t)\\sim f_i(x)$ whenever $t$ is in the $i^{th}$ copy of $[0,1]$ in $\\mc S$ and $t=0$.  \n\n\n\n\n\\begin{lemma}\n\\label{LGT}\nFix $m\\geq 2$. For $i=1,\\ldots, m$, \nlet $A_i$ and $C_i$ be basepointed $CW$-complexes, $B_i=\\prod_{j\\neq i} A_j$, and $E$ be a contractible space.\nSuppose \\seqm{A_i\\times E}{f_i}{C_i} are nullhomotopic maps,\nand $P$ is the homotopy pushout of the maps \n\\seqm{A_i\\times E\\times B_i}{f_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i} for $i=1,\\ldots, m$.\nThen \n$$\n\\cat(P)\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}. \n$$\n\\end{lemma}\n\n\n\\begin{proof}\n\nWe proceed by induction on $m$. Start with $m=2$. By Lemma~$7.1$ in~\\cite{MR3084441}, there is a splitting \n$P\\simeq (\\Sigma A_1\\wedge A_2)\\vee (C_1\\rtimes A_2)\\vee(C_2\\rtimes A_1)$. Thus using Lemma~\\ref{LHalfSmash}, \n$$\n\\cat(P)=\\max\\{\\cat(\\Sigma A_1\\wedge A_2),\\cat(C_1\\rtimes A_2),\\cat(C_2\\rtimes A_1)\\}=\\max\\{1,\\cat(C_1),\\cat(C_2)\\}.\n$$\nThe statement holds when $m=2$.\n\n\nTake $\\mc B_0=\\ast$, $\\mc B_\\ell=\\prod_{j\\leq \\ell} A_j$, $B'_i=\\prod_{j\\neq i,j<m} A_j$, and $B_i$ as basepointed subspaces of $\\mc B=\\prod_j A_j$.\nLet $P'$ be the homotopy pushout of $f_i\\times\\ensuremath{\\mathbbm{1}}_{B'_i}$ for $i=1,\\ldots, m-1$ (these are all maps from $E\\times B_m=E\\times \\mc B_{m-1}$).\nSuppose the lemma holds whenever $m<m'$ for some $m'>2$. Let $m=m'$. Then $\\cat(P')\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_{m-1})\\}$. \nNotice that $P$ is the homotopy pushout of $f_m\\times\\ensuremath{\\mathbbm{1}}_{B_m}$ and the inclusion \\seqm{A_m\\times E\\times B_m}{\\ensuremath{\\mathbbm{1}}_{A_m}\\times g}{A_m\\times P'},\nwhere $g$ is the inclusion $W_{m-1}\\subset P'$, and $W_\\ell=E\\times \\mc B_\\ell\\times\\{1\\}$. \nWe can deform $W_\\ell$ into $W_{\\ell-1}$ in $P'$ as follows. First deform $W_\\ell$ onto $f_\\ell(A_\\ell\\times E)\\times \\mc B_{\\ell-1}$ by moving it\ndown the mapping cylinder $M=((E\\times B_m\\times [0,1])\\coprod (C_\\ell\\times B_\\ell))\/\\sim\\,$ of $P'$ and onto the base $C_\\ell\\times B_\\ell$,\nthen deform it onto $\\ast\\times \\mc B_{\\ell-1}$ in $C_\\ell\\times B_\\ell$ using the nullhomotopy of $f_\\ell$. Finally, \nmove $\\mc B_{\\ell-1}$ back up towards the top of the mapping cylinder $M$ and into $W_{\\ell-1}$. Composing these deformations for $\\ell=m-1,\\ldots,1$\ngives a contraction in $P'$ of $W_{m-1}$ to a point. Thus, $g$ is nullhomotopic, as is $f_m$. \nSince the lemma holds for the base case $m=2$, $\\cat(P)=\\max\\{1,\\cat(P'),\\cat(C_m)\\}\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}$.\n\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{LGluing}\nFix $m\\geq 2$, and for $i=1,\\ldots, m$, let $A_i$, $C_i$, $E$ be basepointed $CW$-complexes, $E$ is path-connected, and let $B_i=\\prod_{j\\neq i} A_j$.\nSuppose \\seqm{A_i\\times E}{f_i}{C_i} are maps such that the restriction $(f_i)_{|A_i\\times\\ast}$ of $f_i$ to $A_i\\times\\ast$ is nullhomotopic,\nand $P$ is the homotopy pushout of the maps \\seqm{A_i\\times E\\times B_i}{f_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i} for $i=1,\\ldots, m$.\nThen \n$$\n\\cat(P)\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}+\\Cat(E). \n$$\n\nMoreover, if each $C_i\\times B_i$ is a subcomplex of some $CW$-complex $X_i$ such that \n$X_i-C_i\\times B_i$ is contractible in $X_i$, \nand $P'$ is the homotopy pushout of the maps $A_i\\times E\\times B_i\\mapaddr{f_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i}\\incladdr{}{X_i}$\nfor $i=1,\\ldots, m$, then also\n$$\n\\cat(P')\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}+\\Cat(E).\n$$ \n\\end{lemma}\n\n\n\\begin{proof}[Proof (part $1$)]\nLet $\\mc B=A_1\\times\\cdots\\times A_m$, and $\\mc D=\\coprod_{i=1,\\ldots,m}(C_i\\times B_i)$,\nand let $\\mc S_t$ for $t<1$ be $m$ copies of the interval $[t,1]$ glued together at the endpoints $1$, \nand $\\mc S'_t$ be its interior, namely, $m$ copies of $(t,1]$ glued at $1$.\n\nLet $k=\\Cat(E)$ and take $E'\\simeq E$ to be such that $k=\\gcat(E')$.\nThen $P$ is homotopy equivalent to the homotopy pushout $Q$ of the maps\n\\seqm{A_i\\times E'\\times B_i}{f'_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i} for $i=1,\\ldots, m$, \nwhere $\\seqm{A_i\\times E'}{f'_i}{C_i}$ is the composite of $f_i$ with the homotopy equivalence\n$\\seqm{A_i\\times E'}{\\ensuremath{\\mathbbm{1}}_{A_i}\\times\\simeq}{A_i\\times E}$.\nSince $E$ is path-connected and $\\gcat()$ is unaffected by attaching an interval $[0,1]$ to a space,\nwe may assume that the homotopy equivalence \\seqm{E'}{\\simeq}{E} is basepointed for some $\\ast\\in E'$. \n\nLet $U_0,\\ldots,U_k$ be an open cover of $E'$ with each $U_i$ a contractible subspace.\nTake $Q_j$ to be the homotopy pushout of \n\\seqm{A_i\\times U_j\\times B_i}{g_{i,j}\\times\\ensuremath{\\mathbbm{1}}_{B_i}}{C_i\\times B_i} for $i=1,\\ldots, m$, \nwhere $g_{i,j}$ is the restriction of $f'_i$ to $A_i\\times U_j$, and let\n$V_j=Q_j-\\mc D\\,\\cong\\,U_j\\times\\mc B\\times \\mc S'_0$. \nSince $g_{i,j}\\times\\ensuremath{\\mathbbm{1}}_{B_i}$ restricts $f'_i\\times\\ensuremath{\\mathbbm{1}}_{B_i}$, \n$Q_j$ is a subspace of $Q$ and $V_j$ is open in $Q$.\nMoreover, we may contract $V_j$ in $Q$ to a point as follows.\nLet $\\mc B_0=\\ast$ and $\\mc B_\\ell=\\prod_{i\\leq\\ell} A_i\\,\\subseteq\\,\\mc B$,\nand take the subspace $W_\\ell=\\ast\\times\\mc B_\\ell\\times\\{1\\}$ of $E'\\times\\mc B\\times\\mc S'_0\\subset Q$.\nWe can deform $W_\\ell$ into $W_{\\ell-1}$ in $Q$, first by deforming $W_\\ell$ onto \n$f'_\\ell(A_\\ell\\times\\ast)\\times\\mc B_{\\ell-1}$ by moving it down the mapping cylinder \n$M=((E'\\times \\mc B\\times [0,1])\\coprod (C_\\ell\\times B_\\ell))\/\\sim\\,$ of $Q$ and onto $C_\\ell\\times B_\\ell$,\nthen deforming it onto $\\ast\\times \\mc B_{\\ell-1}$ in $C_\\ell\\times B_\\ell$ \nusing the nullhomotopy of $(f'_\\ell)_{|A_\\ell\\times\\ast}$, and finally, \nmoving $\\mc B_{\\ell-1}$ back up towards the top of the mapping cylinder $M$ and into $E'\\times \\mc B\\times\\{1\\}$.\nComposing these deformations of $W_\\ell$ into $W_{\\ell-1}$ in $Q$ for $\\ell=m,m-1,\\ldots,1$,\nand deforming $V_j$ onto $W_m$ using contractibility of $U_j$ and $\\mc S'_0$ (onto $1$),\ngives our contraction of $V_j$ in $Q$ to a point.\n\nAssume $\\ast\\in U_0$. \nSince $U_0$ is contractible and $(g_{i,0})_{|A_i\\times\\ast}=(f'_i)_{|A_i\\times\\ast}$ is nullhomotopic,\n$g_{i,0}$ is also nullhomotopic. Lemma~\\ref{LGT} then applies to $Q_0$, namely, we have \n$$\n\\cat(Q_0)\\leq \\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}.\n$$\nLet $\\mc R=\\mc S'_0-\\mc S_{\\frac{1}{2}}\\,\\cong\\,\\coprod_{i=1,\\ldots,m}(0,\\frac{1}{2})$\nand $\\bar{\\mc R}=\\mc S_0-\\mc S_{\\frac{1}{2}}\\,\\cong\\,\\coprod_{i=1,\\ldots,m}[0,\\frac{1}{2})$,\nand consider the open subspace $Q'_0 = Q_0\\cup(E'\\times \\mc B\\times\\mc R)$ of $Q$.\nNotice $Q'_0$ deformation retracts in the weak sense onto $Q_0$ by deformation retracting the subspace of $Q_0$\n$$\n\\paren{(E'\\times \\mc B\\times\\bar{\\mc R})\\,\\coprod\\,\\mc D\\,}\/\\sim\n$$ \nonto $\\mc D$, this being done by contracting each copy of $[0,\\frac{1}{2})$ in the factor $\\bar{\\mc R}$ to $0$, \nat the same time expanding $(U_0\\times\\mc B)\\times \\mc S_{\\frac{1}{2}}$ in $Q'_0$ by expanding each copy of $[\\frac{1}{2},1]$ \nin the factor $\\mc S_{\\frac{1}{2}}$ outwards to $[0,1]$. Then $\\cat(Q'_0)=\\cat(Q_0)$.\nSo take $k'=\\max\\{1,\\cat(C_1),\\ldots,\\cat(C_m)\\}$ and $\\{U'_0,\\ldots,U'_{k'}\\}$ to be a categorical cover for $Q'_0$.\nNotice that $U'_i$ is open in $Q$ since $Q'_0$ is, and $Q=\\bigcup_{j=0}^n Q_j = Q'_0\\cup \\bigcup_{j=1}^n V_j$.\nAs each $V_j$ is open and contractible in $Q$, \nthen $\\{U'_0,\\ldots,U'_{k'},V_1,\\ldots,V_k\\}$ is a categorical cover of $Q$. \nTherefore $\\cat(P)=\\cat(Q)\\leq k'+k$.\n\n\\end{proof}\n\n\n \n\\begin{proof}[Proof (part $2$)]\n\nSince $C_i\\times B_i$ is a subcomplex of $X_i$, $P$ is a subspace of $P'$ with \n$$\nP'-P=\\cdunion{i=1,\\ldots,m}{}{(X_i-C_i\\times B_i)},\n$$ \nso $P'-P$ is open and contractible in $P'$. Notice each $V_j$ is an open (and contractible) subset of $P'$, \nwhile $S_j=U'_j\\cap(\\coprod_{i=1,\\ldots,m} X_i)$ is an open subset of $\\mc D$. \nBy Lemma~\\ref{LOpen}, there exists an open subset $R_j$ of $\\coprod_{i=1,\\ldots,m} X_i$ that deformation retracts onto $S_j$\nsuch that $R_j\\cap \\mc D=S_j$.\nThen $R'_j=R_j\\coprod (U'_j-S_j)$ is an open subset of $P'$ that deformation retracts onto $U'_j$, \nthus is contractible in $P'$. Since $P'-P$ and $V_j$ are both open in $P'$, and $(P'-P)\\cap V_j=\\emptyset$,\nthen the subspace $(P'-P)\\coprod V_j$ is contractible in $P'$. \nWe can therefore take $\\{R'_0,\\ldots,R'_{k'},(V_1\\coprod(P'-P)),V_2,\\ldots,V_k\\}$ as a categorical cover for $P'$,\nso $\\cat(P')\\leq k'+k$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Moment-Angle Complexes}\n\n\nGiven a simplicial complex $K$ on vertex set $[n]$ and \na sequence of of pairs of spaces \n$$\n\\mc S=((X_1,A_1),\\ldots,(X_n,A_n)), \n$$ \n$A_i\\subseteq X_i$, the \\emph{polyhedral product} $\\mc S^K$ is the subspace of $X^{\\times n}$ defined by\n$$\n\\mc S^K=\\bigcup_{\\sigma\\in K}Y^\\sigma_1\\times\\cdots\\times Y^\\sigma_n,\n$$ \nwhere $Y^\\sigma_i=X_i$ if $i\\in\\sigma$, or $Y^\\sigma_i=A_i$ if $i\\nin\\sigma$.\nIf the pairs $(X_i,A_i)$ are all equal to the same pair $(X,A)$, we usually write $\\mc S^K$ as $(X,A)^K$.\nThe \\emph{moment-angle complex} $\\mathcal Z_K$ is defined as the polyhedral product $(D^2,\\ensuremath{\\partial} D^2)^K$,\nand the \\emph{real moment-angle complex} $\\mathbb R\\mathcal Z_K$ is the polyhedral product $(D^1,\\ensuremath{\\partial} D^1)^K$.\n\nThe \\emph{join} of two simplicial complexes $K$ and $L$ is the simplicial complex \n$K\\ast L=\\cset{\\sigma\\sqcup\\tau}{\\sigma\\in K,\\,\\tau\\in L}$,\nand one has $|K\\ast L|\\cong |K|\\ast|L|\\simeq\\Sigma|K|\\wedge|L|$ and $\\z{K\\ast L}\\cong\\mathcal Z_K\\times\\mathcal Z_L$.\nIf $I\\subseteq [n]$, $K_I=\\cset{\\sigma\\in K}{\\sigma\\subseteq I}$ denotes the \\emph{full subcomplex} of $K$ on vertex set $I$,\nin which case $\\z{K_I}$ is a retract of $\\mathcal Z_K$.\nNotice that if $K_I$ and $L_J$ are full subcomplexes of $K$ and $L$, \nthen $K_I\\ast L_J$ is the full subcomplex $(K\\ast L)_{I\\sqcup J}$ of $K\\ast L$.\nAs a convention, we let $\\z{\\emptyset}=\\ast$ when $\\emptyset$ is on empty vertex set. \n\n\n\nWe let $S^0$ denote both the $0$-sphere and the simplicial complex $\\ensuremath{\\partial}\\Delta^1$ consisting of only two vertices. \nGenerally, we assume our simplicial complexes (except $\\emptyset$) are non-empty and have no ghost vertices,\nunless stated otherwise. Under this assumption, it is not difficult to see that the following holds.\n\n\n\n\n\n\\subsection{The Hochster Theorem}\n\\label{SHochster}\n\nWhen $R$ is a field or $\\mb Z$, it was shown in~\\cite{MR1897064,MR2255969,MR2117435,MR0441987} that there are isomorphisms \nof graded commutative algebras\n\\begin{equation}\n\\label{EHochster}\nH^*(\\mathcal Z_K;R)\\cong \\mathrm{Tor}_{R[v_1,\\ldots,v_n]}(R[K],R) \\cong \\cplus{I\\subseteq[n]}{}{\\tilde H^*(\\Sigma^{|I|+1}|K_I|;R)}.\n\\end{equation}\nThe isomorphism on the left is induced by a quasi-isomorphism of DGAs between the Koszul complex of the Stanley-Reisner ring $R[K]$\nand the cellular cochain complex of $\\mathcal Z_K$ with coefficients in $R$. \nThe multiplication on the right is given by maps \\seqm{H^*(K_I)\\otimes H^*(K_J)}{}{H^{*+1}(K_{I\\cup J})} \nthat are zero when $I\\cap J\\neq\\emptyset$, otherwise they are induced by maps\n$\\iota_{I,J}\\ensuremath{\\,\\colon\\,}\\seqm{|K_{I\\cup J}|}{}{|K_I\\ast K_J|\\cong |K_I|\\ast |K_J|\\simeq\\Sigma |K_I|\\wedge |K_J|}$\ngeometrically realizing the canonical inclusions \\incl{K_{I\\cup J}}{}{K_I\\ast K_J}.\nOne can iterate so that any length $\\ell$ product\n\\seqm{\\bigotimes^\\ell_{i=1} H^*(K_{I_i})}{}{H^{*+\\ell-1}(K_{I_1\\cup\\cdots\\cup I_\\ell})} is induced by the inclusion \n$$\n\\iota_{I_1,\\ldots,I_\\ell}\\ensuremath{\\,\\colon\\,}\\incl{|K_{I_1\\cup\\cdots\\cup I_\\ell}|}{}{|K_{I_1}\\ast\\cdots\\ast K_{I_\\ell}|}\n$$ \nwhere the $I_i$'s are mutually disjoint.\n\n\n\\subsection{A Necessary Condition}\n\nSuppose $\\cat(\\mathcal Z_K)\\leq \\ell-1$,\nso cup products of length $l$ vanish in $H^+(\\mathcal Z_K)$.\nThen in light of the Hochster theorem, the inclusions $\\iota_{I_1,\\ldots,I_\\ell}$ must induce trivial maps on cohomology. \nIn fact, their suspensions must be nullhomotopic by the following argument. \n\nLet $\\widehat{\\mathcal Z}_K=\\mathcal Z_K\/\\cset{(x_1,\\ldots,x_n)\\in \\mathcal Z_K}{\\mbox{at least one }x_i=\\ast}$.\nFix $m=|I_1\\cup\\cdots\\cup I_\\ell|$, $Y=\\z{K_{I_1\\cup\\cdots\\cup I_\\ell}}$, and $\\hat Y=\\hz{K_{I_1\\cup\\cdots\\cup I_\\ell}}$. \nSince $Y$ is a retract of $\\mathcal Z_K$, $\\cat(Y)\\leq\\ell-1$.\nRecall from~\\cite{MR516214} that a path-connected basepointed $CW$-complex such as $Y$ satisfies $\\cat(Y)\\leq\\ell-1$ if and only if \nthere is a map \\seqm{Y}{\\psi}{FW_\\ell(Y)} such that the diagonal map \\seqm{Y}{\\vartriangle}{Y^{\\times\\ell}} \nfactors up to homotopy as \\seqmm{Y}{\\psi}{FW_\\ell(Y)}{include}{Y^{\\times\\ell}}.\nHere $FW_\\ell(Y)=\\cset{(y_1,\\ldots,y_\\ell)\\in Y^{\\times \\ell}}{\\mbox{at least one }y_i=\\ast}$ is the fat wedge.  \nThis implies the reduced diagonal map\n$\\bar\\vartriangle\\ensuremath{\\,\\colon\\,}\\seqmmm{Y}{\\vartriangle}{Y^{\\times\\ell}}{}{Y^{\\times\\ell}\/FW_\\ell(Y)}{\\cong}{Y^{\\wedge\\ell}}$\nis nullhomotopic. Then so is\n$\\zeta\\ensuremath{\\,\\colon\\,}\\seqmmm{Y}{\\bar\\vartriangle}{Y^{\\wedge\\ell}}{}{\\bigwedge_j\\z{K_{I_j}}}{}{\\bigwedge_j\\hz{K_{I_j}}}$,\nwhere the second last map is the smash of the coordinate-wise projection maps onto each $\\z{K_{I_j}}$, \nand the last map is the smash of quotient maps. \nThis last nullhomotopic map $\\zeta$ coincides with $\\seqmm{Y}{q}{\\hat Y}{\\hat\\iota}{\\bigwedge_j\\hz{K_{I_j}}}$,\nwhere $q$ is the quotient map and $\\hat\\iota$ is the inclusion given simply by rearranging coordinates. \nMoreover, \n$\\hat\\iota$ is homeomorphic to $\\Sigma^{m+1}\\iota_{I_1,\\ldots,I_\\ell}$ and $\\Sigma q$ has a right homotopy inverse\n(c.f.~\\cite{MR2673742}, and also the proof of Proposition~$2.5$ and pg. $23$ in~\\cite{2014arXiv1409.4462B}).\nIt follows that $\\Sigma^{m+1}\\iota_{I_1,\\ldots,I_\\ell}$ is nullhomotopic.\n\n\n\n\\subsection{Skeleta and Suspension on Coordinates}\n\n\nLet $\\sk{K}{i}$ denote the $i$-skeleton of $K$, and $\\sk{K}{-1}=\\emptyset$ on vertex set $[n]$. \nAn inclusion of simplicial complexes \\incl{L}{}{K} induces a canonical inclusion of $CW$-complexes \\incl{\\mathcal Z_L}{}{\\mathcal Z_K}. \nThis gives $\\zsk{K}{i}$ and $\\zsk{K}{-1}=(\\ensuremath{\\partial} D^2)^{\\times n}=(S^1)^{\\times n}$ as subcomplexes of $\\mathcal Z_K$.\n\n\n\\begin{lemma}[Corollary~$3.3$ in~\\cite{MR3084441}]\n\\label{LGT2}\nIf $K$ is on vertex set $[n]$ with no ghost vertices, then $\\zsk{K}{-1}=(\\ensuremath{\\partial} D^2)^{\\times n}$ is contractible in $\\mathcal Z_K$.~$\\hfill\\square$\n\\end{lemma} \n\n\\begin{lemma}\n\\label{LSkeleta}\nIf $0\\leq l\\leq \\dim K$, then $\\zsk{K}{\\ell}-\\zsk{K}{\\ell-1}$ is contractible in $\\mathcal Z_K$.\n\\end{lemma}\n\n\\begin{proof}\nWe have a decomposition\n$$\n\\zsk{K}{\\ell}-\\zsk{K}{\\ell-1}=\\cdunion{\\sigma\\in K,\\,|\\sigma|=\\ell+1}{}{\\tilde Y_1^\\sigma\\times\\cdots\\times \\tilde Y_n^\\sigma}\n$$\nwhere $\\tilde Y_i^{\\sigma}=D^2-\\ensuremath{\\partial} D^2$ if $i\\in\\sigma$ and $\\tilde Y_i^{\\sigma}=\\ensuremath{\\partial} D^2$ if $i\\nin\\sigma$. \nThis being a disjoint union of open subspaces of $\\zsk{K}{l}$, \neach of which can be deformed into $\\zsk{K}{-1}$ in $\\mathcal Z_K$\n(by contracting $\\tilde Y_i^{\\sigma}$ to a point in $\\ensuremath{\\partial} D^2$ whenever $i\\in\\sigma$). \nThus $\\zsk{K}{\\ell}-\\zsk{K}{\\ell-1}$ can also be deformed into $\\zsk{K}{-1}$.\nThen $\\zsk{K}{\\ell}-\\zsk{K}{\\ell-1}$ is contractible in $\\mathcal Z_K$ by Lemma~\\ref{LGT2}.\n\n\\end{proof}\n\n\\begin{lemma}\n\\label{LSkeleta2}\nIf $-1\\leq j\\leq \\dim K$, then \n$$\n\\cat(\\mathcal Z_K)\\leq \\cat(\\incl{\\zsk{K}{j}}{}{\\mathcal Z_K})+\\dim K-j.\n$$\nIn particular, $\\cat(\\mathcal Z_K)\\leq \\dim K+1$.\n\\end{lemma}\n\n\\begin{remark}\nLemma~\\ref{LSkeleta} and~\\ref{LSkeleta2} can be generalized to any filtration \n$L_j\\subseteq \\cdots\\subseteq L_k=K$ satisfying $\\ensuremath{\\partial}\\sigma\\subseteq L_i$ whenever $\\sigma\\in L_{i+1}$\n(in place of the skeletal filtration).\n\\end{remark}\n\n\n\\begin{proof}\nThe skeletal filtration $\\sk{K}{j}\\subseteq\\cdots\\subseteq\\sk{K}{\\dim K}=K$\ninduces a filtration of subcomplexes $\\zsk{K}{j}\\subseteq\\cdots\\subseteq\\mathcal Z_K$, \nand  for $0\\leq j\\leq \\dim K$, $\\zsk{K}{j}-\\zsk{K}{j-1}$ is contractible in $\\mathcal Z_K$ by Lemma~\\ref{LSkeleta}.\nThe result then follows using Lemma~\\ref{LFiltration}. In particular, when $j=-1$, \nwe get $\\cat(\\mathcal Z_K)\\leq \\dim K+1$ since $\\cat(\\incl{\\zsk{K}{-1}}{}{\\mathcal Z_K})=0$ by Lemma~\\ref{LGT2}.\n\\end{proof}\n\n\n\n\n\\begin{proposition}\n\\label{PCoordSusp}\nLet $\\mc S=((X_1,A_1),\\ldots,(X_n,A_n))$ \nand $\\mc T=((\\Sigma^{m_1}X_1,\\Sigma^{m_1} A_1),\\ldots,(\\Sigma^{m_n} X_n,\\Sigma^{m_n} A_n))$\nbe sequences of pairs of spaces for some integers $m_i$ and connected basepointed $X_i$. \nThen for any $K$ (with no ghost vertices),\n$$\n\\cat(\\mc T^K)\\leq \\cat(\\mc S^K).\n$$\n\\end{proposition}\n\n\\begin{proof}\nLet $K$ be on vertex set $[n]$, $k=\\cat(\\mc S^K)$, and take a categorical cover $\\{U_0,\\ldots,U_k\\}$ of $\\mc S^K$.\nFor any open subset $V$ of $\\mc S^K$, define the following open subset $V^1$ of $\\mc T^K$\n$$\nV^1 = \\cset{((t_1,x_1),\\ldots,(t_n,x_n))\\in \\prod_{i=1}^n \\Sigma^{m_i}X_i}{(x_1,\\ldots,x_n)\\in V,\\,t_i\\in D^{m_i}}.\n$$\nIn particular, $\\mc T^K=(\\mc S^K)^1$. Then $\\{U^1_0,\\ldots,U^1_k\\}$ is an open cover of $\\mc T^K$. \nSince $K$ has no ghost vertices, $A_i\\subseteq X_i$, and each $X_i$ is path-connected, then $\\mc S^K$ is path-connected. \nSince $\\Sigma^{m_i}X_i$ is the reduced suspension of the basepointed space $X_i$, \nwe have identifications $(t,\\ast)\\sim\\ast\\in\\Sigma^{m_i}X_i$. \nThen we can define a contraction of $U^1_i$ in $\\mc T^K$ by contracting $U_i$ in $\\mc S^K$ to a point $p$ \nand homotoping $p$ to the basepoint $(\\ast,\\ldots,\\ast)\\in\\mc S^K$.\nTherefore, $\\{U^1_0,\\ldots,U^1_k\\}$ is a categorical cover of $\\mathcal Z_K$. \n\n\\end{proof}\n\n\n\nNotice that the $(i+1)$-skeleton $\\sk{(\\mathbb R\\mathcal Z_K)}{i+1}$ of $\\mathbb R\\mathcal Z_K$ is equal to $\\rzsk{K}{i}$\n(this is not true for the complex moment-angle complex $\\mathcal Z_K$).\n\n\n\\begin{corollary}\n\\label{CRealMAC}\nFor any $K$ (with no ghost vertices),\n\\begin{equation}\n\\label{mac}\n\\cat(\\mathcal Z_K)\\leq \\cat(\\mathbb R\\mathcal Z_K)\n\\end{equation}\nand if $\\mathbb R\\mathcal Z_K$ is not contractible and $i\\geq 0$, then\n\\begin{equation}\n\\label{RmacCmac}\n\\cat(\\zsk{K}{i})\\leq \\cat(\\rzsk{K}{i})\\leq \\cat(\\mathbb R\\mathcal Z_K).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nInequality~\\eqref{mac} and the first inequality in~\\eqref{RmacCmac} follow from Proposition~\\ref{PCoordSusp}.  \nBy the main corollary of Theorem~$1$ in~\\cite{MR1933583},\nthe $i$-skeleton $\\sk{X}{i}$ of any connected non-contractible $CW$-complex $X$\nsatisfies that $\\cat(\\sk{X}{i})\\leq\\cat(X)$. \nSince $\\sk{(\\mathbb R\\mathcal Z_K)}{i+1}=\\rzsk{K}{i}$ holds for real moment-angle complexes, the last inequality follows.\n\n\\end{proof}\n\n\nIt is plausible that the second bound can be strengthened to $\\cat(\\zsk{K}{i})\\leq \\cat(\\mathcal Z_K)$.\nIn any case, even if it is true, we will sometimes need a sharper bound.\n\nLet $X$ and $Y$ be path-connected paracompact spaces, and $\\mc U=\\{U_0,\\ldots,U_k\\}$ and $\\mc V=\\{V_0,\\ldots,V_{\\ell}\\}$ \nbe categorical covers of $X$ and $Y$, respectively. We recall James' construction of a categorical cover $\\mc W=\\{W_0,\\ldots,W_{k+\\ell}\\}$\nof $X\\times Y$ from the covers $\\mc U$ and $\\mc V$ (see~\\cite{MR516214}, page $333$). \n\nLet $\\{\\pi_j\\}_{j\\in\\{0,\\dots,k\\}}$ be a partition of unity subordinate to the cover $\\mc U$. \nFor any subset $S\\subseteq \\{0,\\dots,k\\}$, define \n$$\nW_{\\mc U}(S)=\\cset{x\\in X}{\\pi_j(x)>\\pi_i(x)\\mbox{ for any }j\\in S\\mbox{ and }i\\nin S},\n$$ \nand for any point $p\\in X$, let \n$$\nS_{\\mc U}(p)=\\cset{j\\in \\{0,\\dots,k\\}}{\\pi_j(p)>0}\n$$\n(since the context is clear, let $W(S)=W_{\\mc U}(S)$ and $S(p)=S_{\\mc U}(p)$).\nThen $W(S)$ is an open subset of $X$ and $X=\\bigcup_{S\\subseteq\\{0,\\dots,k\\}}W(S)$ \n(given $x\\in X$, $x\\in W(S)$ where $S=\\cset{i}{\\,\\pi_i(x)=\\max\\{\\pi_1(x),\\ldots,\\pi_k(x)\\}\\,}$).\nMoreover, $W(S')\\cap W(S)=\\emptyset$ when $S\\nsubseteq S'$ and $S'\\nsubseteq S$ \n(in particular, when $|S|=|S'|$ and $S\\neq S'$), and $W(S)\\subseteq U_j$\nwhenever $j\\in S$. \nTherefore $W(S)$ is contractible in $X$. Then so is the disjoint union of open sets\n\\begin{equation}\n\\label{EUi}\nU'_i=\\cdunionmulti{S=S(p)\\mbox{\\tiny{ for some} }p\\in X}{|S|=i+1}{}{W(S)}.\n\\end{equation}\nSince $W(S)=\\emptyset$ when $S\\neq S(p)$ for every $p\\in X$,\nthe set $\\{U'_0,\\ldots,U'_k\\}$ forms a categorical cover of $X$.\nWe obtain a categorical cover $\\{V'_0,\\ldots,V'_\\ell\\}$ of $Y$ from $\\mc V$ by an analogous construction.\n\n\nNow let $\\bar U_i=U'_{k-i}\\cup\\cdots\\cup U'_k$ and $\\bar V_j=V'_{\\ell-j}\\cup\\cdots\\cup V'_\\ell$,\nand for $-1\\leq s\\leq k+\\ell$, let $C_{-1}=\\emptyset$ and\n$$\nC_s=\\cunionmulti{i+j=s}{i\\leq k,\\,j\\leq\\ell}{}{\\bar U_i\\times\\bar V_j}\n$$\nTake $W_s=C_s-C_{s-1}$. Notice that\n\\begin{equation}\n\\label{EWs}\nW_s=\\cdunionmulti{i+j=s}{i\\leq k,\\,j\\leq\\ell}{}{U'_i\\times V'_j}.\n\\end{equation}\nThis defines a categorical cover $\\mc W$ of $X\\times Y$.\n\nGiven subcomplexes $B\\subseteq Y$ and $A\\subseteq X$,\nconsider the polyhedral product \n$$\n\\mc X^{S^0}=X\\times B \\cup_{A\\times B} A\\times Y\n$$\nover the sequence $\\mc X=((X,A),(Y,B))$.\n\n\n\\begin{lemma}\n\\label{LPoly}\nIf $X-A$ is contractible in $X$ and $Y-B$ is contractible in $Y$, then\n$$\n\\cat(\\mc X^{S^0})\\leq \\cat(A)+\\cat(B)+1.\n$$\n\\end{lemma} \n\n\\begin{proof}\nSuppose we have categorical covers $\\{R_1,\\ldots,R_k\\}$ and $\\{S_1,\\ldots,S_\\ell\\}$\nof $A$, and $B$. By Lemma~\\ref{LOpen}, we have open subsets $U_i\\subseteq X$ and $V_i\\subseteq Y$\nsuch that $U_i$ and $V_i$ deformation retract onto $R_i$ and $S_i$ respectively, \nand $U_i\\cap A=R_i$ and $V_i\\cap B=S_i$ for $i\\geq 1$. Then taking $U_0=X-A$ and $V_0=Y-B$, \n$\\mc U=\\{U_0,\\ldots,U_k\\}$ and $\\mc V=\\{V_0,\\ldots,V_\\ell\\}$ are categorical covers of $X$ and $Y$.\n\nNotice that\n$$\nX\\times Y-U_0\\times V_0=\\mc X^{S^0},\n$$\nand since $R_i=U_i\\cap A=U_i-U_0$ and $S_j=V_j\\cap B=V_j-V_0$ for $i,j\\geq 1$, \n\\begin{equation}\n\\label{EDiff}\nD_{i,j}\\,:=\\,U_i\\times V_j-U_0\\times V_0\\,=\\,(R_i\\times V_j)\\cup_{R_i\\times S_j}(U_i\\times S_j). \n\\end{equation}\nNotice that $D_{i,j}$ is contractible in $\\mc X^{S^0}$ by deformation retracting the factor\n$U_i$ onto $R_i$ and $V_j$ onto $S_j$,\nthen contracting $R_i\\times S_j$ in $A\\times B$.\n\n\nTake the categorical cover $\\mc W=\\{W_0,\\ldots,W_{k+\\ell}\\}$ of $X\\times Y$ constructed from \n$\\mc U$ and $\\mc V$ as above. By~\\eqref{EWs},\n\\begin{align*}\nW_s-U_0\\times V_0 &=\\cdunionmulti{i+j=s}{i\\leq k,\\,j\\leq\\ell}{}{(U'_i\\times V'_j-U_0\\times V_0)},\n\\end{align*}\nand by~\\eqref{EUi},\n\\begin{align*}\nU'_i\\times V'_j-U_0\\times V_0=\\cdunion\n{\\indexsize{\\begin{matrix}\n\\scriptstyle S=S_{\\mc U}(p)\\mbox{\\tiny{ for some} }p\\in X \\cr \n\\scriptstyle T=S_{\\mc V}(q)\\mbox{\\tiny{ for some} }q\\in Y \\cr \n\\scriptstyle |S|=i+1,\\,|T|=j+1\n\\end{matrix}}}\n{}{(W_{\\mc U}(S)\\times W_{\\mc V}(T)-U_0\\times V_0)}.\n\\end{align*}\nThese are disjoint unions of open subsets of $\\mc X^{S^0}$.\nSince $W_{\\mc U}(S)$ is contained in some $U_{i'}$ and\n$W_{\\mc V}(T)$ is contained in some $V_{j'}$, it follows that\n$(W_{\\mc U}(S)\\times W_{\\mc V}(T)-U_0\\times V_0)$ is contained in $D_{i',j'}$, \nso it is contractible in $\\mc X^{S^0}$. \nTherefore, \nso are the disjoint unions $U'_i\\times V'_j-U_0\\times V_0$ and $W_s-U_0\\times V_0$. \nMoreover, since $W_{k+\\ell}=U'_k\\times V'_\\ell$,\nand $U'_k=W_{\\mc U}(\\{0,\\ldots,k\\})$ and $V'_\\ell=W_{\\mc V}(\\{0,\\ldots,\\ell\\})$ \nare contained in $U_{i'}$ and $V_{j'}$ respectively for each $i'\\in\\{0,\\ldots,k\\}$ and $j'\\in\\{0,\\ldots,\\ell\\}$,\n$W_{k+\\ell}-U_0\\times V_0=\\emptyset$. Then\n$$\n\\{(W_0-U_0\\times V_0),\\ldots,(W_{k+\\ell-1}-U_0\\times V_0)\\}\n$$\nis a categorical cover of $\\mc X^{S^0}$.\n\n\\end{proof}\n\n\n\n\n\\begin{corollary}\n\\label{CJoinSkeleton}\nLet $K$ and $L$ be simplicial complexes with $d=\\dim K$ and $d'=\\dim L$ (so $\\dim K\\ast L=d+d'+1$). Then\n$$\n\\cat(\\zsk{(K\\ast L)}{d+d'})\\leq \\cat(\\zsk{K}{d-1})+\\cat(\\zsk{L}{d'-1})+1.\n$$\n\\end{corollary}\n\n\\begin{proof}\nNotice $\\sk{(K\\ast L)}{d+d'}=(K\\ast\\sk{L}{d'-1})\\cup_{(\\sk{K}{d-1}\\ast\\sk{L}{d'-1})} (\\sk{K}{d-1}\\ast L)$, \n$\\z{K\\ast L}=\\mathcal Z_K\\times\\mathcal Z_L$, so\n\\begin{align*}\n\\zsk{(K\\ast L)}{d+d'} &= (\\z{K\\ast\\sk{L}{d'-1}}) \\cup_{\\z{(\\sk{K}{d-1}\\ast\\sk{L}{d'-1})}} (\\z{\\sk{K}{d-1}\\ast L})\\\\\n&= (\\mathcal Z_K\\times\\zsk{L}{d'-1})\\cup_{\\zsk{K}{d-1}\\times\\zsk{L}{d'-1}} (\\zsk{K}{d-1}\\times\\mathcal Z_L),\n\\end{align*}\nand $\\mathcal Z_K-\\zsk{K}{d-1}$ and $\\mathcal Z_L-\\zsk{L}{d'-1}$ are contractible in $\\mathcal Z_K$ and $\\mathcal Z_L$ by Lemma~\\ref{LSkeleta}.\nThe result follows by Lemma~\\ref{LPoly}.\n\n\\end{proof}\n\n\n\n\n\n\\subsection{Missing Face Complexes}\n\nTake $K$ on vertex set $[n]$.\nWe fix the basepoint in the unreduced suspension $\\Sigma|K|=(|K|\\times [0,1]) \/\\sim$ \nto be the tip of the double cone corresponding to $1$ under the indentifications $(x,0)\\sim 0$ and $(x,1)\\sim 1$.\nLet $MF(K)=\\cset{\\sigma\\subseteq[n]}{\\sigma\\nin K,\\,\\ensuremath{\\partial}\\sigma\\subseteq K}$ be the collection of (minimal) missing faces of $K$.\nWe will need a somewhat more flexible alternative to the \\emph{directed missing face complexes} defined in~\\cite{2010arXiv1011.2133G}.\n\n\\begin{definition}\n$K$ on vertex set $[n]$ is called a \\emph{homology missing face complex} (or \\emph{HMF-complex}) if for each non-empty $I\\subseteq [n]$, \n$K_I$ is a simplex or there exists a subcollection $\\mc C_I\\subseteq MF(K_I)$ such that the wedge sum of suspended inclusions\n$$\n\\gamma_I\\ensuremath{\\,\\colon\\,}\\seqm{\\cvee{\\sigma\\in \\mc C_I}{}{\\Sigma|\\ensuremath{\\partial}\\sigma|}}{}{\\Sigma|K_I|}\n$$\ninduces an isomorphism on homology\n(therefore it is a homotopy equivalence since it is a map between suspensions).\n\\end{definition}\n\n\\begin{bigremark}\nGiven $H_*(K_I)$ is torsion-free, since each $\\Sigma|\\ensuremath{\\partial}\\sigma|$ is a sphere, \none needs only to find $\\gamma_I$ that induces surjection on homology in order for $K$ to be an $HMF$-complex.\n\\end{bigremark}\n\n\\begin{proposition}\n\\label{PExtractible}\nIf $K$ is an $HMF$-complex, then $\\mathcal Z_K$ is homotopy equivalent to a wedge of spheres or is contractible.\nTherefore $\\cat(\\mathcal Z_K)\\leq 1$ and $\\Cat(\\mathcal Z_K)\\leq 1$.\n\\end{proposition}\n\n\\begin{proof}\nFor each $I\\subseteq [n]$, either $K_I$ is a simplex, boundary of a simplex, or else for each $\\sigma\\in\\mc C_I$, \nwe can pick an $i_\\sigma\\in I$ such that $\\ensuremath{\\partial}\\sigma\\subseteq K_{I-\\{i_\\sigma\\}}$, \nso each inclusion \\seqm{|\\ensuremath{\\partial}\\sigma|}{}{|K_I|} factors through inclusions \n\\seqmm{|\\ensuremath{\\partial}\\sigma|}{}{|K_{I-\\{i_\\sigma\\}}|}{}{|K_I|}. Take the composite\n$$\nf\\ensuremath{\\,\\colon\\,}\\seqmmm{\\Sigma|K_I|}{\\gamma^{-1}_I}{\\cvee{\\sigma\\in\\mc C_I}{}{\\Sigma|\\ensuremath{\\partial}\\sigma|}}{}\n{\\cvee{i\\in I}{}{\\Sigma|K_{I-\\{i_\\sigma\\}}|}}{}{\\Sigma|K_I|}\n$$\nwhere $\\gamma^{-1}_I$ is a homotopy inverse of $\\gamma_I$,\nthe second last map includes the summand $\\Sigma|\\ensuremath{\\partial}\\sigma|$ into the summand $\\Sigma|K_{I-\\{i_\\sigma\\}}|$,\nand the last map is the standard inclusion on each summand. \nSince the composite of the last two maps is $\\gamma_I$, $f$ is a homotopy equivalence.\nThen $K$ is an \\emph{extractible complex} as defined in~\\cite{2013arXiv1306.6221I}. \nTherefore $\\mathcal Z_K$ is homotopy equivalent to a wedge of spheres or contractible by Corollary~$3.3$ therein.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{Gluings and Connected Sums}\n\nIf $L$ and $K$ are simplicial complexes and $C$ is a full subcomplex common to both $L$ and $K$, then we obtain a new simplicial complex\n$L\\cup_C K$ by gluing $L$ and $K$ along $C$. One can always glue along simplices since they are always full subcomplexes.\nWhen $C=\\emptyset$, $L\\cup_C K$ is just the disjoint union $L\\sqcup K$. \n\nGiven $\\sigma\\in K$, define the \\emph{deletion} of the face $\\sigma$ from $K$ to be the simplicial complex\ngiven by \n$$\nK\\backslash\\sigma=\\cset{\\tau\\in K}{\\sigma\\not\\subseteq\\tau}.\n$$\nIf $\\sigma$ is a common face of $L$ and $K$, define the \\emph{connected sum} $L\\#_\\sigma K$ to be the simplicial complex \n$(L\\backslash\\sigma)\\cup_{\\ensuremath{\\partial}\\sigma}(K\\backslash\\sigma)$.\nIn other words, $L\\#_\\sigma K$ is obtained by deleting $\\sigma$ from $L$ and $K$ and gluing along the boundary $\\ensuremath{\\partial}\\sigma$.\nAs a convention, we let $\\z{\\emptyset}=\\ast$ when $\\emptyset$ is on empty vertex set. \n\n\n\n\n\n\\begin{proposition}\n\\label{PGluing2}\nIf $C$ is a (possibly empty) full subcomplex common to $K_1,\\ldots,K_m$, then \n$$\n\\cat(\\z{K_1\\cup_C\\cdots\\cup_C K_m})\\leq \\max\\{1,\\cat(\\z{K_1}),\\ldots,\\cat(\\z{K_m})\\}+\\Cat(\\z{C}).\n$$\nMoreover, if each $K_i$ is the $d_i-1$ skeleton of some $d_i$ dimensional simplicial complex $\\bar K_i$, \nand $C$ is also a full subcomplex of each $\\bar K_i$, then \n$$\n\\cat(\\z{\\bar K_1\\cup_C\\cdots\\cup_C \\bar K_m})\\leq \\max\\{1,\\cat(\\z{K_1}),\\ldots,\\cat(\\z{K_m})\\}+\\Cat(\\z{C}).\n$$\n\\end{proposition}\n\n\n\\begin{proof\n\nLet $K_i$ be on vertex set $[n_i]$, and $C$ has $\\ell$ vertices. \nIf $C$ is on vertex set $[n_i]$, possibly with ghost vertices, \nthe inclusion \\incl{C}{}{K_i} induces a coordinate-wise inclusion \\incl{(\\ensuremath{\\partial} D^2)^{\\times n_i-\\ell}\\times\\z{C}}{f_i}{\\z{K_i}}.\nBy Lemma~\\ref{LGT2}, $f_i$ is nullhomotopic when restricted to $(\\ensuremath{\\partial} D^2)^{\\times n_i-\\ell}\\times\\ast$.\nLet $N_i=\\Sigma_{j\\neq i}n_j$. Note $\\z{K_1\\cup_C\\cdots\\cup_C K_m}$ is the pushout of\n\\incl{(\\ensuremath{\\partial} D^2)^{\\times n_i-\\ell}\\times\\z{C}\\times(\\ensuremath{\\partial} D^2)^{\\times N_i-\\ell}}{f_i\\times\\ensuremath{\\mathbbm{1}}}{\\z{K_i}\\times(\\ensuremath{\\partial} D^2)^{\\times N_i-\\ell}}\nfor $i=1,\\ldots,m$. \nSince each of these maps are inclusions of subcomplexes, $\\z{K_1\\cup_C\\cdots\\cup_C K_m}$ is homotopy equivalent to the homotopy pushout $P$ \nof these maps. The first inequality therefore follows from the first part of Lemma~\\ref{LGluing}.\n\nBy Lemma~\\ref{LSkeleta}, $\\z{\\bar K_i}-\\z{K_i}$ is contractible in $\\z{\\bar K_i}$,  \nso the second equality follows from the second part of Lemma~\\ref{LGluing}. \n\n\\end{proof}\n\n\\begin{example}\n\\label{EGluing}\nIn particular, when $C$ is a simplex \n$\\cat(\\z{L\\cup_C K})\\leq \\max\\{1,\\cat(\\mathcal Z_L),\\cat(\\mathcal Z_K)\\}$ and\n$\\cat(\\z{\\bar L\\cup_C \\bar K})\\leq \\max\\{1,\\cat(\\mathcal Z_L),\\cat(\\mathcal Z_K)\\}$ \nsince $\\z{C}$ is contractible.\nThese also hold when $C$ is the empty simplex and $\\z{C}=\\ast$\n(in which case $\\bar L\\cup_C \\bar K=\\bar L\\sqcup \\bar K$ and $L\\cup_C K=L\\sqcup K$).\nWhen $C$ is the boundary of a simplex,   \n$\\cat(\\z{L\\cup_C K})\\leq \\max\\{1,\\cat(\\mathcal Z_L),\\cat(\\mathcal Z_K)\\}+1$\nsince $\\z{C}$ here is a sphere.\n\\end{example}\n\nThe bound in Proposition~\\ref{PGluing2} is not always optimal, sometimes far from it. \nIf $K$ and $\\Delta^{n-1}$ are on vertex set $[n]$ and $L$ is formed by gluing $\\Delta^{n-1}$ and $\\{n+1\\}\\ast K$ along $K$,\nthen $\\mathcal Z_L$ is a $co$-$H$-space by~\\cite{2013arXiv1306.6221I} so $\\cat(\\mathcal Z_L)=1$ \n(in fact, it is not difficult to directly show that $\\mathcal Z_L\\simeq\\Sigma^2\\mathcal Z_K$). \nOn the other hand, \nProposition~\\ref{PGluing2} gives $\\cat(\\mathcal Z_L)\\leq \\max\\{1,\\cat(\\mathcal Z_K)\\}+\\Cat(\\mathcal Z_K)$ since $\\z{\\{n+1\\}\\ast K}\\cong D^2\\times\\mathcal Z_K\\simeq\\mathcal Z_K$ and \n$\\z{\\Delta^n}$ is contractible.\n\n\n\n\\begin{corollary}\n\\label{CConnectedSum}\nSuppose\n$$\nK=L_1\\#_{\\sigma_1} L_2\\#_{\\sigma_2}\\cdots\\#_{\\sigma_{k-1}} L_k  \n$$\nwhere $\\dim L_i=d$, $\\sigma_i$ is a $d$-face common to $L_i$ and $L_{i+1}$, \nand $\\sigma_i\\cap\\sigma_j=\\emptyset$ when $i\\neq j$. Then\n$$\n\\cat(\\mathcal Z_K) \\leq \\max\\{1,\\cat(\\zsk{L_1}{d-1}),\\ldots,\\cat(\\zsk{L_k}{d-1})\\}+1.\n$$\n\\end{corollary}\n\n\\begin{proof}\nTake the disjoint unions \n$$\nC=\\ensuremath{\\partial}\\sigma_1\\sqcup\\cdots\\sqcup\\ensuremath{\\partial}\\sigma_{k-1},\n$$ \n$$\nK_1=\\csqdunion{1\\leq 2i+1\\leq k-1}{}{L_{2i+1}}\n$$\n$$\nK_2=\\csqdunion{2\\leq 2i\\leq k-1}{}{L_{2i}},\n$$\nand take the iterated face deletions $K'_1=K_1\\backslash(\\sigma_1\\sqcup\\cdots\\sqcup\\sigma_{k-1})$ and\n$K'_2=K_2\\backslash(\\sigma_1\\sqcup\\cdots\\sqcup\\sigma_{k-1})$. \nThen $C$ is a full subcomplex common to both $K'_1$ and $K'_2$, \nand to both $\\sk{K'_1}{d-1}$ and $\\sk{K'_2}{d-1}$.\nMoreover, $\\sk{K'_1}{d-1}=\\sk{K_1}{d-1}$ and $\\sk{K'_2}{d-1}=\\sk{K_2}{d-1}$,\nand $K=K'_1\\cup_C K'_2$, so by the second part of Proposition~\\ref{PGluing2},\n$$\n\\cat(\\z{K})\\leq \\max\\{1,\\cat(\\zsk{K_1}{d-1}),\\cat(\\zsk{K_2}{d-1})\\}+\\Cat(\\z{C})\n$$\nIt is clear that $C$ is an $HMF$-complex, so $\\z{C}$ is homotopy equivalent to a wedge of spheres \nand $\\Cat(\\z{C})=1$ (alternatively, this follows from Theorem~$10.1$ in~\\cite{MR2321037}).\nMoreover, we can think of $\\zsk{K_1}{d-1}$ as being built up iteratively by gluing \n$\\bigsqcup_{1\\leq 1\\leq j}{}{\\zsk{L_{2i+1}}{d-1}}$ and $\\zsk{L_{2j+3}}{d-1}$ along the empty simplex,\nso iterating the first inequality in Example~\\ref{EGluing},\n$$\n\\cat(\\zsk{K_1}{d-1})\\leq \\max\\{\\,1,\\,\\cat(\\zsk{L_1}{d-1}),\\,\\cat(\\zsk{L_3}{d-1}),\\,\\ldots\\,\\}.\n$$ \nLikewise,\n$\n\\cat(\\zsk{K_2}{d-1})\\leq \\max\\{\\,1,\\,\\cat(\\zsk{L_2}{d-1}),\\,\\cat(\\zsk{L_4}{d-1}),\\,\\ldots\\,\\}.\n$ \nThe inequality in the lemma follows.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Triangulated Spheres}\n\n\n\n\nLet $\\mc C_0=\\{S^0\\}$, $\\mc C_1$, and $\\mc C_2$ consist of all triangulated $0$,$1$, and $2$-spheres, \nand for $d\\geq 3$, let $\\mc C_d$ be the class of triangulated $d$-spheres defined by $K\\in \\mc C_d$ if  \n\\begin{itemize}\n\\item[(1)] $K=L_1\\ast\\cdots\\ast L_k$ for some $L_i\\in\\mc C_{d_i}$, $d_i\\leq 2$, and $d_1+\\cdots+d_k=d-k+1$;  \n\\item[(2)] $K=K_1\\#_{\\sigma_1}\\cdots\\#_{\\sigma_{\\ell-1}}K_\\ell$ where $\\sigma_i$ is a $d$-face common to $K_i$ and $K_{i+1}$ \nwith $\\sigma_i\\cap\\sigma_j=\\emptyset$ when $i\\neq j$, and each $K_i=L_{1,i}\\ast\\cdots\\ast L_{k_i,i}$ \nis of the form (1) such that each $L_{j,i}$ is not the boundary of a simplex.\n\\end{itemize}\n\nThe join $L\\ast L'$ is the simplicial complex $\\cset{\\sigma\\sqcup\\sigma'}{\\sigma\\in L,\\,\\sigma'\\in L'}$, \nand the connected sum $K\\#_{\\sigma} K'$ is given topologically by gluing triangulations $K$ and $K'$ of $S^d$ \nalong a common $d$-face $\\sigma$, and deleting its interior.\n\n\n\n\\begin{bigremark}\nIf $L$ and $L'$ are boundaries $\\ensuremath{\\partial} P^*$ and $\\ensuremath{\\partial} P'^*$ of the duals of  simple polytopes $P$ and $P'$,\nthen $L\\ast L'$ is the boundary of $(P\\times P')^*$, while $L\\#_{\\sigma} L'$ is the boundary of dual $Q^*$, \nwhere $Q$ is obtained by taking the vertex cut at the vertices of $P$ and $P'$ that are dual to $\\sigma$, \ngluing along the new hyperplane and removing it after gluing. \n\\end{bigremark}\n\n\n\nOur goal in this section will be to show the following.\n\n\\begin{theorem}\n\\label{TMain}\nIf $K$ on vertex set $[n]=\\{1,\\ldots,n\\}$ is any triangulated $d$-sphere for $d=0,1,2$, \nor $K\\in \\mc C_d$ for $d\\geq 3$, then the following are equivalent.\n\\begin{itemize}\n\\item[(1)] $K$ is $m$-Golod over $\\ensuremath{\\mathbb{Z}}$;\n\\item[(2)] $\\Nill(\\mathrm{Tor}_{\\ensuremath{\\mathbb{Z}}[v_1,\\ldots,v_n]}(\\ensuremath{\\mathbb{Z}}[K],\\ensuremath{\\mathbb{Z}}))\\leq m+1$ (equivalently $\\cuplen(\\mathcal Z_K)\\leq m$);\n\\item[(3)] for any filtration of full subcomplexes \n$$\n\\ensuremath{\\partial}\\Delta^{d+2-\\ell}=K_{I_\\ell}\\subsetneq K_{I_{\\ell-1}}\\subsetneq\\cdots\\subsetneq K_{I_1}=K\n$$ \nsuch that $|K_{I_i}|\\cong S^{d+1-i}$, we have $\\ell\\leq m$; \n\\item[(4)] $\\cat(\\mathcal Z_K)\\leq m$.\n\\end{itemize}\nMoreover, $1\\leq m\\leq d+1$; that is, $K$ satisfies any of the above for some $m$ which cannot be greater than $d+1$.~$\\hfill\\square$ \n\\end{theorem}\n\n\n\n\n\n\n\\subsection{Well-Behaved Triangulations}\n\n\\begin{definition}\n\\label{DWell}\nWe say a triangulation $K$ of a $d$-sphere $S^d$ on vertex set $[n]$ is \\emph{well-behaved} if \nfor all $I\\subseteq[n]$ and any $k<d$:\n\\begin{itemize}\n\\item[(1)] if $|K_I|\\cong S^k$ and $K_I\\subseteq K_J$ for some $J\\subseteq[n]$ such that $|K_J|\\cong S^{k+1}$, \nthen $|K_{J-I}|\\simeq S^0$;\n\\item[(2)] if $H_*(K_I)\\cong H_*(S^k)$, then there is $I'\\subseteq[n]$ such that $K_{I'}\\subseteq K_I$ and $|K_{I'}|\\cong S^k$;\n\\item[(3)] if $I$ is a face in $K$, then $|K_{[n]-I}|\\simeq\\ast$;\n\\item[(4)] $H_*(K_I)$ is torsion-free.\n\\end{itemize}\n\\end{definition}\n\nAs we will see, conditions $(2)$-$(4)$ are there to keep the well-behaved condition invariant under join and connected sum operations.\n\n\n\\begin{lemma}\n\\label{Ld012Well}\nTriangulations $K$ of $0$, $1$, and $2$-spheres are well-behaved.\n\\end{lemma}\n\\begin{proof}\nThis is trivial for $d=0,1$. When $d=2$, Conditions~$(1)$ and~$(3)$ are also clear.\nSince $|K|$ is a sphere and $|K_I|$ has the homotopy type of a dimension $\\leq 1$ $CW$-complex when \n$I\\subsetneq [n]$, $H_*(K_I)$ is torsion-free. To see that Condition~$(2)$ holds, \nnotice that if $H_*(K_I)\\cong H_*(S^0)$ then we can pick $K_{I'}=S^0\\subseteq K_I$ ($|I'|=2$).\nIf $H_*(K_I)\\cong H_*(S^1)$, we obtain a $1$-dimensional full subcomplex $K_{I''}\\subseteq K_I$\nsuch that $H_*(K_{I''})\\cong H_*(S^1)$ via a sequence of elementary collapses of $2$-faces in $K_I$, \nand then we can choose $K_{I'}\\subseteq K_{I''}$ such that $|K_{I'}|\\cong S^1$.   \n\\end{proof}\n\n\nIf $K$ and $L$ are triangulations of the $d$-sphere $S^d$, then so is $K\\#_\\sigma L$.\nIf $K$ and $L$  are triangulations of $S^d$ and $S^{d'}$, \nthen $K\\ast L$ is a triangulation of $S^{d+d'+1}$ since $|K\\ast L|\\cong |K|\\ast|L|$.\n\n\n\n\n\n\n\\begin{lemma}\n\\label{LCSumWell}\nSuppose $K$ and $L$ are both triangulations of $S^d$, and $\\sigma$ is a $d$-face common to $K$ and $L$.\nIf $K$ and $L$ are well-behaved, then so is $K\\#_\\sigma L$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{LJoinWell}\nIf $K$ and $L$ are well-behaved triangulations of $S^d$ and $S^{d'}$, \nthen $K\\ast L$ is a well-behaved triangulation of $S^{d+d'+1}$.\n\\end{lemma}\n\nWe will need three lemmas before we can prove these.\n\n\n\n\n\n\n\n\\begin{lemma}\n\\label{LUseful}\nSuppose $K$ and $L$ are well-behaved triangulations on vertex sets $\\mc V$ and $\\mc V'$,\nand let $N$ be a full subcomplex of $K\\ast L$. \n\nIf $H_*(N)\\cong H_*(S^k)$ for some $k$, then for some $I\\subseteq\\mc V$ and $J\\subseteq\\mc V'$, \nwe have $N=K_I\\ast L_J$ such that $H_*(K_I)\\cong H_*(S^i)$ and $H_*(L_J)\\cong H_*(S^j)$.\nMoreover, if $|N|\\cong S^k$, then $|K_I|\\cong S^i$ and $|L_J|\\cong S^j$.\n\\end{lemma}\n\n\n\\begin{proof}\n\nSince $N$ is a full subcomplex of $K\\ast L$,\n$N=K_I\\ast L_J$ where $I\\sqcup J$ is the vertex set of $N$. \nSince $K$ and $L$ are well-behaved, \n$H_*(K_I)$ and $H_*(L_J)$ are torsion-free, and since \n\\begin{equation}\n\\label{ETorsion}\n\\tilde H_{*+1}(N)\\cong \\tilde H_{*+1}(|K_I|\\ast|L_J|)\\cong \n\\tilde H_*(|K_I|\\wedge|L_J|)\\cong \\tilde H_*(K_I)\\otimes\\tilde H_*(L_J),\n\\end{equation}\nthen $H_*(N)$ is torsion-free.\n\nIf $I=\\emptyset$ then $N=K_\\emptyset\\ast L_J=\\emptyset\\ast L_J=L_J$, and the lemma follows.\nSimilarly when $J=\\emptyset$. Assume $I\\neq\\emptyset$ and $J\\neq\\emptyset$. \nSuppose $H_*(N)\\cong H_*(S^k)$. Then by~\\eqref{ETorsion}, \nwe must have $H_*(K_I)\\cong H_*(S^i)$ and $H_*(L_J)\\cong H_*(S^j)$ for some $i,j$ such that $i+j+1=k$.\n\nNow suppose, moreover, that we have $|N|\\cong S^k$. Suppose $|K_I|\\not\\cong S^i$. \nSince $K$ is well-behaved and $H_*(K_I)\\cong H_*(S^i)$ , there exists a full subcomplex $K_{I'}\\subseteq K_I$\nsuch that $|K_{I'}|\\cong S^i$, and $K_{I'}\\neq K_I$ since $|K_I|\\not\\cong S^i$.\nThen $H_k(K_{I'}\\ast L_J)\\cong H_k(K_I\\ast L_J)=H_k(N)\\cong\\ensuremath{\\mathbb{Z}}$, \nso under the inlusion $K_{I'}\\ast L_J\\subseteq K_I\\ast L_J=N$, \nthere exists a cycle $\\omega$ in the simplicial chain subgroup $C_k(K_{I'}\\ast L_J)\\subseteq C_k(N)$ that represents  \na generator $\\ensuremath{\\mathbb{Z}}\\subseteq H_k(K_{I'}\\ast L_J)$. But since $\\dim (K_{I'}\\ast L_J)=\\dim N=k$, $K_{I'}\\ast L_J\\neq K_I\\ast L_J$, \nand the $k$-faces of $N$ generate $N$ (since $|N|$ is a sphere), $K_{I'}\\ast L_J$ does not contain all $k$-faces of $N$, \nso $\\omega$ must be a cycle in $C_{k}(N)$ that does not contain all $k$-faces of $N$. \nBut this is impossible since $|N|\\cong S^k$, so any $k$-dimensional cycle must contain all the $k$-faces, \nso we must have $|K_I|\\cong S^i$. By a similar argument, $|L_J|\\cong S^j$.\n\n\n\\end{proof}\n\n\n\n\n\\begin{lemma}\n\\label{LTorsion1}\nLet $K$ be a triangulation of a $d$-sphere $S^d$ on vertex set $[n]$. \nIf $H_*(K_I)$ is torsion-free for some $I\\subseteq[n]$, $I\\neq [n]$, \nthen so is $H_*(K_I\\backslash\\sigma)$ for any $d$-face $\\sigma\\in K_I$,\nand the inclusion \\incl{|\\ensuremath{\\partial}\\sigma|}{}{|K_I\\backslash\\sigma|} \ninduces an injection on homology groups with torsion-free cokernel.\n\\end{lemma}\n\n\\begin{proof}\nSince $|K_I|\\not\\cong|K|\\cong S^d$, \nany cycle in the simplicial chain group $C_d(K)$ consists of all the $d$-faces of $K$,\nand there are no cycles in $C_d(K_I)$ and $C_d(K_I\\backslash\\sigma)$, \nso $H_i(K_I)\\cong H_i(K_I\\backslash\\sigma)=0$ for $i\\geq d$.\nBy the homology long exact sequence for the cofibration sequence \n$|K_I\\backslash\\sigma|\\incladdr{}{|K_I|}\\mapaddr{}{|K_I|\/|K_I\\backslash\\sigma|\\cong S^d}$,\nwe have $H_i(K_I\\backslash\\sigma)\\cong H_i(K_I)$ for $i<d-1$, and a short exact sequence\n$$\n\\seqmmmm{0}{}{H_d(S^d)}{}{H_{d-1}(K_I\\backslash\\sigma)}{}{H_{d-1}(K_I)}{}{0}.\n$$\nTherefore $H_{d-1}(K_I\\backslash\\sigma)$ is torsion-free since $H_{d-1}(K_I)$ and $H_d(S^d)\\cong\\mb Z$ are,\nand the second map in the sequence has torsion-free cokernel.\nSince $|K_I|$ is the homotopy cofiber of \\incl{S^{d-1}\\cong|\\ensuremath{\\partial}\\sigma|}{}{|K_I\\backslash\\sigma|},\nthe second map is induced by this inclusion in degree $d-1$, \nso it induces an injection with torsion-free cokernel on each homology group.\n\\end{proof}\n\n\n\n\\begin{lemma}\n\\label{LTorsion2}\nSuppose $K$ and $L$ are triangulations of $S^d$ on vertex sets $\\mc V$ and $\\mc V'$,\n$\\sigma$ is a $d$-face common to $K$ and $L$, and take $K\\#_{\\sigma} L$ on vertex set $\\mc W$. \n\nIf $H_*(K_I)$ and $H_*(L_J)$ are torsion-free for each $I\\subseteq\\mc V$ and $J\\subseteq\\mc V'$, \nthen $H_*((K\\#_{\\sigma} L)_{\\mc I})$ is torsion-free for each subset $\\mc I\\subseteq\\mc W$.\n\\end{lemma}\n\n\\begin{proof}\nLet $Z=K\\#_{\\sigma} L$.\nGiven $\\mc I\\subseteq\\mc W$, and $I=\\mc V\\cap\\mc I$ and $J=\\mc V'\\cap\\mc I$, \nwe have $|Z_{\\mc I}|$ is homotopy equivalent to: \n(a) $|K_I|\\vee |L_J|$ when $\\mc I\\cap\\sigma\\neq\\emptyset$ and $\\sigma\\not\\subseteq\\mc I$;\n(b) $|K_I|\\coprod |L_J|$ when $\\mc I\\cap\\sigma=\\emptyset$;\n(c) otherwise it is equal to $|K_I|\\#_{\\sigma}|L_J|$ when $\\sigma\\subseteq\\mc I$. \nTherefore $H_*(Z_{\\mc I})$ is torsion free in the first two cases, or the last case when $\\mc I=\\mc W$. \nTo see that it is torsion-free in the last case when $\\mc I\\neq\\mc W$, \nnotice that one of $I\\neq\\mc V$ or $J\\neq\\mc V'$. If $I=\\mc V$ or $J=\\mc V'$, \nthen $|Z_{\\mc I}|\\cong |L_J|$ or $|Z_{\\mc I}|\\cong |K_I|$ respectively, and we are done. \nAssume $I\\neq\\mc V$ and $J\\neq\\mc V'$. \nNotice that $|K_I\\backslash\\sigma|\\cup|L_J\\backslash\\sigma|=|Z_{\\mc I}|$, \n$|K_I\\backslash\\sigma|\\cap|L_J\\backslash\\sigma|=|\\ensuremath{\\partial}\\sigma|$, and by Lemma~\\ref{LTorsion1},\n$H_*(K_I\\backslash\\sigma)$ and $H_*(L_J\\backslash\\sigma)$ are torsion-free,\nwith the inclusions \\incl{|\\ensuremath{\\partial}\\sigma|}{}{|K_I\\backslash\\sigma|} and \\incl{|\\ensuremath{\\partial}\\sigma|}{}{|L_J\\backslash\\sigma|}  \ninducing injections on homology groups with torsion-free cokernel. \nThen it follows by the Mayer-Veitoris sequence for $(|Z_{\\mc I}|,|K_I\\backslash\\sigma|,|L_J\\backslash\\sigma|)$\nthat $H_*(Z_{\\mc I})$ is torsion free. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{LCSumWell}]\nLet $Z=K\\#_\\sigma L$ on vertex set $\\mc W$, and $K$ and $L$ be on vertex sets $\\mc V$ and $\\mc V'$.\nBy Lemma~\\ref{LTorsion2}, any full subcomplex $Z_{\\mc I}$ has torsion-free homology,\nso condition~$(4)$ in Definition~\\ref{DWell} holds for $Z_{\\mc I}$.\n\nSince $Z$ is $K\\backslash\\sigma$ and $L\\backslash\\sigma$ glued along the boundary of the $d$-simplex $\\ensuremath{\\partial}\\sigma$, \n$\\mc I$ is a face in $K$ or $L$ whenever it is a face in $Z$.\nThen $K$ and $L$ being well-behaved, \nwe have $|Z_{\\mc W-\\mc I}|\\simeq |K_{\\mc V-\\mc I}|\\vee|L_{\\mc V'-\\mc I}|\\simeq\\ast$ when $\\mc I\\cap\\sigma\\neq\\emptyset$. \nSince $|L\\backslash\\sigma|\\cong |K\\backslash\\sigma|\\cong D^d$,\nthen $|Z_{\\mc W-\\mc I}|\\cong |K_{\\mc V-\\mc I}|\\simeq\\ast$ or $|Z_{\\mc W-\\mc I}|\\cong |L_{\\mc V'-\\mc I}|\\simeq\\ast$\nwhen $\\mc I\\cap\\sigma=\\emptyset$. Condition~$(3)$ holds as well.\n\nIf $|Z_{\\mc I}|\\cong S^k$ for some $k<d$, \nthen $Z_{\\mc I}$ is either a full subcomplex of one of $K$ or $L$, or else $Z_{\\mc I}=\\ensuremath{\\partial}\\sigma$\n(otherwise $Z_{\\mc I}$ would have a $(k-1)$-face contained in three $k$-faces)\nso the only case to check is the last one $Z_{\\mc I}=\\ensuremath{\\partial}\\sigma$. \nHere, condition~$(2)$ in Definition~\\ref{DWell} clearly holds. \nSince $K$ and $L$ are well-behaved, $|K_{\\mc V-\\sigma}|\\simeq|L_{\\mc V'-\\sigma}|\\simeq\\ast$,\nso it follows that $|Z_{\\mc W-\\sigma}|\\simeq S^0$, and condition~$(1)$ holds as well.\n\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{LJoinWell}]\n\nLet $K$ and $L$ be on vertex sets $\\mc V$ and $\\mc V'$, $Z=K\\ast L$ be on vertex set $\\mc W=\\mc V\\sqcup\\mc V'$,\n and let $\\mc I\\subseteq \\mc W$. \nThen $Z_{\\mc I}=K_I\\ast L_J$ where $I\\sqcup J=\\mc I$, and since $H_*(K_I)$ and $H_*(L_J)$ are torsion-free,\n$\\tilde H_{*+1}(Z_{\\mc I})\\cong \\tilde H_*(K_I)\\otimes\\tilde H_*(L_J)$ is torsion-free.    \n\nIf $\\mc I$ is a face in $Z$, then $I$ and $J$ are faces in $K$ and $L$,\nso we have $|K_{\\mc V-I}|\\simeq\\ast$ and $|L_{\\mc V'-J}|\\simeq\\ast$ since $K$ and $L$ are well-behaved.\nTherefore $|Z_{\\mc W-\\mc I}|=|K_{\\mc V-I}\\ast L_{\\mc V'-J}|\\simeq\\ast$,\nand condition~$(3)$ of Definition~\\ref{DWell} holds.\n\nSuppose $H_*(Z_{\\mc I})\\cong H_*(S^k)$. \nSuppose $I\\neq\\emptyset$ and $J\\neq\\emptyset$. \nBy Lemma~\\ref{LUseful}, $H_*(K_I)\\cong H_*(S^i)$ and $H_*(L_J)\\cong H_*(S^j)$ for some $i$,$j$ such that $i+j+1=k$.\nSince $K$ and $L$ and well-behaved, this implies there are $K_{I'}\\subseteq K_I$ and $L_{J'}\\subseteq L_J$ \nsuch that $|K_I|\\cong S^i$ are $|L_J|\\cong S^j$. \nThen \n$Z_{I'\\sqcup J'}=K_{I'}\\ast L_{J'}\\subseteq K_I\\ast K_J=Z_{\\mc I}$\nand $|Z_{I'\\sqcup J'}|\\cong S^{i+j+1}$.\nThus condition~$(2)$ holds when $I\\neq\\emptyset$ and $J\\neq\\emptyset$.  \n\nNow consider that case where $I=\\emptyset$ or $J=\\emptyset$. \nWithout loss of generality, suppose $I=\\emptyset$.\nWe have $Z_{\\mc I}=K_\\emptyset\\ast L_J=\\emptyset\\ast L_J=L_J$,\nso if $H_*(Z_{\\mc I})\\cong H_*(S^k)$, then $H_*(L_J)\\cong S^k$, and we have $L_{J'}\\subseteq L_J=Z_{\\mc I}$\nsuch that $|L_{J'}|\\cong S^k$ since $L$ is well-behaved.\nThus condition~$(2)$ holds when $I=\\emptyset$ or $J=\\emptyset$.  \n\n\n\nLet $S^{-1}=\\emptyset$.\nSuppose $|Z_{\\mc I}|\\cong S^k$ and $Z_{\\mc I}\\subseteq Z_{\\mc J}$ such that $|Z_{\\mc J}|\\cong S^{k+1}$.\nThen $Z_{\\mc J}=K_{I'}\\ast L_{J'}$ where $I'\\sqcup J'=\\mc J\\subsetneq\\mc I=I\\sqcup J$, \nso $K_I\\subseteq K_{I'}$ and $L_J\\subseteq L_{J'}$,\nand by Lemma~\\ref{LUseful}, we must have $|K_I|\\cong S^i$, $|L_J|\\cong S^j$, $|K_{I'}|\\cong S^{i'}$, $|L_{J'}|\\cong S^{j'}$, \nwhere $i+j+1=k$ and $i'+j'+1=k+1$. Since $K_I\\subseteq K_{I'}$ and $L_J\\subseteq L_{J'}$, then $i\\leq i'$ and $j\\leq j'$,\nso either (a) $i'=i$ and $j'=j+1$; or (b) $i'=i+1$ and $j'=j$. Without any loss of generality, assume (a) holds. \nThen $K_I=K_{I'}$ (i.e. $I=I'$), $K_{I-I'}=\\emptyset$, and\n$$\n|Z_{\\mc J-\\mc I}|=|K_{I'-I}\\ast L_{J'-J}|=|L_{J'-J}|\\simeq S^0,\n$$\nwith the last homotopy equivalence since $L$ is well-behaved. \nIt follows that condition~$(1)$ of Definition~\\ref{DWell} holds. \n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Spherical Filtrations and Filtration Length}\n\n\n\n\n\n\\begin{definition}\nGiven a triangulation of a $d$-sphere $K$ on vertex set $[n]$, \nsuppose we have a filtration of full subcomplexes\n$$\n\\ensuremath{\\partial}\\Delta^{d+2-\\ell}=K_{I_\\ell}\\subsetneq\\cdots\\subsetneq K_{I_1}=K\n$$ \nsuch that $K_{I_i}$ is a triangulation of a $(d+1-i)$-sphere.\nThen we say that this is a \\emph{spherical filtration} of $K$ of length $\\ell$. \n\\end{definition}\n\n\\begin{remark}\nImplicitly, $I_\\ell\\subsetneq\\cdots\\subsetneq I_1=[n]$ and $|I_\\ell|=d+3-\\ell$. \n\\end{remark}\n\n\\begin{definition}\nFor any triangulated sphere $K$, \ndefine the \\emph{filtration length} $\\filt(K)$ to be the largest integer $\\ell$ \nsuch that $K$ admits a spherical filtration of length $\\ell$.\n\\end{definition}\n\n\nFor example, $\\filt(\\ensuremath{\\partial}\\Delta^{d+1})=1$. \nGenerally, there are the following bounds with respect to joins, connected sums, and cup product length. \n\n\n\n\\begin{lemma}\n\\label{LFiltCSum}\nIf $K$ and $L$ are both triangulations of $S^d$, and $\\sigma$ is a $d$-face common to $K$ and $L$, \nthen \n$$\n\\filt(K\\#_\\sigma L)\\geq \\max\\{2,\\filt(K),\\filt(L)\\}.\n$$\n\\end{lemma}\n\n\\begin{proof}\nAs in the proof of Lemma~\\ref{LCSumWell},\na full subcomplex $Z_{\\mc I}$ of $K\\#_\\sigma L$ satisfying $|Z_{\\mc I}|\\cong S^k$ for some $k<d$ must either be a full subcomplex\nof exactly one of $K$ or $L$, or else $Z_{\\mc I}=\\ensuremath{\\partial}\\sigma$. \nMoreover, $K\\#_\\sigma L$ always has the length $2$ spherical filtration $\\ensuremath{\\partial}\\sigma\\subsetneq K\\#_\\sigma L$.\nThe lemma follows immediately. \n\\end{proof}\n\n\n\n\n\n\n\n\\begin{lemma}\n\\label{LFiltJoin}\nIf $K$ and $L$ are any triangulated spheres, then \n$$\n\\filt(K\\ast L)\\geq \\filt(K)+\\filt(L).\n$$\n\\end{lemma}\n\n\n\n\\begin{proof}\n\nLet $K$ and $L$ be on vertex sets $[n]$ and $[m]$.\nLet $d=\\dim K$, $d'=\\dim L'$, $\\ell=\\filt(K)$, $\\ell'=\\filt(L)$, and take \n$\\ensuremath{\\partial}\\Delta^{d+2-\\ell}=K_{I_\\ell}\\subsetneq\\cdots\\subsetneq K_{I_1}=K$ and\n$\\ensuremath{\\partial}\\Delta^{d'+2-\\ell'}=L_{J_{\\ell'}}\\subsetneq\\cdots\\subsetneq L_{J_1}=L$ \nto be spherical filtrations of $K$ and $L$.\n\nSince $|K_{I_i}\\ast L_{J_j}|\\cong |K_{I_i}|\\ast|L_{J_j}|\\cong S^{d+1-i}\\ast S^{d'+1-j}\\cong S^{d+d'-i-j+3}$,\nand  $|K_{I_i}\\ast L_{J_j}|$ is a full subcomplex of $|K_{I_{i'}}\\ast L_{J_{j'}}|$ when $i\\leq i'$ and $j\\leq j'$,\nwe have a length $\\ell+\\ell'$ spherical filtration of $K\\ast L$\n$$\n\\ensuremath{\\partial}\\Delta^{d+2-\\ell}\n\\subsetneq(K_{I_\\ell}\\ast L_{J_{\\ell'}})\\subsetneq\\cdots\n\\subsetneq (K_{I_\\ell}\\ast L_{J_2})\n\\subsetneq (K_{I_\\ell}\\ast L_{J_1})\n\\subsetneq (K_{I_{\\ell-1}}\\ast L_{J_1})\n\\subsetneq\\cdots\\subsetneq (K_{I_1}\\ast L_{J_1})=K\\ast L, \n$$\nwhere $\\ensuremath{\\partial}\\Delta^{d+2-\\ell}$ is the full subcomplex of \n$K_{I_\\ell}\\ast L_{J_{\\ell'}}=\\ensuremath{\\partial}\\Delta^{d+2-\\ell}\\ast\\ensuremath{\\partial}\\Delta^{d'+2-\\ell'}$\non the vertices of $\\ensuremath{\\partial}\\Delta^{d+2-\\ell}$. Therefore $\\filt(K\\ast L)\\geq\\ell+\\ell'$. \n\n\\end{proof}\n\n\n\n\\vspace{5mm}\n\n\n\n\n\n\n\n\\begin{lemma}\n\\label{LFiltCup}\nIf $K$ is a well-behaved triangulation of $S^d$, then $\\filt(K)\\leq\\cuplen(\\mathcal Z_K)$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\ell=\\filt(K)$ and $\\ensuremath{\\partial}\\Delta^{d+1-\\ell}=K_{I_\\ell}\\subsetneq\\cdots\\subsetneq K_{I_1}=K$ be a spherical filtration\nof length $\\ell$. \n\nLet $J_{i+1}=I_i-I_{i+1}$ for $i<\\ell$. Since $K$ is well-behaved, $|K_{J_{i+1}}|\\simeq S^0$,\nso $|K_{J_{i+1}}|\\cong D_i\\coprod D'_i$ for some contractible subcomplexes $D_i$ and $D'_i$ of $|K_{I_i}|$.\nTake the inclusion induced by \\incl{K_{I_i}}{}{K_{I_{i+1}}\\ast K_{J_{i+1}}}\n$$\n\\iota_i\\ensuremath{\\,\\colon\\,}\\incl{|K_{I_i}|}{}{|K_{I_{i+1}}\\ast K_{J_{i+1}}|\\cong |K_{I_{i+1}}|\\ast |K_{J_{i+1}}|},\n$$ \nand let $h_i\\colon\\seqm{|K_{I_{i+1}}|\\ast |K_{J_{i+1}}|}{}{|K_{I_{i+1}}|\\ast S^0}$\nbe the join of the identity map on the left factor with the map collapsing $D_i$ and $D'_i$ to $-1$ and $1$ \nin $S^0=\\{-1,1\\}$ on the right factor. Then $h_i$ is a homotopy equivalence since $D_i$ and $D'_i$ are contractible. \nNotice $h_i\\circ\\iota_i$ is the quotient map that collapses $D_i$ to a point and $D'_i$ to another point.\nSince $D_i$ and $D'_i$ are disjoint contractible subcomplexes of $|K_{I_i}|$, \n$h_i\\circ\\iota_i$ is a homotopy equivalence. Therefore, $\\iota_i$ is a homotopy equivalence.\n\nTake the composite of inclusions\n\\begin{equation}\n\\label{EComposite}\n|K|=|K_1|\\incladdr{\\iota_1}{|K_{I_2}\\ast K_{J_2}|}\\incladdr{\\iota'_2}{|K_{I_3}\\ast K_{J_3}\\ast K_{J_2}|}\\incladdr{\\iota'_3}{\\cdots}\n\\incladdr{\\iota'_{\\ell-1}}{|K_{I_\\ell}\\ast K_{J_\\ell}\\ast\\cdots\\ast K_{J_2}|}.\n\\end{equation}\nThe $i^{th}$ map ($i\\geq 2$) in this composite  \n$$\n\\iota'_i\\ensuremath{\\,\\colon\\,}\n\\incl{|K_{I_i}\\ast K_{J_i}\\ast\\cdots\\ast K_{J_2}|}{}{|K_{I_{i+1}}\\ast K_{J_{i+1}}\\ast K_{J_i}\\ast\\cdots\\ast K_{J_2}|}\n$$\nis the join of the homotopy equivalence $\\iota_i$ and the identity \\seqm{|K_{J_i}\\ast\\cdots\\ast K_{J_2}|}{\\ensuremath{\\mathbbm{1}}_i}{|K_{J_i}\\ast\\cdots\\ast K_{J_2}|}, \nand since each $|K_{J_j}|\\simeq S^0$, then $\\ensuremath{\\mathbbm{1}}_i$ is homotopy equivalent to the identity \\seqm{S^{i-2}}{\\ensuremath{\\mathbbm{1}}}{S^{i-2}}.\nEach $\\iota'_i$ (and $\\iota_1$) is therefore a homotopy equivalence, and then so is composite~\\eqref{EComposite}\n(in fact, since $|K_{I_i}|\\cong S^{d+1-i}$, $\\iota_i$ and~\\eqref{EComposite} are homotopy equivalent to homeomorphisms \n\\seqm{S^{d+1-i}}{\\cong}{S^{d+1-i}} and \\seqm{S^d}{\\cong}{S^d}). \nTherefore~\\eqref{EComposite} induces a non-trivial map on cohomology,\nand so the Hochster theorem implies there is a non-trivial length $\\ell$ cup product in $H^+(\\mathcal Z_K)$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Triangulated $d$-spheres for $d=0,1,2$}\nThe only triangulated $0$-sphere is $S^0$, and the only triangulated $1$-sphere with $n\\geq 3$ vertices is the $n$-gon. \nWe will say that $C$ is a \\emph{chordless cycle} in $K$ if $C$ is a full subcomplex $K_I$ of $K$, \nand $C$ is an $m$-gon for some $m\\geq 3$. \nWhen $K$ is a graph, this is the same as $C$ being an induced cycle of $G$.\n$K$ is said to be \\emph{chordal} if it contains no chordless cycles with $4$ or more vertices.\n\n\n\\begin{lemma} \n\\label{LCup1}\nIf $K$ is a triangulation of $S^1$ on vertex set $[n]$, then\n\\begin{itemize}\n\\item[(i)] $\\filt(K)\\geq 2$ whenever $K$ has at least $4$ vertices.\n\\end{itemize}\nIf $K$ is a triangulation of $S^2$, then \n\\begin{itemize}\n\\item[(ii)] $\\filt(K)\\geq 2$ whenever $K$ has a chordless cycle;\n\\item[(iii)] $\\filt(K)\\geq 3$ whenever $K$ has a chordless cycle with at least $4$ vertices.\n\\end{itemize}\n\\end{lemma}\n\n\n\\begin{proof}\n\nLet $|K|\\cong S^1$. If $K$ has at least $n\\geq 4$ vertices, \nthen $K$ being an $n$-gon means we can take $I'\\subset [n]$, $|I'|=2$, \nsuch that $|K_{I'}|=S^0$. Then $S^0\\subsetneq K$ is a length $2$ spherical filtration of $K$. \n\nLet $|K|\\cong S^2$, and $C=K_I$ be a chordless cycle for some $I\\subset [n]$. \nWe have $|K_I|\\cong S^1$ and $K_I=\\ensuremath{\\partial}\\Delta^2$ when $K_I$ has $3$ vertices, \nin which case $K_I\\subsetneq K$ is a length $2$ spherical filtration.\nOtherwise, when $K_I$ has at least $4$ vertices, $\\filt(K)\\geq 3$ since \nthere is a spherical filtration $S^0\\subsetneq K_I\\subsetneq K$ by part~$(i)$.\n\\end{proof}\n\n\n\n\\begin{proposition} \n\\label{Pd012}\nLet $K$ be a triangulated $d$-sphere, $d=0,1,2$. Then \n$$\n1\\leq\\filt(K)=\\cuplen(\\mathcal Z_K)=\\cat(\\mathcal Z_K)\\leq d+1.\n$$\nIn particular, letting $m=\\cat(\\mathcal Z_K)$, when $d=1$ we have $m=1$ iff $K=\\ensuremath{\\partial}\\Delta^2$, and $m=2$ iff $K$ at least $4$-vertices;\nand when $d=2$ we have $m=1$ iff $K=\\ensuremath{\\partial}\\Delta^3$, $m=2$ iff $K$ has a chordless cycle but none with more than $3$ vertices,\nand $m=3$ iff $K$ has a chordless cycle with at least $4$ vertices.\n\n\\end{proposition}\n\n\\begin{proof} \n\n\nThe case $d=0$ is immediate since $S^0$ is the only triangulated $0$-sphere, and $\\z{S^0}\\cong S^3$. \nSince $K$ is well-behaved by Lemma~\\ref{Ld012Well},\nwe have by Lemma~\\ref{LSkeleta2} and Lemma~\\ref{LFiltCup}\n$$\n1\\leq\\filt(K)\\leq\\cuplen(\\mathcal Z_K)\\leq\\cat(\\mathcal Z_K)\\leq d+1,\n$$ \nso it remains to show that $\\cat(\\mathcal Z_K)\\leq\\filt(K)$. \n\nFix $d=1$. Using Lemma~\\ref{LCup1}, if $K$ has at least $4$ vertices, \nthen $\\filt(K)\\geq 2$, so $\\cat(\\mathcal Z_K)\\leq\\filt(K)$ since $\\cat(\\mathcal Z_K)\\leq d+1=2$. \nOtherwise, if $K$ has $3$-vertices, then $K=\\ensuremath{\\partial}\\Delta^2$ and $\\mathcal Z_K\\cong S^5$, so $\\filt(K)=\\cat(\\mathcal Z_K)=1$. \n\nNow fix $d=2$. \nBy Lemma~\\ref{LCup1}, $\\filt(K)\\geq 3$ whenever $K$ has chordless cycles with at least $4$ vertices, \nso $\\cat(\\mathcal Z_K)\\leq\\filt(K)$ since $\\cat(\\mathcal Z_K)\\leq d+1=3$. \nOn the other hand, suppose $K$ has chordless cycles, but none with more than $3$ vertices. \nThen $\\filt(K)=2$ by Lemma~\\ref{LCup1}, and the $1$-skeleton $\\sk{K}{1}$ is a chordal graph. \nThe chordal property is closed under vertex deletion (taking full subcomplexes).\nMoreover, recall from~\\cite{MR0186421} that chordal graphs have a \\emph{total elimination ordering}, that is,\nthey can be built up one vertex $v$ at a time in some order such that at each step the neighbours of $v$ form a clique. \nInducting on this ordering, one sees that chordal graphs are $HMF$-complexes,\ntherefore $\\cat(\\zsk{K}{1})\\leq 1$ by Proposition~\\ref{PExtractible} (this also follows the main result in~\\cite{2015arXiv150608970I}).\nUsing Lemma~\\ref{LSkeleta2}, we have $\\cat(\\mathcal Z_K)\\leq \\cat(\\zsk{K}{1})+1\\leq 2=\\filt(K)$.\nOtherwise, $K=\\ensuremath{\\partial}\\Delta^3$ when $K$ has no chordless cycles at all, so we have $\\mathcal Z_K\\cong S^7$ and $\\filt(K)=\\cat(\\mathcal Z_K)=1$. \n\n\\end{proof}\n\n\n\n\\begin{corollary}\n\\label{Cd012}\nLet $K$ be a triangulated $d$-sphere, $d=1,2$, then \n$$\n\\cat(\\zsk{K}{d-1})\\leq\\max\\{1,\\cat(\\mathcal Z_K)-1\\}=\\max\\{1,\\filt(K)-1\\}.\n$$\n\\end{corollary}\n\n\\begin{proof}\nThe last equality $\\filt(K)=\\cat(\\mathcal Z_K)$ is from the previous proposition. Let $m=\\cat(\\mathcal Z_K)$. \nUsing Proposition~\\ref{Pd012}, the statement simplifies to $\\cat(\\zsk{K}{0})\\leq 1$ when $d=1$, or $d=2$ and $m=1$.\nThis is true since $\\zsk{K}{0}$ has the homotopy type of a wedge of spheres by~\\cite{MR2138475}, \nor by Proposition~\\ref{PExtractible} (as $\\sk{K}{0}$ is a collection of disjoint points).\nFix $d=2$. As in the proof of Propostion~\\ref{Pd012}, $\\cat(\\zsk{K}{1})\\leq 1<m$ when $m=2$, since $\\filt(K)=2$. \nThis last inequality also holds when $m=3$ since $\\cat(\\zsk{K}{1})\\leq \\dim\\sk{K}{1}+1=2$ is always true by Lemma~\\ref{LSkeleta2}. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Triangulated $d$-spheres for $d\\geq 3$}\n\n\n\n\n\\begin{proposition}\n\\label{Pdinfinity}\nSuppose $K\\in\\mc C_d$, $d\\geq 0$. Then \n$$\n1\\leq\\filt(K)=\\cuplen(\\mathcal Z_K)=\\cat(\\mathcal Z_K)\\leq d+1.\n$$\n\\end{proposition}\n\n\n\n\\begin{proof}\nTriangulations of $0,1,2$-spheres are well-behaved by Lemma~\\ref{Ld012Well}, \nso iterating Lemmas~\\ref{LJoinWell} and~\\ref{LCSumWell}, any $K\\in\\mc C_d$ is well-behaved.\nThen by Lemmas~\\ref{LSkeleta2} and~\\ref{LFiltCup},\n$$\n1\\leq\\filt(K)\\leq\\cuplen(\\mathcal Z_K)\\leq\\cat(\\mathcal Z_K)\\leq d+1.\n$$ \nIt remains to show that $\\cat(\\mathcal Z_K)\\leq\\filt(K)$. The $d=0,1,2$ case is Proposition~\\ref{Pd012}.\n\nSuppose $K=L_1\\ast\\cdots\\ast L_k\\in \\mc C_d$ for some $L_i\\in\\mc C_{d_i}$, $d_i\\leq 2$, and $d_1+\\cdots+d_k=d-k-1$.\nThen $\\mathcal Z_K=\\z{L_1}\\times\\cdots\\times\\z{L_k}$, and so using Proposition~\\ref{Pd012} and iterating Lemma~\\ref{LFiltJoin},\n$$\n\\cat(\\mathcal Z_K)\\leq \\csum{i=1,\\ldots,k}{}{\\cat(\\z{L_i})} = \\csum{i=1,\\ldots,k}{}{\\filt(L_i)} \\leq \\filt(L_1\\ast\\cdots\\ast L_k)=\\filt(K).\n$$ \nMoreover, iterating Corollary~\\ref{CJoinSkeleton}, and using Corollary~\\ref{Cd012}, when each $L_i\\neq S^0$\n$$\n\\cat(\\zsk{K}{d-1})\\leq \\csum{i=1,\\ldots,k}{}{\\cat(\\zsk{L_i}{d_i-1})}+k-1\\leq\\csum{i=1,\\ldots,k}{}{\\max\\{1,\\filt(L_i)-1\\}}+k-1,\n$$\nand when each $L_i\\neq\\ensuremath{\\partial}\\Delta^{d_i+1}$, we have $\\filt(L_i)>1$, therefore \n\\begin{equation}\n\\label{ESk}\n\\cat(\\zsk{K}{d-1})\\leq \\paren{\\csum{i=1,\\ldots,k}{}{\\filt(L_i)}}-1\\leq \\filt(L_1\\ast\\cdots\\ast L_k)-1=\\filt(K)-1,\n\\end{equation}\nthe second inequality by iterating Lemma~\\ref{LFiltJoin}.\n\n\nSuppose $K=K_1\\#_{\\sigma_1}\\cdots\\#_{\\sigma_{\\ell-1}}K_\\ell$ where each $K_i=L_{1,i}\\ast\\cdots\\ast L_{k_i,i}$ \nis a join of the above form such that each $L_{j,i}$ is not the boundary of a simplex,\nand $\\sigma_i$ is a $d$-face common to $K_i$ and $K_{i+1}$ with $\\sigma_i\\cap\\sigma_j=\\emptyset$ when $i\\neq j$.\nBy Corollary~\\ref{CConnectedSum} and inequality~\\eqref{ESk}, we have \n\\begin{align*}\n\\cat(\\mathcal Z_K) & \\leq \\max\\{1,\\cat(\\zsk{K_1}{d-1}),\\ldots,\\cat(\\zsk{K_\\ell}{d-1})\\}+1\\\\\n& \\leq \\max\\{1,\\filt(K_1)-1,\\ldots,\\filt(K_\\ell)-1\\}+1\\\\\n& = \\max\\{2,\\filt(K_1),\\ldots,\\filt(K_\\ell)\\}\\\\\n& = \\filt(K_1\\#_{\\sigma_1}\\cdots\\#_{\\sigma_{\\ell-1}}K_\\ell)\\\\\n& = \\filt(K),\n\\end{align*}\nwhere the second last inequality follows from iterating Lemma~\\ref{LFiltCSum}.\n\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{TMain}}\n\nThe following is an immediate consequence of Theorem~$4.4$ in~\\cite{MR1644063} and the fact that \n\\emph{category weight} $cwgt()$ as defined there is bounded below by $1$, \nand linearly below with respect to cup products.\n\n\n\\begin{theorem}[Rudyak~\\cite{MR1644063}]\nIf $\\cat(X)\\leq m$, then\n\\begin{itemize}\n\\item[(1)] $\\cuplen(X)\\leq m$;\n\\item[(2)] Massey products $\\vbr{v_1,\\ldots,v_k}$ vanish in $H^*(X)$ \nwhenever $v_i=a_1\\cdots a_{m_i}$ and $v_j=b_1\\cdots b_{m_j}$, and $m_i+m_j>m$, for some odd $i$ and even $j$ \nand $a_s,b_t\\in H^+(X)$.~$\\hfill\\square$\n\\end{itemize}\n\\end{theorem}\n\nBy our remarks in Section~\\ref{SHochster}, and by definition $\\cuplen(X)=\\Nill H^*(X)-1$, we obtain\n\n\\begin{proposition}\n\\label{PRudyak}\nIf $\\cat(\\mathcal Z_K)\\leq m$, then $K$ is $m$-Golod.~$\\hfill\\square$ \n\\end{proposition}\n\n\nNow Theorem~\\ref{TMain} follows from Propositions~\\ref{Pd012} and~\\ref{Pdinfinity}, \nand the fact that $\\cuplen(\\mathcal Z_K)\\leq m$ when $K$ is $m$-Golod (by definition).\n\n\n\n\n\n\n\n\n\n\n\\section{Further Applications}\n\n\n\n\n\n\\subsection{Fullerenes}\nA \\emph{fullerene} $P$ is a simple $3$-polytope all of whose $2$-faces are pentagons and hexagons. \nThese are mathematical idealisations of physical fullerenes - \nspherical molecules of carbon such that each carbon atom belongs to three carbon rings, and each carbon ring is either a pentagon or hexagon. \n\n\nThe authors in~\\cite{2015arXiv151103624F} have shown that the cohomology ring of moment-angle complexes is a complete combinatorial invariant of fullerenes,\nwhile Buchstaber and Erokhovets~\\cite{2015arXiv151002948B,fullerene} show that the finer details of their cohomology encode many interesting properties of fullerenes.\nFor example, if $P^\\ast$ is the dual of $P$, then the bigraded Betti numbers of $\\z{\\ensuremath{\\partial} P^\\ast}$ count the number $k$-\\emph{belts} in $P$. \nHere, a $k$-belt of a simple polytope such as $P$ is a sequence of $2$-faces $(F_1,\\ldots,F_k)$ such that\n$F_k\\cap F_1\\neq\\emptyset$, $F_i\\cap F_{i+1}\\neq\\emptyset$ for $1\\leq i\\leq k-1$, and all other intersections are empty.\nNotice that the $k$-belts of $P$ correspond to full subcomplexes of $\\ensuremath{\\partial} P^\\ast$ that are $n$-gons.  \nBut since fullerenes can have no $3$-belts~\\cite{2015arXiv151002948B,fullerene}, $\\ensuremath{\\partial} P^\\ast$ must only have $n$-gons as full subcomplexes for $n\\geq 4$.\nMoreover, since $\\ensuremath{\\partial} P^\\ast$ is a triangulated $2$-sphere that is not a boundary of the $2$-simplex, it must have at least one such $n$-gon as a full subcomplex. \nThus, $\\filt(\\ensuremath{\\partial} P^\\ast)=3$, so by Theorem~\\ref{TMain}, \n\\begin{proposition}\nFor fullerences $P$, $\\cat(\\z{\\ensuremath{\\partial} P^\\ast})=3$ and $\\ensuremath{\\partial} P^\\ast$ is $3$-Golod.\nIn particular, all Massey products consisting of decomposable elements in $H^+(\\z{\\ensuremath{\\partial} P^\\ast})$ must vanish.~$\\hfill\\square$\n\\end{proposition}\n\n\n\n\n\n\n\\subsection{Neighbourly Complexes}\n\nFor any finite simply-connected $CW$-complex $X$, let \n$$\n\\hd(X)=\\max\\br{\\,\\max\\cset{i}{\\tilde H^i(X)\\otimes\\mb Q\\neq 0},\\,\\max\\cset{i}{Torsion(\\tilde H^{i-1}(X))\\neq 0}\\,}\n$$ \nand \n$$\n\\hc(X)=\\min\\cset{i}{\\tilde H^{i+1}(X)\\neq 0}.\n$$\nThese coincide with the dimension and connectivity of $X$ up to homotopy equivalence.\nIt is well known (c.f.~\\cite{MR516214}) that $X$ satisfies \n\\begin{equation}\n\\label{EOld}\n\\cat(X)\\leq \\frac{\\hd(X)}{\\hc(X)+1}.\n\\end{equation} \n\nA version of the Hochster formula also holds for real moment-angle complexes, namely,\n\\begin{equation}\nH^*(\\mathbb R\\mathcal Z_K)\\cong\\bigoplus_{I\\subseteq[n]}\\tilde H^*(\\Sigma|K_I|). \n\\end{equation} \nThus, \n$$\n\\hd(\\mathbb R\\mathcal Z_K)=1+\\max\\cset{\\,\\hd(|K_I|)}{I\\subseteq[n]\\,}\\,\\leq\\, 1+\\dim K\n$$\nand\n$$\n\\hc(\\mathbb R\\mathcal Z_K)=1+\\min\\cset{\\,\\hc(|K_I|)}{I\\subseteq[n]\\,},\n$$\nand using the inequality $\\cat(\\mathcal Z_K)\\leq\\cat(\\mathbb R\\mathcal Z_K)$ from Corollary~\\ref{CRealMAC}, \n\\begin{proposition}\n\\label{PImproved}\nIt holds that \n$$\\cat(\\mathcal Z_K)\\leq \\frac{\\hd(\\mathbb R\\mathcal Z_K)}{\\hc(\\mathbb R\\mathcal Z_K)}.$$ \n\\end{proposition} \nComparing the Hochster formula for $H^*(\\mathbb R\\mathcal Z_K)$ to the Hochster formula for $H^*(\\mathcal Z_K)$ in Section~\\ref{SHochster}, \none sees that the inequality $\\frac{\\hd(\\mathbb R\\mathcal Z_K)}{\\hc(\\mathbb R\\mathcal Z_K)}\\leq \\frac{\\hd(\\mathcal Z_K)}{\\hc(\\mathcal Z_K)}$ usually holds, \nwith the disparity between these two often being very large. In such case, the bound in Proposition~\\ref{PImproved}\nis an improvement over what one would get by applying~\\eqref{EOld} directly to $X=\\mathcal Z_K$.\n\nConsider, for instance, the case of $k$-neighbourly complexes. \nA simplicial complex $K$ on vertex set $[n]$ is said to be $k$-\\emph{neighbourly} if every subset of $k$ or less vertices in $[n]$ is a face of $K$.\nIn this case $H_i(K_I)=0$ for $i\\leq k-2$ and each $I\\subseteq[n]$, so $\\hc(\\mathbb R\\mathcal Z_K)\\geq k-1$. Therefore\nwe have the following result.\n\n\\begin{theorem}\nIf $K$ is $k$-neighbourly, $\\cat(\\mathcal Z_K)\\leq \\frac{1+\\dim K}{k}$. In particular, $K$ is $\\paren{\\frac{1+\\dim K}{k}}$-Golod.~$\\hfill\\square$\n\\end{theorem}\n\n\\begin{example}\nSuppose $K$ is $\\ceil{\\frac{n}{2}}$-neighbourly. \nIf $K\\neq\\Delta^{n-1}$, then $\\dim K\\leq n-2$.\nThus, $\\cat(\\mathcal Z_K)\\leq 1$ ($\\mathcal Z_K$ is a $co$-$H$-space) and $K$ is $1$-Golod.\n\\end{example}\n\n\n\n\n\n\n\n\n\n\\subsection{Simplicial Wedges}\n\n\nWe recall the \\emph{simplicial wedge} construction defined in~\\cite{MR593648,MR3426378}. \nLet $K$ be a simplicial complex on vertex set $\\{v_1,...,v_n\\}$, and for any face $\\sigma\\in K$, \ndefine the \\emph{link} of $\\sigma$ in $K$ the subcomplex of $K$ given by\n$$\n\\link_K(\\sigma)=\\cset{\\tau\\in K}{\\tau\\cap\\sigma=\\emptyset,\\,\\tau\\cup\\sigma\\in K}.\n$$\nBy \\emph{doubling} a vertex $v_i$ in $K$, we obtain a new simplicial complex $K(v_i)$ on vertex set \n$$\n\\{v_1,\\ldots,v_{i-1},v_{i1},v_{i2},v_{i+1},\\ldots,v_n\\}\n$$ \ndefined by\n$$\nK(v_i) = (v_{i1},v_{i2})\\ast \\link_K(v_i)\\cup_{\\{v_{i1},v_{i2}\\}\\ast\\link_K(v_i)} \\{v_{i1},v_{i2}\\}\\ast K\\backslash\\{v_i\\},\n$$\nwhere $(v_{i1},v_{i2})$ is the $1$-simplex with vertices $\\{v_{i1},v_{i2}\\}$.\nOne can of course iterate this construction by reapplying the doubling operation to successive complexes,\nand the order of vertices on which this is done is irrelevant. \nTo this end, take any sequence $J=(j_1,\\ldots,j_n)$ of non-negative integers, \nlet $u_j$ be the $j^{th}$ vertex in the sequence $v_1,v_{12},\\ldots,v_{1j_1},v_2,\\ldots,v_n,v_{n2},\\ldots,v_{nj_n}$\nand $N=j_1+\\cdots+j_n$, and define \n$$\nK(J)=K_N\n$$ \nwhere $K_{j+1}=K_j(u_{j+1})$ and $K_0=K$.\nIn algebraic terms, the Stanley-Reisner ideal of $K(J)$ is obtained from the Stanley-Reisner ideal of $K$ by replacing each vertex $v_i$ \nby $v_{i1},v_{i2},\\ldots,v_{ij_i}$ in each monomial. \nThis construction arises in combinatorics (see~\\cite{MR593648}) and has the important property that if $K$ is the boundary of the dual of $d$-polytope, \nthen $K(J)$ is the boundary of the dual of a $(d+N)$-polytope.\n\n\n\n\n\\begin{theorem}\n\\label{TSW} For any $J$, $\\cat(\\z{K(J)})\\leq \\cat(\\mathcal Z_K)$.\n\\end{theorem}\n\n\n\\begin{proof}\nLet $(D^J,S^J)$ be the sequence of pairs $((D^{2j_1+2},S^{2j_1+1}),\\ldots,(D^{2j_n+2},S^{2j_n+1}))$.\nBy~\\cite{MR3426378,MR3084441}, there is a homeomorphism\n$$\n\\z{K(J)}\\,=\\,(D^2,S^1)^{K(J)}\\,\\cong\\,(D^J,S^J)^K,\n$$\nand by Proposition~\\ref{PCoordSusp}, $\\cat((D^J,S^J)^K)\\leq \\cat((D^2,S^1)^K)=\\cat(\\mathcal Z_K)$.\n\\end{proof}\n\n\nThis result becomes algebraically useful when a good bound on $\\cat(\\mathcal Z_K)$ is known.\nFor instance, there are many examples of complexes $K$ for which $\\cat(\\mathcal Z_K)=1$, \nduals of sequential Cohen Macaulay and shellable complexes, and chordal flag complexes to name a few~\\cite{MR3461047,2013arXiv1306.6221I}.\nIn each of these examples $\\cat(\\z{K(J)})\\leq 1$, so $K(J)$ is Golod. \nGenerally, $K(J)$ is at least $(1+\\dim K)$-Golod since $\\cat(\\mathcal Z_K)\\leq 1+\\dim K$, \neven though $\\dim K(J)-\\dim K$ can be arbitrarily large.\n\nNotice $K(J)$ is a triangulation of a $(d+N)$-sphere whenever $K$ is a triangulation of a $d$-sphere.\nCombining Theorem~\\ref{TSW} and Proposition~\\ref{PRudyak},\nthe range of spheres for which Theorem~\\ref{TMain} holds generalizes as follows. \n\n\n\\begin{corollary}\nLet $K$ be be any triangulated $d$-sphere for $d=0,1,2$, or $K\\in \\mc C_d$ when $d\\geq 3$,\nand let $m=\\filt(K)$ (equivalently $m=\\cuplen(\\mathcal Z_K))$. Then $\\cat(\\z{K(J)})\\leq m$ and $K(J)$ is $m$-Golod.~$\\hfill\\square$\n\\end{corollary}\n\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}