n-1$). By (\ref{bgcellw}), $$ \bideg(D_II_J0_LT_P1_Q)=(j-p,2(i+p)). $$ Now we calculate $\chi_r(\wk)$ using (\ref{chipw}) and (\ref{ci}): $$ \chi_r(\wk)=\mathop{\sum_{i,j,l,p}}\limits_{i+p=r,l\ge1}(-1)^{j-p} f_{i+j+l-1}\textstyle\binom{i+j+l}i\binom{j+l}l\binom{m-i-j-l}p. $$ Substituting $s=i+j+l$ above we obtain \begin{align*} \chi_r(\wk)&=\mathop{\sum_{l,s,p}}\limits_{l\ge1}(-1)^{s-r-l} f_{s-1}\textstyle{\binom{s}{r-p}\binom{s-r+p}l\binom{m-s}p}\\ &=\sum_{s,p}\Bigl((-1)^{s-r}f_{s-1}{\textstyle\binom{s}{r-p}\binom{m-s}p} \sum_{l\ge1}(-1)^l{\textstyle\binom{s-r+p}l}\Bigl) \end{align*} Since $$ \sum_{l\ge1}(-1)^l{\textstyle\binom{s-r+p}l}= \left\{ \begin{aligned} -1,&\quad s>r-p,\\ 0,&\quad s\le r-p \end{aligned}, \right. $$ we get \begin{align*} \chi_r(\wk)&=-\mathop{\sum_{s,p}}\limits_{s>r-p}(-1)^{s-r} f_{s-1}{\textstyle\binom{s}{r-p}\binom{m-s}p}\\ &=-\sum_{s,p}(-1)^{r-s} f_{s-1}{\textstyle\binom{s}{r-p}\binom{m-s}p}+ \sum_s(-1)^{r-s}f_{s-1}{\textstyle\binom{m-s}{r-s}}. \end{align*} The second sum in the above formula is exactly $\chi_r(\zk)$ (see~(\ref{chipzk})). To calculate the first sum we observe that $\sum_p\bin{s}{r-p}\bin{m-s}p=\bin mr$ (this follows from calculating the coefficient of $\alpha^r$ in both sides of the identity $(1+\alpha)^s(1+\alpha)^{m-s}=(1+\alpha)^m$). Hence, $$ \chi_r(\wk)=-\sum_s(-1)^{r-s}f_{s-1}{\textstyle\binom mr}+\chi_r(\zk)= (-1)^r{\textstyle\binom mr}\bigl( \chi(K)-1 \bigr)+\chi_r(\zk), $$ since $-\sum_s(-1)^sf_{s-1}=\chi(K)-1$ (remember that $f_{-1}=1$). Finally, using~(\ref{hchi}) we calculate \begin{align*} \chi(\wk;t)=\sum_{r=0}^m\chi_r(\wk)t^{2r}= \sum_{r=0}^m(-1)^r{\textstyle\binom mr}\bigl( \chi(K)-1 \bigr)t^{2r}+ \sum_{r=0}^m\chi_r(\zk)t^{2r}\\ =\bigl( \chi(K)-1 \bigr)(1-t^2)^m+ (1-t^2)^{m-n}(h_0+h_1t^2+\cdots+h_nt^{2n}). \end{align*} \end{proof} Suppose now that $K$ is an orientable simplicial manifold. It is easy to see that in this case $W_K$ is also orientable. Hence, there are relative Poincar\'e duality isomorphisms: \begin{equation}\label{rpd} H_k(W_K)\cong H^{m+n-k}(W_K,\partial W_K), \quad k=0,\ldots,m. \end{equation} \begin{corollary}[Generalised Dehn--Sommerville equations] \label{DSsm} The following relations hold for the $h$-vector $(h_0,h_1,\ldots,h_n)$ of any orientable simplicial manifold~$K^{n-1}$: $$ h_{n-i}-h_i=(-1)^i\bigl(\chi(K^{n-1})-\chi(S^{n-1})\bigr) {\textstyle\binom ni},\quad i=0,1,\ldots,n, $$ where $\chi(S^{n-1})=1+(-1)^{n-1}$ is the Euler characteristic of an $(n-1)$-sphere. \end{corollary} \begin{proof} By Theorem~\ref{homotwk}, $H_k(W_K)=H_k(\wk)$ and $H^{m+n-k}(W_K,\partial_c W_K)= H^{m+n-k}(\zk,T^m)$. Moreover, it can be seen in the same way as in Corollary~\ref{bpd} that relative Poincar\'e duality isomorphisms~(\ref{rpd}) regard the bigraded structures in the (co)homology of $\wk$ and $(\zk,T^m)$. It follows that \begin{gather} \notag b_{-q,2p}(\wk)=b_{-(m-n)+q,2(m-p)}(\zk,T^m),\\ \notag \chi_p(\wk)=(-1)^{m-n}\chi_{m-p}(\zk,T^m),\\ \label{relchid} \chi(\wk;t)=(-1)^{m-n}t^{2m}\chi(\zk,T^m;\textstyle\frac1t). \end{gather} Using (\ref{rhchi}) we calculate \begin{multline*} (-1)^{m-n}t^{2m}\chi(\zk,T^m;{\textstyle\frac1t})\\ =(-1)^{m-n}t^{2m}(1-t^{-2})^{m-n}(h_0+h_1t^{-2}+\cdots+h_nt^{-2n})\\ -(-1)^{m-n}t^{2m}(1-t^{-2})^m\\ =(1-t^2)^{m-n}(h_0t^{2n}+h_1t^{2n-2}+\cdots+h_n)+ (-1)^{n-1}(1-t^2)^m. \end{multline*} Substituting the formula for $\chi(\wk;t)$ from Theorem~\ref{chitwk} and the above expression into~(\ref{relchid}) we obtain \begin{multline*} (1-t^2)^{m-n}(h_0+h_1t^2+\cdots+h_nt^{2n})+\bigl(\chi(K)-1\bigr) (1-t^2)^m\\=(1-t^2)^{m-n}(h_0t^{2n}+h_1t^{2n-2}+\cdots+h_n)+ (-1)^{n-1}(1-t^2)^m. \end{multline*} Calculating the coefficient of $t^{2i}$ in both sides after dividing the above identity by $(1-t^2)^{m-n}$, we obtain $h_{n-i}-h_i=(-1)^i\bigl(\chi(K^{n-1})-\chi(S^{n-1})\bigr) {\textstyle\binom ni}$, as needed. \end{proof} If $|K|=S^{n-1}$ or $n-1$ is odd, Corollary~\ref{DSsm} gives the classical equations $h_{n-i}=h_i$. \begin{corollary} If $K^{n-1}$ is a simplicial manifold with $h$-vector $(h_0,\ldots,h_n)$ then $$ h_{n-i}-h_i=(-1)^i(h_n-1){\textstyle\binom ni},\quad i=0,1,\ldots,n. $$ \end{corollary} \begin{proof} Since $\chi(K^{n-1})=1+(-1)^{n-1}h_n$, $\chi(S^{n-1})=1+(-1)^{n-1}$, we have $$ \chi(K^{n-1})-\chi(S^{n-1})=(-1)^{n-1}(h_n-1)=(h_n-1) $$ (the coefficient $(-1)^{n-1}$ can be dropped since for odd $n-1$ the left hand side is zero). \end{proof} \begin{corollary} For any $(n-1)$-dimensional orientable simplicial manifold the numbers $h_{n-i}-h_i$, $i=0,1,\ldots,n$, are homotopy invariants. In particular, they do not depend on a triangulation. \end{corollary} In the particular case of $PL$-manifolds the topological invariance of numbers $h_{n-i}-h_i$ was firstly observed by Pachner in~\cite[(7.11)]{Pac2}. \begin{figure} \begin{center} \begin{picture}(30,30) \multiput(0,0)(0,10){4}{\line(1,0){30}} \multiput(0,0)(10,0){4}{\line(0,1){30}} \put(0,20){\line(1,1){10}} \put(0,10){\line(1,1){20}} \put(0,0){\line(1,1){30}} \put(10,0){\line(1,1){20}} \put(20,0){\line(1,1){10}} \end{picture} \end{center} \caption{Triangulation of $T^2$ with $\mb f=(9,27,18)$, $\mb h=(1,6,12,-1)$.} \end{figure} \begin{example} Consider triangulations of the 2-torus~$T^2$. We have $n=3$, $\chi(T^2)=0$. From $\chi(K^{n-1})=1+(-1)^{n-1}h_n$ we deduce $h_3=-1$. Corollary~\ref{DSsm} gives $$ h_3-h_0=-2,\quad h_2-h_1=6. $$ For instance, the triangulation on Figure~10 has $f_0=9$ vertices, $f_1=27$ edges and $f_2=18$ triangles. The corresponding $h$-vector is $(1,6,12,-1)$. \end{example} \section{Subspace arrangements and cohomology rings of their complements} \subsection{Summary of results on the cohomology of general arrangement complements} \label{gene} An {\it arrangement\/} is a finite set $\A=\{L_1,\ldots,L_r\}$ of planes (affine subspaces) in some affine space (either real or complex). For any arrangement $\A=\{L_1,\ldots,L_r\}$ in $\C^m$ define its {\it support\/} $|\A|$ as $$ |\A|:=\bigcup_{i=1}^rL_i\subset\C^m, $$ and its {\it complement\/} $U(\A)$ as $$ U(\A):=\C^m\setminus|\A|, $$ and similarly for arrangements in~$\R^m$. Arrangements and their complements play a pivotal r\^ole in many constructions of combinatorics, algebraic and symplectic geometry etc.; they also arise as configuration spaces for different classical mechanical systems. In the study of arrangements it is very important to get a sufficiently detailed description of the topology of complements $U(\A)$ (this includes number of connected components, homotopy type, homology groups, cohomology ring etc.). A host of elegant results in this direction appeared during the last three decades, however, the whole picture is far from being complete. The theory ascends to work of Arnold~\cite{Ar}, which described the classifying space for the braid group $B_n$ as the complement of arrangement of all diagonal hyperplanes $\{z_i=z_j\}$ in~$\C^n$. The cohomology ring of this complement was also calculated there. This result was generalised by Brieskorn~\cite{Br} and motivated the further development of the theory of {\it complex hyperplane arrangements\/} (i.e. arrangements of codimension-one complex affine subspaces). One of the main results here is the following. \begin{theorem}[\cite{Ar}, \cite{Br}, \cite{OS}] Let $\A=\{L_1,\ldots,L_r\}$ be an arrangement of complex hyperplanes in~$\C^m$, and the hyperplane $L_i$ is the zero set of linear function $l_i$, $i=1,\ldots,r$. Then the integer cohomology algebra of the complement $\C^m\setminus|\A|$ is isomorphic to the algebra generated by closed differential 1-forms $\frac 1{2\pi i}\frac{dl_i}{l_i}$. \end{theorem} \noindent Relations between the forms $\frac 1{2\pi i}\frac{dl_i}{l_i}$ are also explicitly described. In the case of diagonal hyperplanes $\{z_i=z_j\}$ we have the forms $\omega_{ij}=\frac1{2\pi i}\frac{d(z_i-z_j)}{z_i-z_j}$, which are subject to the following relations: $$ \omega_{ij}\wedge\omega_{jk}+\omega_{jk}\wedge\omega_{ki}+ \omega_{ki}\wedge\omega_{ij}=0. $$ The theory of complex hyperplane arrangements is probably the most well understood part of the whole study. Several surveys and monographs are available; we mention just~\cite{OT}, where further references can be found. Relationships of {\it real\/} hyperplane arrangements with polytopes and {\it oriented matroids\/} are discussed in~\cite[Lecture~7]{Zi}. In the general situation, the Goresky--MacPherson theorem~\cite[Part~III]{GM} expresses the cohomology groups $H^{i}(U(\mathcal A))$ (without ring structure) as a sum of homology groups of subcomplexes of a certain simplicial complex. We formulate this result below. For a detailed survey of general arrangements we refer to~\cite{Bj}. Some important results in this direction can be found in monograph~\cite{Va}. Let $\A=\{L_1,\ldots,L_r\}$ be an arrangement of planes in~$\R^n$. The intersections $$ v=L_{i_1}\cap\dots\cap L_{i_k} $$ form a poset $(\mathcal P,<)$ with respect to the inclusion (i.e. $vv}=\{x\in\P\::\:x>v\}. $$ \begin{theorem}[{Goresky and MacPherson~\cite[Part~III]{GM}}]\label{GMf} The following formula holds for the homology of the complement~$U(\A)${\rm:} $$ H_i\bigl(U(\A);\Z\bigr)=\bigoplus_{v\in\P} H^{n-d(v)-i-1}\bigl(K(\P_{>v}),K(\P_{v,T});\Z\bigr), $$ with the agreement that $H^{-1}(\emptyset,\emptyset)=\Z$. \end{theorem} \noindent The proof of this theorem uses the {\it stratified Morse theory\/} developed in~\cite{GM}. \begin{remark} The homology groups of a complex arrangement in $\C^n$ can be calculated by regarding it as a real arrangement in~$\R^{2n}$. \end{remark} The cohomology {\it rings\/} of the complements of arrangements are much more subtle. In general, the integer cohomology ring of $U(\A)$ {\it is not\/} determined by the poset~$\P$. An approach to calculating the cohomology algebra of the complement $U(\A)$ was proposed by De Concini and Procesi~\cite{dCP}. In particular, they proved that the {\it rational\/} cohomology ring of $U(\A)$ is determined by the combinatorics of intersections. This result was extended by Yuzvinsky in~\cite{Yu}. \subsection{Coordinate subspace arrangements and cohomology of~$\zk$.} \label{coor} An arrangement $\A=\{L_1,\ldots,L_r\}$ is called {\it coordinate\/} if every plane~$L_i$, $i=1,\ldots,r$, is a coordinate subspace. In this section we apply the results of chapter~4 to cohomology algebras of the complements of complex coordinate subspace arrangements. The case of real coordinate arrangements is discussed at the end of the section. Any coordinate subspace of $\C^m$ has the form \begin{equation} \label{li} L_I=\{(z_1,\ldots,z_m)\in\C^m\::\:z_{i_1}=\cdots=z_{i_k}=0\}, \end{equation} where $I=\{i_1,\ldots,i_k\}$ is a subset of~$[m]$. Obviously, $\dim L_I=m-\#I$. \begin{construction}\label{casim} For each simplicial complex $K$ on the set $[m]$ define the complex coordinate subspace arrangement $\mathcal{CA}(K)$ as the set of subspaces $L_I$ such that $I$ is not a simplex of~$K$: $$ \mathcal{CA}(K)=\{L_I\::\:I\notin K\}. $$ Denote the complement of $\mathcal{CA}(K)$ by~$U(K)$, that is \begin{equation} \label{compl} U(K)=\C^m\setminus\bigcup_{I\notin K}L_I. \end{equation} If $K'\subset K$ is a subcomplex, then $U(K')\subset U(K)$. \end{construction} \begin{proposition} The assignment $K\mapsto U(K)$ defines a one-to-one order-preserving correspondence between simplicial complexes on the set~$[m]$ and complements of coordinate subspace arrangements in~$\C^m$. \end{proposition} \begin{proof} Suppose $\mathcal{CA}$ is a coordinate subspace arrangement in~$\C^m$. Define \begin{equation}\label{ka} K(\mathcal{CA}):=\{I\subset[m]\::\:L_I\not\subset|\mathcal{CA}|\}. \end{equation} Obviously, $K(\mathcal{CA})$ is a simplicial complex. By the construction, $K(\mathcal{CA})$ depends only on $|\mathcal{CA}|$ (i.e. on $U(\mathcal{CA})$) and $U(K(\mathcal{CA}))=U(\mathcal{CA})$, whence the proposition follows. \end{proof} If a coordinate subspace arrangement $\mathcal A$ contains a hyperplane, say $\{z_i=0\}$, then its complement $U(\mathcal A)$ is factorised as $U(\mathcal A_0)\times\C^{*}$, where $\mathcal A_0$ is a coordinate subspace arrangement in the hyperplane $\{z_i=0\}$ and $\C^{*}=\C\setminus\{0\}$. Thus, for any coordinate subspace arrangement $\mathcal A$ the complement $U(\mathcal A)$ decomposes as $$ U(\mathcal A)=U(\mathcal A')\times(\C^{*})^k, $$ were $\mathcal A'$ is a coordinate arrangement in $\C^{m-k}$ that does not contain hyperplanes. On the other hand,~(\ref{ka}) shows that $\mathcal{CA}$ contains the hyperplane $\{z_i=0\}$ if and only if $\{i\}$ is not a vertex of $K(\mathcal{CA})$. It follows that $U(K)$ is the complement of a coordinate arrangement without hyperplanes if and only if the vertex set of $K$ is the whole~$[m]$. Keeping in mind these remarks, we restrict ourselves to coordinate subspace arrangements without hyperplanes and simplicial complexes on the vertex set~$[m]$. \begin{remark} In terms of Construction \ref{nsc} we have $U(K)=K_\bullet(\C,\C^*)$. \end{remark} \begin{example} \label{uk} 1. If $K=\D^{m-1}$ (the $(m-1)$-simplex) then $U(K)=\C^m$. 2. If $K=\partial\D^{m-1}$ (the boundary of simplex) then $U(K)=\C^m\setminus\{0\}$. 3. If $K$ is a disjoint union of $m$ vertices, then $U(K)$ is obtained by removing all codimension-two coordinate subspaces $z_i=z_j=0$, $i,j=1,\ldots,m$, from~$\C^m$. \end{example} The action of the algebraic torus $(\C^*)^m$ on $\C^m$ descends to~$U(K)$. In particular, the standard action of the torus $T^m$ is defined on~$U(K)$. The quotient $U(K)/T^m$ can be identified with $U(K)\cap\R^m_+$, where $\R^m_+$ is regarded as a subset of~$\C^m$. \begin{lemma} \label{zu} $\cc(K)\subset U(K)\cap\R^m_+$ and $\zk\subset U(K)$ (see Construction~{\rm\ref{cck}} and~{\rm(\ref{zkwk})}). \end{lemma} \begin{proof} Take $y=(y_1,\ldots,y_m)\in\cc(K)$. Let $I=\{i_1,\ldots,i_k\}$ be the maximal subset of $[m]$ such that $y\in L_I\cap\R^n_+$ (i.e. $y_{i_1}=\dots=y_{i_k}=0$). Then it follows from the definition of $\cc(K)$ (see~(\ref{fcck})) that $I$ is a simplex of~$K$. Hence, $L_I\notin\mathcal{CA}(K)$ and $y\in U(K)$. Thus, the first statement is proved. Since $\cc(K)$ is the quotient of~$\zk$, the second assertion follows from the first one. \end{proof} \begin{theorem} \label{he1} There is an equivariant deformation retraction $U(K)\to\zk$. \end{theorem} \begin{proof} First, we construct a deformation retraction $r:U(K)\cap\R^m_+\to\cc(K)$. This is done inductively. We start from the boundary complex of an $(m-1)$-simplex and remove simplices of positive dimensions until we obtain~$K$. On each step we construct a deformation retraction, and the composite map would be a required retraction~$r$. If $K=\partial\D^{m-1}$ is the boundary complex of an $(m-1)$-simplex, then $U(K)\cap\R^m_+=\R^m_+\setminus\{0\}$. In this case the retraction $r$ is shown on Figure~11. \begin{figure} \begin{picture}(120,45) \put(45,5){\circle{2}} \put(80,5){\circle*{2}} \put(45,40){\circle*{2}} \put(80,40){\circle*{2}} \put(46,5){\vector(1,0){24}} \put(46,5){\line(1,0){34}} \put(45.8,5.2){\vector(4,1){24}} \put(45.8,5.2){\line(4,1){34}} \put(45.5,5.5){\vector(2,1){24}} \put(45.5,5.5){\line(2,1){34}} \put(45.8,5.8){\vector(4,3){24}} \put(45.8,5.8){\line(4,3){34}} \put(46,6){\vector(1,1){24}} \put(46,6){\line(1,1){34}} \put(45,6){\vector(0,1){24}} \put(45,6){\line(0,1){34}} \put(45.2,5.8){\vector(1,4){6}} \put(45.2,5.8){\line(1,4){8.5}} \put(45.5,5.5){\vector(1,2){12}} \put(45.5,5.5){\line(1,2){17}} \put(45.8,5.2){\vector(3,4){18}} \put(45.8,5.2){\line(3,4){26}} \linethickness{1mm} \put(80,5){\line(0,1){35}} \put(45,40){\line(1,0){35}} \end{picture} \caption{The retraction $r:U(K)\cap\R^m_+\to\cc(K)$ for $K=\partial\D^{m-1}$.} \end{figure} Now suppose that $K$ is obtained by removing one $(k-1)$-dimensional simplex $J=\{j_1,\ldots,j_k\}$ from simplicial complex~$K'$, that is $K\cup J=K'$. By the inductive hypothesis, the there is a deformation retraction $r':U(K')\cap\R^m_+\to\cc(K')$. Let $a\in\R^m_+$ be the point with coordinates $y_{j_1}=\ldots=y_{j_k}=0$ and $y_i=1$ for $i\notin J$. Since $J$ is not a simplex of~$K$, we have $a\notin U(K)\cap\R^m_+$. At the same time, $a\in C_J$ (see~(\ref{ijface})). Hence, we may apply the retraction from Figure~11 on the face $C_J\subset I^m$, with centre at~$a$. Denote this retraction by~$r_J$. Then $r=r_J\circ r'$ is the required deformation retraction. The deformation retraction $r:U(K)\cap\R^m_+\to\cc(K)$ is covered by an equivariant deformation retraction $U(K)\to\zk$. \end{proof} In the case $K=K_P$ (i.e. $K$ is a polytopal simplicial sphere corresponding to a simple polytope~$P^n$) the deformation retraction $U(K_P)\to\zp$ from Theorem~\ref{he1} can be realised as the orbit map for an action of a contractible group. We denote $U(P^n):=U(K_P)$. Set $$ \R^m_>=\{(y_1,\ldots,y_m)\in\R^m\::\:y_i>0,\; i=1,\ldots,m\}\subset\R^m_+. $$ Then $\R^m_>$ is a group with respect to the multiplication, and it acts on $\R^m$, $\C^m$ and $U(P^n)$ by coordinatewise multiplication. There is the isomorphism $\exp:\R^m\to\R^m_>$ between the additive and the multiplicative groups taking $(y_1,\ldots,y_m)\in\R^m$ to $(e^{y_1},\ldots,e^{y_m})\in\R^m_>$. Let us consider the $m\times(m-n)$-matrix $W$ introduced in Construction~\ref{dist} for any simple polytope~(\ref{ptope}). \begin{proposition} \label{wprop} For any vertex $v=F_{i_1}\cap\cdots\cap F_{i_n}$ of $P^n$ the maximal minor $W_{\hat i_1\ldots\hat i_n}$ which is obtained by deleting $n$ rows $i_1,\ldots,i_n$ from $W$ is non-degenerate: $\det W_{\hat i_1\ldots \hat i_n}\ne0$. \end{proposition} \begin{proof} If $\det W_{\hat i_1\ldots \hat i_n}=0$ then vectors $\mb l_{i_1},\ldots,\mb l_{i_n}$ (see~(\ref{ptope})) are linearly dependent, which is impossible. \end{proof} The matrix $W$ defines the subgroup \begin{equation}\label{rw} R_W=\bigr\{(e^{w_{11}\tau_1+\cdots+w_{1,m-n}\tau_{m-n}},\ldots, e^{w_{m1}\tau_1+\cdots+w_{m,m-n}\tau_{m-n}})\bigl\}\subset\R^m_>, \end{equation} where the parameters $\tau_1,\ldots,\tau_{m-n}$ vary over~$\R^{m-n}$. Obviously, $R_W\cong\R^{m-n}_>$. \begin{theorem}[{\cite[Theorem~2.3]{BP4}}] \label{zpos} The subgroup $R_W\subset\R^m_>$ acts freely on $U(P^n)\subset\C^m$. The composite map $\zp\hookrightarrow U(P^n)\to U(P^n)/R_W$ of the embedding $i_e$ (Lemma~{\rm\ref{ie}}) and the orbit map is an equivariant diffeomorphism (with respect to the $T^m$-actions). \end{theorem} \begin{proof} A point from $\C^m$ has the non-trivial isotropy subgroup with respect to the action of $\R^m_>$ on $\C^m$ if and only if at least one of its coordinates vanishes. It follows from~(\ref{compl}) that if a point $x\in U(P^n)$ has some zero coordinates, then the corresponding facets of $P^n$ have at least one common vertex $v\in P^n$. Let $v=F_{i_1}\cap\cdots\cap F_{i_n}$. The isotropy subgroup of the point $x$ with respect to the action of the subgroup $R_W$ is non-trivial only if some linear combination of columns of $W$ lies in the coordinate subspace spanned by $\mb e_{i_1},\ldots,\mb e_{i_n}$. But this implies that $\det W_{\hat i_1\ldots \hat i_n}=0$, which contradicts Proposition~\ref{wprop}. Thus, $R_W$ acts freely on $U(P^n)$. To prove the second part of the theorem it is sufficient to show that each orbit of the action of $R_W$ on $U(P^n)\subset\C^m$ intersects the image $i_e(\zp)$ at a single point. Since the embedding $i_e$ is equivariant with respect to the $T^m$-actions, the latter statement is equivalent to that each orbit of the action of $R_W$ on $U(P^n)\cap\R^m_+$ intersects the image $i_P(P^n)$ (see Theorem~\ref{thcubpol}) at a single point. Let $y\in i_P(P^n)\subset\R^m$. Then $y=(y_1,\ldots,y_m)$ lies in some $n$-face $i_P(C^n_v)$ of the cube $I^m\subset\R^m$, see~(\ref{cubpolmap}). We need to show that the $(m-n)$-dimensional subspace spanned by the vectors $(w_{11}y_1,\ldots,w_{m1}y_m)^t,\ldots,(w_{1,m-n}y_1,\ldots,w_{m,m-n}y_m)^t$ is in general position with the $n$-face $i_P(C^n_v)$ of~$I^m$. This follows directly from~(\ref{cubpolmap}) and Proposition~\ref{wprop}. \end{proof} Suppose now that $P^n$ is a lattice simple polytope, and let $M_P$ be the corresponding toric variety (Construction~\ref{nf}). Along with the real subgroup $R_W\subset\R^m_>$~(\ref{rw}) we define $$ C_W=\bigl\{(e^{w_{11}\phi_1+\cdots+w_{1,m-n}\phi_{m-n}},\ldots, e^{w_{m1}\phi_1+\cdots+w_{m,m-n}\phi_{m-n}})\bigr\} \subset (\C^{*})^m, $$ where the parameters $\phi_1,\ldots,\phi_{m-n}$ vary over~$\C^{m-n}$. Obviously, $C_W\cong(\C^*)^{m-n}$. It is shown in \cite{Au}, \cite{Ba},~\cite{Co2} that $C_W$ acts freely on $U(P^n)$ and the toric variety $M_P$ can be identified with the orbit space (or {\it geometric quotient\/}) $U(P^n)/C_W$. Thus, we have the following commutative diagram: \begin{equation}\label{uptv} \begin{CD} U(P^n) @>R_W\cong\R^{m-n}_{>}>> \zp\\ @VC_W\cong(\C^{*})^{m-n}VV @VVT^{m-n}V\\ M_P @= M_P. \end{CD} \end{equation} \begin{remark} It can be shown~\cite[Theorem~2.1]{Co2} that {\it any\/} toric variety $M_\Sigma$ corresponding to a fan $\Sigma\subset\R^n$ with $m$ one-dimensional cones can be identified with the universal categorical quotient $U(\mathcal{CA}_\Sigma)/G$, where $U(\mathcal{CA}_\Sigma)$ is the complement of a certain coordinate arrangement (determined by the fan~$\Sigma$) and $G\cong(\C^*)^{m-n}$. The categorical quotient becomes the geometric quotient if and only if the fan $\Sigma$ is simplicial. In this case $U(\mathcal{CA}_\Sigma)=U(K_\Sigma)$. \end{remark} On the other hand, if the projective toric variety $M_P$ is non-singular, then $M_P$ is a symplectic manifold of dimension~$2n$, and the action of $T^n$ on it is Hamiltonian~\cite{Au}. In this case the diagram~(\ref{uptv}) displays $M_P$ as the result of the process of {\it symplectic reduction\/}. Namely, let $H_W\cong T^{m-n}$ be the maximal compact subgroup in~$C_W$, and $\mu:\C^m\to\R^{m-n}$ the {\it moment map\/} for the Hamiltonian action of $H_W$ on~$\C^m$. Then for any regular value $a\in\R^{m-n}$ of the map $\mu$ there is the following diffeomorphism: $$ \mu^{-1}(a)/H_W\longrightarrow U(P^n)/C_W=M_P $$ (details can be found in~\cite{Au}). In this situation $\mu^{-1}(a)$ is exactly our manifold~$\zp$. This gives us another interpretation of the manifold $\zp$ as the level surface for the moment map (in the case when $P^n$ can be realised as the quotient of a non-singular projective toric variety). \begin{example} Let $P^n=\D^n$ (the $n$-simplex). Then $m=n+1$, $U(P^n)=\C^{n+1}\setminus\{0\}$. $R_W\cong\R_>$, $C_W\cong\C^{*}$ and $H_W\cong S^1$ are the diagonal subgroups in $\R^{n+1}_>$, $(\C^{*})^{n+1}$ and $T^{m+1}$ respectively (see Example~\ref{simdist}). Hence, $\zp\cong S^{2n+1}=(\C^{n+1}\setminus\{0\})/\R_>$ and $M_P=(\C^{n+1}\setminus\{0\})/\C^{*}=\C P^n$. The moment map $\mu:\C^m\to\R$ takes $(z_1,\ldots,z_m)\in\C^m$ to $\frac12(|z_1|^2+\ldots+|z_m|^2)$, and for $a\ne0$ we have $\mu^{-1}(a)\cong S^{2n+1}\cong\zk$. \end{example} The previous discussion illustrates the importance of calculating the cohomology of subspace arrangement complements. \begin{theorem}[Buchstaber and Panov] \label{cohar} The following isomorphism of graded algebras holds: \begin{align*} H^{*}\bigl(U(K)\bigr) &\cong\Tor_{\k[v_1,\ldots,v_m]}\bigl(\k(K),\k\bigr)\\ &=H\bigl[\L[u_1,\ldots,u_m]\otimes\k(K),d\bigr]. \end{align*} \end{theorem} \begin{proof} This follows from theorems~\ref{he1}, \ref{cohom1} and \ref{cohom2}. \end{proof} Theorem~\ref{cohar} provides an extremely effective way to calculate the cohomology algebra of the complement of any complex coordinate subspace arrangement. The De Concini and Procesi~\cite{dCP} and Yuzvinsky~\cite{Yu} rational models of the cohomology algebra of an arrangement complement also can be interpreted as an application of the Koszul resolution. However, these author did not discuss the relationships with the Stanley--Reisner ring in the case of coordinate subspace arrangements. \begin{problem} Calculate the cohomology algebra {\sl with $\Z$ coefficients\/} of a coordinate subspace arrangement complement and describe its relationships with the corresponding $\Tor$-algebra $\Tor_{\Z[v_1,\ldots,v_m]}(\Z(K),\Z)$. \end{problem} \begin{example} Let $K$ be a disjoint union of $m$ vertices. Then $U(K)$ is obtained by removing all codimension-two coordinate subspaces $z_i=z_j=0$, $i,j=1,\ldots,m$ from $\C^m$ (see Example~\ref{uk}). The face ring is $\k(K)=\k[v_1,\ldots,v_m]/\mathcal I_K$, where $\mathcal I_K$ is generated by monomials $v_iv_j$, $i\ne j$. An easy calculation using Corollary~\ref{cohar} shows that the subspace of cocycles in $\k(K)\otimes\Lambda[u_1,\ldots,u_m]$ has the basis consisting of monomials $v_{i_1}u_{i_2}u_{i_3}\cdots u_{i_k}$ with $k\ge2$, $i_p\ne i_q$ for $p\ne q$. Since $\deg(v_{i_1}u_{i_2}u_{i_3}\cdots u_{i_k})=k+1$, the space of $(k+1)$-dimensional cocycles has dimension $m\binom{m-1}{k-1}$. The space of $(k+1)$-dimensional coboundaries is $\binom mk$-dimensional (it is spanned by the coboundaries of the form $d(u_{i_1}\cdots u_{i_k})$). Hence, \begin{align*} &\dim H^{0}\bigl(U(K)\bigr)=1,\quad H^{1}\bigl(U(K)\bigr)=H^{2}\bigl(U(K)\bigr)=0,\\ &\dim H^{k+1}\bigl(U(K)\bigr)= m\bin{m-1}{k-1}-\bin mk=(k-1)\bin mk,\quad2\le k\le m, \end{align*} and the multiplication in the cohomology is trivial. In particular, for $m=3$ we have 6 three-dimensional cohomology classes $[v_iu_j]$, $i\ne j$ subject to 3 relations $[v_iu_j]=[v_ju_i]$, and 3 four-dimensional cohomology classes $[v_1u_2u_3]$, $[v_2u_1u_3]$, $[v_3u_1u_2]$ subject to one relation $$ [v_1u_2u_3]-[v_2u_1u_3]+[v_3u_1u_2]=0. $$ Hence, $\dim H^{3}(U(K))=3$, $\dim H^4(U(K))=2$, and the multiplication is trivial. \end{example} \begin{example} Let $K$ be the boundary of an $m$-gon, $m>3$. Then $$ U(K)=\C^m\setminus\bigcup_{i-j\ne0,1\mod m}\{z_i=z_j=0\}. $$ By Theorem~\ref{cohar}, the cohomology ring of $H^*(U(K);\k)$ is isomorphic to the ring described in Example~\ref{mgon}. \end{example} As it is shown in~\cite{GPW}, in the case of arrangements of {\it real\/} coordinate subspaces only {\it additive\/} analogue of our Theorem~\ref{cohar} holds. Namely, let us consider the polynomial ring $\k[x_1,\ldots,x_m]$ with $\deg x_i=1$, $i=1,\ldots,m$. Then the graded structure in the face ring $\k(K)$ changes accordingly. The Betti numbers of the real coordinate subspace arrangement $U_\R(K)$ are calculated by means of the following result. \begin{theorem}[{\cite[Theorem~3.1]{GPW}}] The following isomorphism hold: $$ H^p\bigl(U_\R(K)\bigr) \cong\sum_{-i+j=p}\Tor^{-i,j}_{\k[x_1,\ldots,x_m]}\bigl(\k(K),\k\bigr) =H^{-i,j}\bigl[\L[u_1,\ldots,u_m]\otimes\k(K),d\bigr], $$ where $\bideg u_i=(-1,1)$, $\bideg v_i=(0,1)$, $du_i=x_i$, $dx_i=0$. \end{theorem} As it was observed in~\cite{GPW}, there is {\it no\/} multiplicative isomorphism analogous to Theorem~\ref{cohar} in the case of real arrangements, that is, the algebras $H^*(U_\R(K))$ and $\Tor_{\k[x_1,\ldots,x_m]}(\k(K),\k)$ are not isomorphic in general. The paper~\cite{GPW} also contains the formulation of the first multiplicative isomorphism of our Theorem~\ref{cohar} for complex coordinate subspace arrangements (see~\cite[Theorem~3.6]{GPW}), with a reference to yet unpublished paper by Babson and Chan. Until now, we have used the description of coordinate subspaces by means of equations (see~(\ref{li})). On the other hand, a coordinate subspace can be defined as the linear span of some subset of the standard basis $\{\mb e_1,\ldots,\mb e_m\}$. This leads to the dual approach to the description of coordinate subspace arrangements, which corresponds to the passage from simplicial complex $K$ to the dual complex $\widehat{K}$ (Example~\ref{dual}). This approach was used in~\cite{dL}. It was shown there that the summands in the Goresky--MacPherson formula in the coordinate subspace arrangement case are homology groups of links of simplices of~$\widehat{K}$. This allowed to interpret the product of cohomology classes of the complement of a coordinate subspace arrangement (either real or complex) in terms of the combinatorics of links of simplices in~$\widehat{K}$ (see~\cite[Theorem~1.1]{dL}). We mention that our theorems~\ref{cohom2} and~\ref{he1} show that the Goresky--\-Mac\-Pher\-son result (Theorem~\ref{GMf}) in the case of coordinate subspace arrangements is equivalent to the Hochster theorem (Theorem~\ref{hoch}). \subsection{Diagonal subspace arrangements and cohomology of loop space~$\O\zk$.} \label{diag} In this section we establish relationships between the results of~\cite{PRW} on the cohomology of real diagonal arrangement complements and the cohomology of the loop spaces $\O\bk$ and~$\O\zk$. For each subset $I=\{i_1,\ldots,i_k\}\subset[m]$ define the {\it diagonal subspace\/} $D_I$ in $\R^m$ as $$ D_I=\{(y_1,\ldots,y_m)\in\R^m\::\:y_{i_1}=\cdots=y_{i_k}\}. $$ Diagonal subspaces in $\C^m$ are defined similarly. An arrangement of planes $\A=\{L_1,\ldots,L_r\}$ (either real or complex) is called {\it diagonal\/} if all planes $L_i$, $i=1,\ldots,r$, are diagonal subspaces. The classical example of a diagonal subspace arrangement is given by the arrangement of all diagonal hyperplanes $\{z_i=z_j\}$ in~$\C^m$; its complement is the classifying space for the braid group~$B_m$, see~\cite{Ar}. \begin{construction}\label{dasim} Given a simplicial complex $K$ on the vertex set~$[m]$, introduce the diagonal subspace arrangement $\mathcal{DA}(K)$ as the set of subspaces $D_I$ such that $I$ is not a simplex of~$K$: $$ \mathcal{DA}(K)=\{D_I\::\:I\notin K\}. $$ Denote the complement of the arrangement $\mathcal{DA}(K)$ by~$M(K)$. \end{construction} The proof of the following statement is similar to the proof of the corresponding statement for coordinate subspace arrangements from section~\ref{coor}. \begin{proposition} The assignment $K\mapsto M(K)$ defines a one-to-one order-preserving correspondence between simplicial complexes on the vertex set~$[m]$ and the complements of diagonal subspace arrangements in~$\R^m$. \end{proposition} Here we still assume that $\k$ is a field. The multigraded (or $\N^m$-graded) structure in the ring $\k[v_1,\ldots,v_m]$ (Construction~\ref{mgrad}) defines an $\N^m$-grading in the Stanley--Reisner ring~$\k(K)$. The monomial $v_1^{i_1}\cdots v_m^{i_m}$ acquires the multidegree $(2i_1,\ldots,2i_m)$. Let us consider the modules $\Tor_{\k(K)}(\k,\k)$. They can be calculated, for example, by means of the minimal free resolution (Example~\ref{minimal}) of the field~$\k$ regarded as a $\k(K)$-module. The minimal resolution also carries a natural $\N^m$-grading, and we denote the subgroup of elements of multidegree $(2i_1,\ldots,2i_m)$ in $\Tor_{\k(K)}(\k,\k)$ by $\Tor_{\k(K)}(\k,\k)_{(2i_1,\ldots,2i_m)}$. \begin{theorem}[{\cite[Theorem 1.3]{PRW}}]\label{dacoh} The following isomorphism holds for the cohomology groups of the complement $M(K)$ of a real diagonal subspace arrangement: $$ H^i\bigl(M(K);\k\bigr)\cong\Tor^{-(m-i)}_{\k(K)}(\k,\k)_{(2,\ldots,2)}. $$ \end{theorem} \begin{remark} Instead of simplicial complexes $K$ on the vertex set~$[m]$ the authors of~\cite{PRW} considered square-free monomial ideals $\mathcal I\subset\k[v_1,\ldots,v_m]$. Proposition~\ref{sfmi} shows that these two approaches are equivalent. \end{remark} \begin{theorem}\label{looptor} The following additive isomorphism holds: $$ H^*(\O\bk;\k)\cong\Tor_{\k(K)}(\k,\k). $$ \end{theorem} \begin{proof} Let us consider the Eilenberg--Moore spectral sequence of the Serre fibration $P\to SR(K)$ with fibre $\O SR(K)$, where $SR(K)$ is the Stanley--Reisner space (Definition~\ref{srspace}) and $P$ is the path space over~$SR(K)$. By Corollary~\ref{onefib}, \begin{equation}\label{de2t} E_2=\Tor_{H^*(SR(K))}\bigl(H^*(P),\k\bigr)\cong\Tor_{\k(K)}(\k,\k), \end{equation} and the spectral sequence converges to $\Tor_{C^*(SR(K))}(C^*(P),\k)\cong H^*(\O SR(K))$. Since $P$ is contractible, there is a cochain equivalence $C^*(P)\simeq\k$. We have $C^*(SR(K))\cong\k(K)$. Therefore, $$ \Tor_{C^*(SR(K))}\bigl(C^*(P),\k\bigr)\cong\Tor_{\k(K)}(\k,\k), $$ which together with (\ref{de2t}) shows that the spectral sequence collapses ate the $E_2$ term. Hence, $H^*(\O SR(K))\cong\Tor_{\k(K)}(\k,\k)$. Finally, Theorem~\ref{homeq1} shows that $H^*(\O SR(K))\cong H^*(\O\bk)$, which concludes the proof. \end{proof} \begin{proposition}\label{loopi} The following isomorphism of algebras holds $$ H^*(\O\bk)\cong H^*(\O\zk)\otimes\L[u_1,\ldots,u_m]. $$ \end{proposition} \begin{proof} Consider the bundle $\bk\to BT^m$ with fibre~$\zk$. It is not hard to prove that the corresponding loop bundle $\O\bk\to T^m$ with fibre $\O\zk$ is trivial (note that $\O BT^m\cong T^m$). To finish the proof it remains to mention that $H^*(T^m)\cong\L[u_1,\ldots,u_m]$. \end{proof} Theorems \ref{he1} and \ref{cohar} give an application of the theory of moment-angle complexes to calculating the cohomology ring of a coordinate subspace arrangement complement. 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