% May 11, 1997 % latex2e \documentclass[12pt]{amsart} \makeatletter \let\atopwithdelims\@@atopwithdelims \makeatother \headheight=8pt \textheight=624pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \topmargin=0pt \def\ovA{{\overline{A}}} \def\Da{{\Delta}} \def\aA{{\mathcal A}} \def\bB{{\mathcal B}} \def\cC{{\mathcal C}} \def\dD{{\mathcal D}} \def\eE{{\mathcal E}} \def\fF{{\mathcal F}} \def\hH{{\mathcal H}} \def\pP{{\mathcal P}} \def\uU{{\mathcal U}} \def\tT{{\mathcal T}} \def\ZZ{{\mathbb Z}} \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} %\def\RR{{\Bbb R}} \def\Lat{{\mathcal Lat}} \def\la{{\lambda}} \def\da{{\Da}} \def\al{\alpha} \def\lie{\ell ie} %\def\NN{{\it N}} %\def\NN{{\bf N}} %\def\ZZ{{\it Z}} %\def\ZZ{{\bf Z}} %\def\RR{{\it R}} %\def\KK{{\it K}} %\def\CC{{\it C}} %\def\HH{{\it H}} \def\RP{{\it R}{P}} \def\CP{{\it C}{P}} \def\HP{{\it H}{P}} \def\int#1{\mathop{#1}\nolimits_{}^{\circ}} \def\hra{\hookrightarrow} \def\lra{\rightarrow} \def\lmt{\mapsto} \def\sse{\subseteq} \def\sm{{\setminus}} \def\sta,{\mbox{\rm star}_\Delta} \def\Int{\mbox{\rm Int}} \def\wh#1{\widehat{#1}} \def\id{{\mbox{\rm id}}} \def\#{\sharp} \def\coprod{\biguplus} \def\codim{{\rm codim}} \def\sign{{\rm sign}} \def\Cut{{\rm Cut}} \def\rank{{\rm rank}} \def\log{{\rm log}} \def\exp{{\rm exp}} %\newcommand{\qed}{\hfill \mbox{ \small $\Box$} \\ \vspace{1mm} } \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\lp}{\left\{} \newcommand{\rp}{\right\}} \newcommand{\dl}[1]{\Delta(#1)} \newcommand{\de}{\Delta} \newcommand{\dee}[2]{\de_{#1}^{#2}} \newcommand{\ra}{\rightarrow} \newcommand{\ov}[1]{\overline{#1}} \newcommand{\ti}[1]{\tilde{#1}} \newcommand{\mb}[1]{\mu_{#1}(\hat{0},\hat{1})} \newcommand{\ho}{\hat{0}} \newcommand{\me}[2]{\mu_{#1}(\ho,#2)} \newcommand{\meg}{\me{M^g}{F}} \newcommand{\mega}{\me{L^{g^\ast}}{G}} \newcommand{\snk}{\Sigma_{n,k}} \newcommand{\osnk}{\ov{\Sigma}_{n,k}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{fact}[theorem]{Fact} %\newtheorem{definition}[theorem]{Definition} %\newtheorem{remark}[theorem]{Remark} \newenvironment{Proof}{\smallskip\noindent\mbox{\sf Proof: } }{\qed\medskip} \newenvironment{example}{\smallskip\noindent\mbox{\sf Example: } }{\medskip} \newenvironment{remark}{\smallskip\noindent\mbox{\sf Remark: } }{\medskip} \newenvironment{definition}{\smallskip\noindent\mbox{\sf Definition: } }{\medskip} \begin{document} \title{Complexes of not $i$-Connected Graphs} \author[E. Babson]{Eric Babson} \thanks{Babson, Bj\"orner, Linusson and Welker were partially supported by MSRI. Babson was supported by a National Science Foundation postdoctoral fellowship. Linusson was supported by a Swedish Natural Sciences Research Council (NFR) postdoctoral fellowship. Welker was supported by Deutsche Forschungsgemeinschaft (DFG). Research at MSRI is supported in part by NSF grant DMS-9022140. } \address{\hskip-\parindent Eric Babson, Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070, USA.} \email{babson@msri.org} \author[A. Bj\"orner]{Anders Bj\"orner} \address{\hskip-\parindent Anders Bj\"orner, Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden} \email{bjorner@math.kth.se} \author[S. Linusson]{Svante Linusson} \address{\hskip-\parindent Svante Linusson, Department of Mathematics, Stockholms Universitet, S-106 91 Stockholm, Sweden} \email{linusson@matematik.su.se} \author[J. Shareshian]{John Shareshian} \address{\hskip-\parindent John Shareshian, Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070, USA.} \email{shareshi@msri.org} \author[V. Welker]{Volkmar Welker} \address{\hskip-\parindent Volkmar Welker, Fachbereich 6, Mathematik, Universit\"at GH-Essen, D-45117 Essen, Germany} \email{welker@exp-math.uni-essen.de} \keywords{Connected graphs, complexes of graphs, matching, homotopy type, simplicial resolution, knot invariant, $S_n$-character} %\subjclass{} \date{} \begin{abstract} Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev \cite{Vas93-1,Vas94,Vas97}. In this paper we study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We show that the complex of not $2$-connected graphs has the homotopy type of a wedge of $(n-2)!$ spheres of dimension $2n-5$. This answers one of the questions raised by Vassiliev \cite{Vas97} in connection with knot invariants. For this case the $S_n$-action on the homology of the complex is also determined. For complexes of not $2$-connected $k$-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not $(n-2)$-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. %The latter complexes and their homology are of interest in %various parts of mathematics (see \cite{BLVZ92}). For not $(n-3)$-connected graphs we provide a formula for the generating function of the Euler characteristic. \end{abstract} \maketitle \section{Introduction} In this paper we study the homotopy type and homology of simplicial complexes whose simplices are the edge sets of not $i$-connected graphs and hypergraphs on $n$ vertices. The case $i=1$ is already well understood (see Proposition \ref{discon}), and here we begin the examination of the topological structure of such complexes for $i \geq 2$. Although our point of view is mainly combinatorial, our original motivation for studying these complexes comes from the theory of Vassiliev invariants in knot theory. By determining the homotopy type of the complex of not $2$-connected graphs on $n$ vertices we answer a question posed by V. Vassiliev in \cite{Vas97}, where he presents a new approach to Vassiliev knot invariants using a filtration of the simplicial resolution of the space of not-knots as in \cite{Vas94}. More precisely, he studies the space $\Sigma$ of maps $f : S^1 \rightarrow \RR^3$ such that $f(S^1)$ has multiple points or cusps. The simplicial resolution $\widetilde{\Sigma}$ of $\Sigma$ is obtained roughly speaking as follows: singular knots are resolved by blowing up each $r$-fold self-intersection to an $({r\choose 2}-1)$-simplex, and similarly for the set of cusps. A suitable filtration (see \cite{Vas97}) of $\widetilde{\Sigma}$, combinatorially defined in terms of these simplices, gives rise to a spectral sequence that contains the homology of the complex of not $2$-connected graphs on $n$ vertices as a basic ingredient. Our work continues the already fruitful interaction between the theory of Vassiliev invariants and questions in topological and homological combinatorics of graph complexes (see \cite{Vas93-1}). The study of complexes of not $i$-connected graphs has intriguing combinatorial and algebraic aspects as well. For example, such aspects become apparent when considering the complex of not $(n-2)$-connected graphs on $n$ vertices. In Section \ref{match} this complex is shown to be Alexander dual to the complex of partial matchings of the complete graph on $n$ vertices. These matching complexes, along with complexes of partial matchings of bipartite graphs, have previously been studied for other reasons, see \cite{BLVZ92}. In each case for which we calculate the Betti numbers, we detect nontrivial homology. For $(n-3)$-connected graphs (see Section \ref{notn3}) and for most complexes of not $2$-connected hypergraphs (see Section \ref{notconnhyper}) we have been unable to compute the Betti numbers explicitly, but we do determine the generating function of their reduced Euler characteristics. The homology is seen to be nontrivial in almost all of these cases. Surprisingly, these non-vanishing phenomena are suggested by a result motivated by a conjecture in complexity theory. The conjecture states that complexes of graphs on $n$ vertices having some non-trivial monotone graph property -- like being not $i$-connected -- are evasive (see for example \cite{KahSakStu84}). Kahn, Saks \& Sturtevant \cite{KahSakStu84} showed that non-evasive complexes are contractible. In many naturally arising cases, including those examined here, the converse is true and evasive complexes in fact have non-vanishing reduced Euler characteristics. In Section \ref{rep} we study the action of the symmetric group on the complex of not $2$-connected graphs induced by its natural action on the vertices. This action induces a representation of $S_n$ on the homology groups of the complex, which we determine. Using the representation, we deduce upper bounds on the number of Vassiliev invariants of a given bi-order. This representation coincides with a recently well studied representation which appears in the work of Robinson \& Whitehouse \cite{RobWhi94,Whi94-2}, Kontsevich \cite{Kon93}, Getzler \& Kapranov \cite{GetKap95}, Mathieu \cite{Mat96}, Hanlon \& Stanley \cite{Han96,HanSta95} and Sundaram \cite{Sun96}. \medskip \noindent {\sf Acknowledgment:} We are grateful to V. Vassiliev for inspiring discussions and hints, which sparked our interest and initiated this research. All computer calculations presented in this paper were performed using a Mathematica$^{\text{\copyright}}$ package designed by Vic Reiner and a C-Program by Frank Heckenbach. \section{Preliminaries} \label{prel} We now introduce the basic concepts used in this paper. By a graph $G = (V(G),E(G))$ we mean a loopless graph without multiple edges on the vertex set $V(G)$ and with edge set $E(G) \subseteq \binom{V(G)}{2}$. Our standard vertex set will be the set $[n]:=\{1,2,\dots,n\}$. A graph $G$ is called {\it connected} if for any two distinct vertices $v, v' \in V(G)$ there is a path from $v$ to $v'$ in $G$, that is, a sequence of edges $\{ v_1, v_2 \}$, $\{ v_2, v_3 \}$, $\ldots$, $\{v_{l-1},v_l\}$ $\in E(G)$ such that $v = v_1$ and $v' = v_{l}$. Such a path will sometimes be denoted by $v_1,v_2,\ldots,v_l$. The {\it size} of a graph $G$ is $|V(G)|$. A graph $G$ is called {\it $i$-connected}, for a number $i$ such that $02$, let $x$ be a source in $D_M(\Sigma)$, which must exist since $D_M(\Sigma)$ contains no directed cycle. It is easy to see that $x$ must be a free face of $\Sigma$ which is properly contained in a unique face $y \in \Sigma$. Now $\Sigma$ is collapsible to the complex obtained by removing $x$ and $y$, and we can apply the inductive hypothesis. \end{Proof} We call a perfect matching of the type described in Proposition \ref{forman} an {\it acyclic perfect matching} on $D(\Sigma)$. Our goal is to produce an acyclic perfect matching on $D(\de(k-1,k))$. The following easy result will be useful. \begin{lemma} Let $\Sigma$ be a simplicial complex, let $M$ be a matching on $D(\Sigma)$ and let $F_0 \ra F_1 \ra \ldots \ra F_r \ra F_0$ be a directed cycle in $D_M(\Sigma)$. Then there is some dimension $d$ such that $dim(F_i) \in \lp d,d+1 \rp$ for all $i \in \lb r \rb$. \label{twolev} \end{lemma} \begin{Proof} If the $F_i$ have more than two distinct dimensions then some $F_i$ must be incident to two upward directed edges. This contradicts the fact that $M$ is a matching, and the result follows immediately. \end{Proof} Before proceeding with the proof of Lemma \ref{dkkcon} we make some technical definitions. \begin{definition} Consider separable graphs on the vertex set $[n]$. \begin{itemize} \item[(1)] We denote the set of cutpoints of such a graph $G$ by $\Cut(G)$. \item[(2)] For fixed $k \in \lp 3,\ldots,n-1 \rp$, let \begin{itemize} \item[(a)] $I(k):=\lp G \in \de(k-1,k)~|~N_G(1)=\emptyset \rp$. \item[(b)] $J(k):=\lp G \in \de(k-1,k)~|~N_G(1) \neq \emptyset \mbox{ and } \Cut(G+1k) \neq \lp 1 \rp \rp$. \item[(c)] $F(k):=\lp G \in \de(k-1,k)~|~\Cut(G+1k)=\lp 1 \rp \rp$. \end{itemize} \end{itemize} \end{definition} \noindent Note that $\de(k-1,k)$ is the disjoint union of $I(k)$, $J(k)$ and $F(k)$, and that both $I(k)$ and $I(k) \cup J(k)$ are subcomplexes of $\de(k-1,k)$. \smallskip The following lemma implies Lemma \ref{dkkcon}, and therefore completes the proof of Theorem \ref{main}. \begin{lemma} For any $k \in \lp 3,\ldots,n-1 \rp$, $D(\de(k-1,k))$ admits an acyclic perfect matching. \label{apm} \end{lemma} \noindent \begin{Proof} This proof will be carried out in three steps. We will construct an acyclic perfect matching first for $D(I(k))$, then for $D(I(k) \cup J(k))$, and finally for $D(\de(k-1,k))$. \medskip \noindent {\sf Step 1:} $D(I(k))$ admits an acyclic perfect matching. Note that $I(k)$ contains a unique maximal face, namely the complete graph on $\lp 2,\ldots,n \rp$. Thus $I(k)$ is a simplex and it is easy to see that the matching $M=\lp G+23 \rightarrow G \setminus 23 | G \in I(k) \rp$ is an acyclic perfect matching on $D(I(k))$. \medskip \noindent {\sf Step 2:} $D(I(k) \cup J(k))$ admits an acyclic perfect matching. It suffices to show that there exists a matching $M^\ast$ consisting of edges between elements of $J(k)$ which covers all the elements of $J(k)$, and such that $D_{M^\ast}(I(k) \cup J(k))$ is acyclic. If $M^\ast$ is such a matching, let $M^\circ$ be an acyclic perfect matching on $D(I(k))$ and set $M=M^\ast \cup M^\circ$. Then $M$ is a perfect matching on $D(I(k) \cup J(k))$ which contains no edges between $I(k)$ and $J(k)$, so that any directed cycle in $D_M(I(k))$ cannot cover points from both $I(k)$ and $J(k)$. It follows immediately that $M$ is acyclic. \smallskip Now let $G \in J(k)$ and let $c \in \Cut(G+1k)$, $c\neq 1$. Let $x=\min\lp N_G(1) \rp$. If $xk \in E(G)$ then clearly $G\setminus xk \in J(k)$. If $c \not\in \lp x,k \rp$ then since $1k$ and $1x$ are edges of $G+1k$, $x$ and $k$ lie in the same connected component of $(G+1k)-c$. If $c \in \lp x,k \rp$ then clearly $c$ is a cutpoint of $G+xk+1k$. In any case, $c \in \Cut(G+xk+1k)$ and $G+xk \in J(k)$. \smallskip Let $M^\ast$ consist of all edges $G+xk \rightarrow G \setminus xk$, where $x$ is determined as above. Clearly $M^\ast$ is a matching which covers all points in $J(k)$. Assume for contradiction that $A_1 \ra B_1 \ra A_2 \ra B_2 \ra \ldots \ra B_r \ra A_1$ is a directed cycle in $D_{M^\ast}(I(k) \cup J(k))$. Clearly all the $A_i$ and all the $B_i$ are in $J(k)$, and by Lemma \ref{twolev} we may assume that for each $i$ there are edges $\alpha_i$ and $\beta_i$ such that $B_i=A_i+\alpha_i$ and $A_{i+1}=B_i \setminus \beta_i$. Thus $A=A+\alpha_1 \setminus \beta_1+\ldots +\alpha_r\setminus \beta_r$ and $\lp \alpha_i \rp=\lp \beta_i \rp$. By the definition of $M^\ast$, no $\alpha_i =x_i k$ contains $1$, so no $\beta_i$ contains $1$. It follows that $N_{A_1}(1)=N_{B_1}(1)=N_{A_2}(1)=\ldots= N_{B_r}(1)$. By the choice of the $x_i$'s this forces $\alpha_1=\alpha_2=\ldots=\alpha_r$, which is clearly impossible. \medskip \noindent {\sf Step 3:} $D(\de(k-1,k))$ admits an acyclic perfect matching. As in Step 2, it suffices to produce a matching $M^\ast$ on edges connecting elements of $F(k)$ which covers all points in $F(k)$ and such that $D_{M^\ast}(\de(k-1,k))$ is acyclic. \smallskip Let $G \in F(k)$. Then $(G+1k)-1$ splits into connected components $C_1, \ldots,C_s$ such that for each $i \in \lb s \rb$ the subgraph of $G+1k$ induced on $V(C_i) \cup \lp 1 \rp$ is $2$-connected. We may assume that $n \in V(C_1)$. Note that since $k1$. Indeed, let $1,x,\ldots,n$ be a shortest path from $1$ to $n$ in $G+1k$. Clearly $x \in S(G)$. Since the subgraph of $G+1k$ induced on $V(C_1) \cup \lp 1 \rp$ is $2$-connected and $x \neq n$, there exists a path from $1$ to $n$ in this graph which does not contain $x$. Let $1,y,\ldots,n$ be a shortest such path. Then $y \in S(G)$. \smallskip Let $x,y$ be the two smallest elements of $S(G)$. If $xy \not\in G$ then clearly $G+xy \in F(k)$ and $S(G+xy)=S(G)$. Now assume $xy \in G$ and let $H$ be the subgraph of $G \setminus xy+1k$ induced on $V(C_1) \cup \lp 1 \rp$. If $d$ is a cutpoint of $H$ then $x$ and $y$ are in different components of $H-d$ (otherwise $d$ is a cutpoint of the subgraph of $G+1k$ induced on $V(C_1) \cup \lp 1 \rp$). However, there is a cycle $1,x,\ldots,y,1$ in $H$. Thus there is no such cutpoint $d$ and $H$ is $2$-connected. It follows that $G \setminus xy \in F(k)$ and $S(G \setminus xy)=S(G)$. \smallskip Now, let $M^\ast$ consist of the edges $G+xy \ra G \setminus xy$ where $x,y$ are determined as above. Then $M^\ast$ is a matching which consists of edges connecting points in $F(k)$ and covers all points in $F(k)$. It remains to show that $D_{M^\ast}(\de(k-1,k))$ is acyclic. \smallskip Assume for contradiction that $A_1 \ra B_1 \ra A_2 \ra \ldots \ra B_r \ra A_1$ is a directed cycle in $D_{M^\ast}(\de(k-1,k))$. As in Step 2, we may assume that there are edges $\alpha_i$ and $\beta_i$ such that $B_i=A_i+\alpha_i$, $A_{i+1}=B_i \setminus \beta_i$ and $\lp \alpha_i \rp=\lp \beta_i \rp$. \smallskip By the definition of $M^\ast$, each $\alpha_i$ connects two elements of $N_{B_i+1k}(1)=N_{A_i+1k}(1)$, so no $\beta_i$ contains $1$. Thus $N_{A_i}(1) =N_{B_j}(1)$ for all $i,j$, and each $\beta_i$ connects two elements of $N_{B_i+1k}(1)=N_{A_{i+1}+1k}(1)$. Write $\alpha_1=xy$. Then $\beta_1 \neq xy$, and in $A_2+1k$, $x$ and $y$ are still the two smallest neighbors of $1$ which are contained in paths from $1$ to $n$ which intersect $N_{A_2+1k}(1)$ exactly once. Thus $\alpha_2=xy=\alpha_1$, giving the desired contradiction. \end{Proof} %\section{Homology Representations of $S_n$} \section{The character for the action of $S_n$ on $\ti{H}_{2n-5}(\dee{n}{2})$} \label{rep} \label{homrep} In view of Theorem \ref{main} it is natural to investigate the representation of the symmetric group $S_n$ on the only non-zero homology group of $\dee{n}{2}$, induced by the obvious action. In this section we consider homology with complex coefficients, hence all representations are over $\CC$. In many of the computations below, we actually determine character values for the representation of $S_n$ on the only non-zero homology group of $\Delta(\ov{\Lat(\dee{n}{2})})$, which is easily seen to be the same as the representation described above. \medskip \begin{definition} \begin{itemize} \item[(i)] We denote by $\omega_n^2$ the character of $S_n$ given by $g \mapsto Trace(g,\ti{H}_{2n-5}(\dee{n}{2}))$. \item[(ii)] Let $C_n$ be a cyclic subgroup of $S_n$ generated by a full $n$-cycle. We denote by $\lie_n$ the character of $S_n$ induced from the character on $C_n$ which takes the value $e^{\frac{2 \pi i}{n}}$ on a fixed generator. It is well known (see e.g. \cite[Chapter 8]{Reu93}) that $\lie_n$ is the character of $S_n$ on the multigraded piece of the free Lie algebra generated by $n$ variables. \end{itemize} \end{definition} For the rest of this section we let $S_{n-1}$ be the stabilizer of the point $1$ in the natural action of $S_n$ on the set $[n]$. \begin{theorem} \label{char} The character $\omega_n^2$ is given by $$\omega_n^2 = \lie_{n-1} \uparrow_{S_{n-1}}^{S_n} - \lie_{n}.$$ \end{theorem} The proof will follow a sequence of lemmas establishing the main steps. \begin{lemma} \label{first} If $g \in S_{n-1}$ then $\omega_n^2(g)=\lie_{n-1}(g).$ \label{resg1} \end{lemma} \begin{Proof} It is easily seen that the map $\phi:\ov{\Lat(\dee{n}{2})} \rightarrow \ov{B_{n-1} \times \Pi_{n-1}}$, defined in the previous section, commutes with the actions of $S_{n-1}$ on the two posets. Thus the induced map on homology is $S_{n-1}$-equivariant and is an $S_{n-1}$-module isomorphism by Lemma \ref{hom:equiv}. Thus, the characters of $S_{n-1}$ on the homology of $\Delta_n^2$ and on the homology of $\Delta(\ov{B_{n-1} \times \Pi_{n-1}})$ coincide. By an equivariant version of Proposition \ref{homcom} (see \cite{Wel90-1}), $\Delta(\ov{B_{n-1} \times \Pi_{n-1}})$ has the $S_{n-1}$-homotopy type of $\Sigma ( \Delta(\ov{B_{n-1}}) * \Delta(\ov{\Pi_{n-1}})),$ where the group $S_{n-1}$ acts diagonally on $\Delta(\ov{B_{n-1}}) * \Delta(\ov{\Pi_{n-1}})$. Thus the character of $S_{n-1}$ on the homology of $\Delta(\ov{B_{n-1} \times \Pi_{n-1}})$ is given by the product of the characters of $S_{n-1}$ on $\widetilde{H}_*(\Delta(\ov{B_{n-1}}))$ and $\widetilde{H}_*(\Delta(\ov{\Pi_{n-1}}))$. The character of $S_{n-1}$ on $\widetilde{H}_*(\Delta(\ov{B_{n-1}}))$ is rather easily seen to be the sign-character of $S_{n-1}$ (see \cite{Sta82}). The character of $S_{n-1}$ on $\widetilde{H}_*(\Delta(\ov{\Pi_{n-1}}))$ was determined in \cite{Sta82} as $sign_{n-1} \cdot \lie_{n-1}$. This implies the assertion. \end{Proof} \noindent Since every element of $S_n$ which has a fixed point is conjugate to an element of $S_{n-1}$, it remains to determine $\omega_n^2(g)$ for all fixed-point-free $g \in S_n$. \medskip \begin{definition} Let $g \in S_n$. We denote by $L^g$ the poset of faces of $\dee{n}{2}$ which are fixed by $g$, and by $g^\ast$ the element of $S_{n+1}$ which fixes $n+1$ and acts as $g$ does on $\lb n \rb$. \end{definition} \noindent Write $\ho$ for the empty graph in $L^g$, which is the unique minimum element of $L^g$, and for any poset $P$ let $\mu_{P}$ be the M\"obius function on $P$. \begin{lemma} For $g \in S_n$, $\omega_n^2(g)=\displaystyle{\sum_{G \in L^g}\mu_{L^g}(\ho,G)}$. \label{hall} \end{lemma} \noindent \begin{Proof} It is well-known (see e.g. \cite[(13.5)]{Bjo91}) that if a group acts on a bounded poset $P$ then for any group element $g$ we have \[ \mb{P^g}=\sum_i(-1)^iTr(g,\ti{H}_i(\de(\ov{P}))). \] In the case under consideration, the only nonzero reduced homology group is the one in dimension $2n-5$, so the lemma follows immediately from the definition of the M\"obius function. \end{Proof} The next two lemmas will be used to determine $\omega_n^2(g)$ when $g$ is fixed-point-free. \begin{lemma} Let $G$ be a graph whose automorphism group acts transitively on $V(G)$. If $G$ is connected then $G$ is $2$-connected. \label{cyctrans} \end{lemma} \begin{Proof} %Assume for contradiction that $G$ is a counterexample to the lemma and %let $c \in Cut(G)$. Let $C_1,C_2,\ldots,C_r$ be the connected components %of $G-c$, and let $c_i=|C_i|$. We may assume that $c_1 \geq c_2 \geq %\ldots \geq c_r$. Let $g \in Aut(G)$. Then $c^g \in Cut(G)$ and $G-c^g$ %has $r$ components of sizes $c_1,\ldots,c_r$. Say $c^g \not\in C_1$. %Then $C_1 \cup \lp c \rp$ is contained in a connected component of $G-c^g$, %which thus has size at least $c_1+1$, which is impossible. Thus $c^g %\in C_1$. Since $Aut(G)$ is transitive, $V(G) \setminus \lp c \rp=C_1$, %which contradicts $c \in Cut(G)$. Let $v$ be a leaf of some spanning tree in the connected graph $G$. Then $v$ is not a cutpoint. Since Aut($G$) is transitive on vertices there cannot be any other cutpoints. Hence $G$ is $2$-connected. \end{Proof} \begin{lemma} Let $g \in S_n$ be fixed-point-free. Write $g$ as a product of disjoint cycles, $g=g_1 \ldots g_r$. Let $V_i=supp(g_i)$. Let $G \in L^g$ be connected and let $x \in \Cut(G)$ with $x \in V_j$. Then there exists some connected component $C$ of $G-x$ such that $V_j \setminus \lp x \rp \subseteq C$ and $C \cap V_i \neq \emptyset$ for all $i \in \lb r \rb$. \label{bigcomp} \end{lemma} \noindent \begin{Proof} Let $G_j$ be the graph on $V_j$ such that an edge $yz$ is in $E(G_j)$ if $yz \in E(G)$ or if there is a path $P$ from $y$ to $z$ in $E(G)$ such that $P \cap V_j=\lp y,z \rp$. Since $G$ is connected, so is $G_j$. Also, the group generated by $g_j$ is a group of automorphisms of $G_j$ which acts transitively on $V_j$. By Lemma \ref{cyctrans}, $G_j$ is $2$-connected. It follows that all elements of $V_j \setminus \lp x \rp$ are in the same connected component of $G-x$. Now for $i \neq j$, let $P$ be a path of shortest length connecting some $y \in V_i$ with some $z \in V_j$. If $z=x$ replace $P$ with $g(P)$. Now $P$ contains no vertices from $V_i \cup V_j$ other than $y$ and $z \neq x$. Thus $P$ is a path in $G-x$ and $y$ lies in the component of $G-x$ containing $V_j \setminus \lp x \rp$. \end{Proof} We can now determine the values of $\omega_n^2$ on fixed-point-free elements of $S_n$. \begin{lemma} \label{second} Let $g \in S_n$ be fixed-point-free. Then $\omega_n^2(g)=-\omega_{n+1}^2(g^\ast)$. \label{wfpf} \end{lemma} \begin{Proof} As usual we write $\ho$ for the empty graph. By Lemma \ref{hall} we have \[ \omega_{n+1}^2(g^\ast)=\sum_{G \in L^{g^\ast}}\mega. \] Let $M^g$ be the poset of all graphs on $\lb n \rb$ which are fixed by $g$. Note that if $G \in L^{g^\ast}$ then $G - (n+1) \in M^g$. For $F \in M^g$ let $D(F)$ be the set of all $G \in L^{g^\ast}$ such that $G-(n+1)=F$. We have \[ \omega_{n+1}^2(g^\ast)=\sum_{F \in M^g}\sum_{G \in D(F)}\mega. \] Any $G \in L^{g^\ast}$ is a union of $\langle g^\ast \rangle$-orbits on ${{\lb n+1 \rb} \choose {2}}$. Let $o(G)$ be the number of such orbits. It is easy to see that $\mega=(-1)^{o(G)}$. Let $p(G)$ be the number of such orbits containing edges covering the point $n+1$. Applying the previous argument to $M^g$, we get for any $F \in M^g$ \[ \sum_{G \in D(F)}\mega=\meg\sum_{G \in D(F)}(-1)^{p(G)}. \] We will examine this sum for each $F \in M^g$, looking separately at the cases where $F$ is disconnected, connected but not $2$-connected, and $2$-connected. Write $g$ as a product of disjoint cycles, $g=g_1 \ldots g_r$ and let $V_i=supp(g_i)$. Note that if $v \in V_i$ and $G \in L^{g^\ast}$ with $\lp v,n+1 \rp \in G$, then the $\langle g^\ast \rangle$-orbit containing $\{v,n+1\}$ consists of the edges $\{w,n+1\}$ for all $w \in V_i$, and is contained in $E(G)$. Also, $p(G)$ is simply the number of such orbits. Let $O(g)$ be the set of all such orbits, and for $S \subseteq O(g)$ let $G(S)$ be the graph induced on the edges which are contained in elements of $S$. For $F \in M^g$ define \[ \Sigma(F):=\lp S \subseteq O(g): F \cup G(S) \in L^{g^\ast} \rp. \] Note that $\Sigma(F)$ is a simplicial complex on $O(g)$. Let $P(F)=\Lat(\Sigma(F)) \setminus \lp \hat{1} \rp$. By the above arguments we have \[ \sum_{G \in D(F)}\mega=\meg\sum_{S \in P(F)}\mu_{P(F)}(\hat{0},S). \] \medskip We now examine the three cases. \medskip \noindent {\sc Case 1:} $F$ is not connected. Then $P(F)$ is the Boolean algebra on $O(g)$, since $n+1$ is a cutpoint of $F \cup G(S)$ for all $S \subseteq O(g)$. It follows immediately that \[ \sum_{G \in D(F)}\mega=0. \] \medskip \noindent {\sf Case 2:} $F$ is connected but not $2$-connected. We will use the block decomposition described in Proposition \ref{sep} of the following section. Given a connected but not $2$-connected graph $F \in M^g$, let $T(F)$ be the bipartite graph whose vertices are the vertices of $F$ and the blocks of $F$, with $\{ v,W_i\}$ an edge if and only if $v \in W_i$. It is easy to see that $T(F)$ is a tree and that $\langle g \rangle$ is a group of automorphisms of $T(F)$ which preserves each part of the given bipartition. It follows that $g$ fixes a vertex of $T(F)$ (see \cite{Lov93}). Since $g$ fixes no vertex of $F$, $g$ must fix some block $B$ of $F$. This means that there is some nonempty $J \subset \lb r \rb$ such that $B=\cup_{j \in J}V_j$. Let $S$ be the set of all orbits in $O(g)$ which contain edges that include vertices in $B$. We will show that every maximal element of $P(F)$ contains $S$, from which it follows immediately that \[ \sum_{G \in D(F)}\mega=0. \] Let $G \in D(F)$ and let $c \in \Cut(G)$. Since $F$ is connected, $c \neq n+1$. Also, if $N_G(n+1) \neq \emptyset$ then since $g$ is fixed-point-free $c$ must be a cutpoint of $F$. If $N_G(n+1)=\emptyset$ then every $x \in \lb n \rb$ cuts $G$, so in any case we may assume $c \in \Cut(F)$. Let $c \in V_i$. By Lemma \ref{bigcomp}, there is some connected component $C$ of $F-c$ which contains $V_i \setminus \lp c \rp$ and at least one element of each $V_j$. Since $B$ is $2$-connected and $B \cap C \neq \emptyset$, we must have $B \subseteq C$. We will now show that $c$ must be a cutpoint of $G \cup S$. If $N_G(n+1)=\emptyset$ then adding $S$ to $G$ simply moves the previously isolated point $n+1$ into the connected component of $G-c$ which contains $C$. However, there is a component of $F-c$ besides $C$, which remains separated from $C$ in $G-c$. Now, assume that $N_G(n+1) \neq \emptyset$. Then there exists some set $I$ such that $N_G(n+1)=\cup_{i \in I}V_i$. The component of $G-c$ containing $C$ contains elements of each $V_i$, and it follows that $n+1$ must also be in this component. Thus, adding $S$ to $G$ does not reduce the number of components of $G-c$. \medskip \noindent {\sf Case 3:} $F$ is $2$-connected. In this case the only $G \in D(F)$ is that for which $N_G(n+1)=\emptyset$. Indeed, since each $V_i$ has at least two elements we cannot have $|N_G(n+1)|=1$, and the claim follows. Thus \[ \sum_{G \in D(F)}\mega=\mu_{M^g}(\ho,F). \] \smallskip Let $K^g$ be the set of $2$-connected graphs in $M^g$. Combining the information from the three cases we have shown that \[ \omega_{n+1}^2(g^\ast)=\sum_{F \in K^g}\mu_{M^g}(\ho,F). \] By definition of the M\"obius function and the fact that $M^g$ has a maximum element, we have \[ \sum_{F \in K^g}\mu_{M^g}(\ho,F)=-\sum_{F \in L^g}\mu_{L^g}(\ho,F) =-\omega_n^2(g), \] and the proof is complete. \end{Proof} \medskip \noindent {\sf Proof of Theorem \ref{char}:} Set $\rho_n = \lie_{n-1} \uparrow_{S_{n-1}}^{S_n} - \lie_n$. We must show that $\rho_n(g) = \omega_n^2(g)$ for all $g \in S_n$. By the definition of induced characters, if $g \in S_n$ is not the product of disjoint cycles of the same length then $\lie_n(g) = 0$. We will assume from now on that any $g \in S_n$ which fixes a point is contained in $S_{n-1}$ (so by our convention it fixes the point $1$). By the definition of induced characters and Theorem \ref{main}, we have $$\rho_n(\id) = (n-2)!~ [S_n:S_{n-1}] - (n-1)! = (n-2)! = \omega_n^2(\id).$$ If $g \neq \id$ and $g$ has at least two fixed points, then $\lie_{n-1}(g) = \lie_n(g) = 0$, so $\rho_n(g) = \omega_n^2(g)$ by Lemma \ref{first}. If $g \neq \id$ has exactly one fixed point, then $\lie_n(g) = 0$. For $h \in S_n$ we have $g^h := h^{-1}gh \in S_{n-1}$ if and only if $h \in S_{n-1}$. By the definition of induced characters and Lemma \ref{first}, $$\rho_n(g) = \lie_{n-1} \uparrow_{S_{n-1}}^{S_n}(g) = \frac{1}{(n-1)!} \sum_{h \in S_{n-1}} \lie_{n-1}(g^h) = \lie_{n-1}(g) = \omega_n(g).$$ If $g \in S_n$ has no fixed points then $\lie_{n-1} \uparrow_{S_{n-1}}^{S_n} (g) = 0$ and $\rho_n(g) = - \lie_n(g)$. As before, let $g^*$ be the element of $S_{n+1}$ which fixes $n+1$ and acts as $g$ does on $[n]$. We have shown above that $\omega_{n+1}^2(g^*) = \lie_n(g)$. Hence, by Lemma \ref{second}, $\rho_n(G) = \omega_n^2(g)$. \qed \medskip According to Vassiliev, the number of linearly independent knot invariants of bi-order $(n,n-1)$, modulo lower bi-order invariants, is bounded from above by the multiplicity of the trivial representation in the restriction of $\omega_n^2$ to the cyclic group $C_n$ generated by $(12 \cdots n)$. See \cite{Vas97} for all details. As a corollary of Theorem \ref{char} we obtain a formula for this multiplicity. We write $\langle \xi,1 \rangle$ for the multiplicity of the trivial character in any character $\xi$ of $C_n$. \begin{corollary} \label{trivmult} \[ \langle \omega_n^2 \downarrow^{S_n}_{C_n},1 \rangle=(n-2)! - \frac{1}{n}\sum_{d|n}\mu(d)\phi(d) (\frac{n}{d}-1)! \hskip2pt d^{\frac{n}{d}-1} \] \end{corollary} \begin{Proof} As a consequence of a result by Hanlon \cite{Han82} (see also \cite{Sta82}), it is straightforward to show that \[ \langle \lie_n\downarrow^{S_n}_{C_n},1 \rangle= \frac{1}{n}\sum_{d|n}\mu(d)\phi(d) (\frac{n}{d}-1)! \hskip1pt d^{\frac{n}{d}-1}, \] where $\mu$ is the usual number-theoretic M\"obius function and $\phi$ is Euler's function. On the other hand, $$\langle \lie_{n-1}\uparrow_{S_{n-1}}^{S_n} \downarrow_{C_n}^{S_n},1 \rangle = \frac{1}{n} \sum_{g \in C_n} \lie_{n-1} \uparrow_{S_{n-1}}^{S_n}(g) = \frac{1}{n} \hskip2pt \lie_{n-1} \uparrow_{S_{n-1}}^{S_n} (\id) = (n-2)!.$$ Now the assertion follows immediately from Theorem \ref{char}. \end{Proof} The values of $w_n = \langle \omega_n^2\downarrow^{S_n}_{C_n},1 \rangle$ for small $n$ are given in the table below. \medskip \begin{center} $\begin{array}{|c||c|c|c|c|c|c|c|c|c|} \hline n & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline \hline w_n & 1 & 1 & 2 & 6 & 18 & 96 & 564 & 4,072 & 32,990\\ \hline \end{array}$ \end{center} \centerline{{\sf Table 1:} Multiplicity $w_n$ of the trivial character in $\omega_n^2 \downarrow_{C_n}^{S_n}$} \medskip The character $\omega_n^2$ and the tensor product of $\omega_n^2$ with the sign character have recently appeared in various different settings, see Section \ref{questionrep}. \section{The lattice of block-closed graphs} %structure of separable graphs and a complex homotopy %equivalent to $\Delta_{n,k}^2$} \label{closed} In this section we will obtain information on the topology of $\Delta_{n,k}^2$ by producing a lattice $\snk$ such that $\Delta(\ov{\snk})$ is homotopy equivalent to $\Delta_{n,k}^2$ and examining the structure of $\snk$. For lattice and poset terminology not explained in Section \ref{notation} we refer to \cite{Sta86}. We begin by recalling some elements of the well known structure theory of separable graphs, which appears e.g. in \cite{Lov93}. \begin{definition} Let $G$ be any graph. A {\em block} of $G$ is a subset $W$ of $V(G)$ such that the subgraph of $G$ induced on $W$ is $2$-connected or $W$ is a singleton or a pair of points connected by an edge, and the subgraph of $G$ induced on any proper superset of $W$ is separable. We will say that $G$ is {\em block-closed} if the subgraph induced on each block is a clique. \end{definition} Given a graph $G$, say that $e\equiv e'$ for two of its edges $e$ and $e'$ if they both lie in some circuit of $G$. This is easily seen to be an equivalence relation on $E(G)$. If $W$ is the set of nodes underlying an equivalence class then $W$ is a block, and all non-singleton blocks correspond to equivalence classes of edges in this way. {From} this it is easy to derive the following basic facts about the ``block decomposition'' of $G$, see \cite{Lov93} for more details. \begin{proposition} Let $G$ be a graph. Then there exists a unique decomposition of $V(G)$ into blocks $W_1,\ldots,W_r$, and if $i \neq j$ we have $|W_i \cap W_j| \leq 1$. Moreover, if $B_G$ is the graph with vertex set $\lp w_1,\ldots,w_r \rp$ such that $\lp w_i,w_j \rp \in E(B_G)$ if and only if $|W_i \cap W_j|=1$, then $B_G$ is a forest (that is, $B_G$ contains no cycles). \label{sep} \end{proposition} Note that if $K$ is a $k$-graph with underlying graph $G$, then every block of $G$ has size at least $k$ or is a single vertex. \begin{definition} Let $K$ be a $k$-graph with underlying graph $G$, and let $W_1,\ldots,W_r$ be the blocks of $G$. We define $K^*$ to be the $k$-graph which induces the complete $k$-graph on each $W_i$ and contains no other hyperedges. We also define $\snk$ to be the poset of all graphs on vertex set $\lb n \rb$ in which every block is either an isolated vertex or a clique of size at least $k$, ordered by inclusion. \end{definition} The first part of the following lemma is immediate from the definition, and the second follows via a standard argument for closure operators on lattices. \begin{lemma} \begin{itemize} \item[(i)] The map $K \mapsto K^*$ defines a closure operator on ${\Lat(\Delta_{n,k}^2)}$ whose image is isomorphic to $\snk$. \item[(ii)] $\snk$ is a lattice. \label{clop} \end{itemize} \end{lemma} The meet operation in the lattice $\snk$ is intersection of edge-sets followed by deletion of the edges in all blocks of size smaller than $k$. Note that the elements of $\Sigma_{n,2}$ are the block-closed graphs, and that we have a tower of embeddings as subposets (not sublattices): $$\snk\subseteq \cdots \subseteq \Sigma_{n,3} \subseteq \Sigma_{n,2}.$$ Hence, in view of the following result the topology of all the complexes $\Delta_{n,k}^2$ is encoded into the lattice $\Sigma_{n,2}$ of block-closed graphs. \begin{theorem} The complexes $\Delta_{n,k}^2$ and $\Delta(\ov{\snk})$ are homotopy equivalent. \label{dnksnk} \end{theorem} \begin{Proof} $K^*$ is the complete graph (the top element of $\snk$) if and only if $K$ is $2$-connected. Hence, the map $K \mapsto K^*$ restricts to a closure operator on $\ov{\Lat(\Delta_{n,k}^2)}$ whose image is isomorphic to $\osnk$. The theorem then follows from Corollary \ref{closure}. \end{Proof} We will now investigate the structure of $\snk$. The next two lemmas follow immediately from the definition of $\snk$. We write $\hat{0}$ for the empty graph, which is the minimum element of $\snk$, and $\hat{1}$ for the complete graph, which is its maximum. \begin{lemma} Let $G,H \in \snk$. Then $G$ covers $H$ if and only if one of the following conditions holds: \begin{itemize} \item[(i)] $E(G) \setminus E(H)$ is a clique on $k$ vertices belonging to $k$ pairwise different components of $H$. \item[(ii)] $E(G) \setminus E(H)$ is a complete bipartite graph on parts $A$ and $B$, and there is a vertex $v$ such that $A \cup \lp v \rp$ and $B \cup \lp v \rp$ are blocks in $H$. \item[(iii)] Only if $k>2$: $E(G) \setminus E(H)$ is a star (that is, a connected graph with at most one vertex of degree more than one), and the vertices of degree one in this star form a block in $H$ belonging to a component of $H$ distinct from that of the center of the star. \end{itemize} \label{covsnk} \end{lemma} The three types of coverings can informally be described as follows: \begin{itemize} \item[(i)] select a vertex from each of $k$ pairwise disjoint components of $H$ and then create a $k$-clique on these vertices; \item[(ii)] complete the union of two overlapping blocks of $H$ to a clique; \item[(iii)] for $k>2$: select a block and a vertex from different components of $H$ and complete their union to a clique. \end{itemize} The lattices $\snk$ are neither upper nor lower semimodular. However, they exhibit a recursive structure on lower intervals, and certain upper intervals are upper semimodular, as the following lemma shows. \begin{lemma} Let $G\in \snk$. \begin{itemize} \item[(i)] If $G$ has $r$ non-singleton blocks of sizes $m_1, \dots , m_r$ then the interval $[\hat{0},G]$ is isomorphic to the direct product $\Sigma_{m_1,k} \times \cdots \times \Sigma_{m_r,k}$ \item[(ii)] If $G$ is connected then the interval $[G,\hat{1}]$ is isomorphic to a direct product of partition lattices. More precisely, suppose that $G$ has $s$ cutpoints and that the $i$-th cutpoint lies in $t_i \geq 2$ blocks. Then, $[G,\hat{1}] \cong \Pi_{t_1} \times \cdots \times \Pi_{t_s}$. \end{itemize} \label{interv} \end{lemma} The following description of the coatoms of $\snk$, that is, the elements which are covered by $\hat{1}$, follows immediately from the two preceding lemmas. \begin{lemma} Let $M$ be a coatom of $\snk$. Then one of the following conditions holds: \begin{itemize} \item[(i)] $M$ is connected and has two blocks of size $l,m$ with $k \leq l \leq m \leq n-k+1$ and $l+m=n+1$. In this case, the interval $\lb \hat{0},M \rb$ is isomorphic to $\Sigma_{l,k} \times \Sigma_{m,k}$. \item[(ii)] $M$ consists of an $(n-1)$-clique and an isolated vertex. In this case, $k>2$ and the interval $\lb \hat{0},M \rb$ is isomorphic to $\Sigma_{n-1,k}$. \end{itemize} \label{cosnk} \end{lemma} For any graph $G$ let $c(G)$ be the number of connected components, and $b(G)$ the number of blocks of size $\geq 2$. \begin{theorem} \begin{itemize} \item[(i)] The lattice $\Sigma_{n,2}$ is graded with rank function $$\rho(G)=2n-2c(G)-b(G).$$ In particular, its length is $\rho(\hat{1})=2n-3$. \item[(ii)] The lattice $\Sigma_{n,3}$ is graded with rank function $$\rho(G)=n-c(G)-b(G).$$ In particular, its length is $\rho(\hat{1})=n-2$. \item[(iii)] If $k>3$ and $n<2k-1$, then $\snk$ is isomorphic to the lower-truncated Boolean algebra $\{A\subseteq [n]: |A|\geq k\}\cup \{ \emptyset \}.$ In particular, $\snk$ is graded of length $n-k+1$. \item[(iv)] If $k>3$ and $n \geq 2k-1$, then $\ell$ is the length of a maximal chain of $\snk$ if and only if $$\ell =(n-2)-t(k-3), \mbox{ for some } 1\leq t\leq \lfloor \frac{n-1}{k-1} \rfloor .$$ In particular, $\snk$ is of length $n-k+1$ and is not graded. \item[(v)] If $k>3$ then $G \in \ov{\snk}$ is contained in a chain of length $n-k+1$ if and only if $G$ consists of a clique of size $l \geq k$ and $n-l$ isolated vertices. \end{itemize} \label{chsnk} \end{theorem} \begin{Proof} For claims (i) and (ii) it suffices to check that the given rank functions increase by $1$ for each type of covering given in Lemma \ref{covsnk} and take value zero at the empty graph. Claim (iii) is clear from the definition. Claims (iv) and (v) are implied by the following description of the maximal chains in $\snk$. We will here view $\snk$ as a subposet of $\Sigma_{n,3}$, and we let $\rho$ denote the restriction of the rank function of claim (ii) from $\Sigma_{n,3}$ to $\snk$. A maximal chain from $\hat{0}$ to $\hat{1}$ in $\snk$ is a sequence of covering steps. By Lemma \ref{covsnk} there are three possibilities for each step. The rank function $\rho$ will increase by $1$ for coverings of types (ii) or (iii), and by $k-2$ for coverings of type (i). Hence, the length of a maximal chain must be $n-2-t(k-3)$, where $t$ is the number of covering steps of type (i). Note that $t\geq 1$ since the first covering in the chain must be of type (i), and that $t\leq \lfloor \frac{n-1}{k-1} \rfloor$ since each step of type (i) reduces the number of connected components by $k-1$ and the total reduction of components along the whole chain is $n-1$. Now, suppose that $1\leq t\leq \lfloor \frac{n-1}{k-1} \rfloor$. A maximal chain of length $n-2-t(k-3)$ is constructed as follows. First perform a sequence of $t$ covering steps of type (i) producing the graph with $k$-cliques on the sets $\{1,\dots,k\}$, $\{k,\dots,2k-1\}$, $\dots$ $\{(t-1)k-(t-2),\dots,tk-(t-1)\}$. Then continue from there via a sequence of $t-1$ covering steps of type (ii) leading to the graph with a $(tk-(t-1))$-clique on the set $[tk-(t-1)]$. Finally, $n-(tk-(t-1))$ covering steps of type (iii) will lead to the complete graph. The total number of steps taken, i.e. the length of the constructed chain, is $t+(t-1)+n-(tk-(t-1))=n-2-t(k-3)$. %All the claims follow from Lemma \ref{cosnk}, using induction on $n$. %Except for the claim about shortest chains in (iv), the base case is %$n=k$, which is trivial, and the induction step is easy. If $k>3$ %and $n=2k-1$ then any $G \in \osnk$ consists of either a clique of %size $l \geq k$ and $n-l$ isolated vertices or two $k$-cliques which %intersect in a point. Using Lemma \ref{covsnk}, it is easily seen that %a maximal chain in $\snk$ %which does not include a graph of the second type has length $n-k+1$, %while a maximal chain including a graph of the second type has length %three, and the claim holds in this case. Now assume $n>2k-1$. We will %show that if $M$ is a coatom in $\snk$ then every maximal chain in %$\lb \hat{0},M \rb$ has length at least %$(n-2)-(k-3)\lfloor \frac{n-1}{k-1} \rfloor-1$, and that there is some %coatom which gives a chain of exactly this length. If $M$ consists of %an $(n-1)$-clique and an isolated vertex, then our claim follows %immediately by inductive hypothesis. Now assume $M$ consists of two cliques %of sizes $l,m$ which intersect in a single vertex, with $l+m=n+1$. We %examine the three possible cases, using the inductive hypothesis and %the structure of $\lb \hat{0},M \rb$ as described in Lemma \ref{cosnk}. %\begin{itemize} %\item If $2k-1 \leq l \leq m$ then the shortest maximal chain in %$\lb \hat{0},M \rb$ has length %\[ %(n-3)- %(k-3)(\lfloor \frac{l-1}{k-1} \rfloor + \lfloor \frac{m-1}{k-1} \rfloor). %\] %\item If $l<2k-1 \leq m$ then the shortest maximal chain in %$\lb \hat{0},M \rb$ has length %\[ %n-k-(k-3)\lfloor \frac{m-1}{k-1} \rfloor= %(n-2)-(k-3)(\lfloor \frac{m-1}{k-1} \rfloor +1)-1. %\] %\item If $l \leq m<2k-1$ then the shortest maximal chain in %$\lb \hat{0},M \rb$ has length $n-2-2(k-3)$. %\end{itemize} %In all cases, the shortest chain is as long as necessary. Moreover, if %$M$ consists of a $k$-clique and an $(n-k+1)$-clique which intersect in %a unique vertex, then the shortest chain in $\lb \hat{0},M \rb$ has %length $(n-2)-(k-3)\lfloor \frac{n-1}{k-1} \rfloor-1$. \end{Proof} The above result yields some nontrivial information about the topology of $\Delta_{n,k}^2$. For instance, part (i) shows that the order complex of $\ov{\Sigma_{n,2}}$ is pure of dimension $2n-5$. With Theorem \ref{dnksnk} this implies that the homology of $\Delta_{n,2}^2$ vanishes in dimensions greater than $2n-5$ and is free in dimension $2n-5$. Of course, in this case we already have more precise knowledge from Theorem \ref{main}. By similar reasoning we can conclude the following new information about the $k=3$ case from part (ii) of Theorem \ref{chsnk}. \begin{theorem} $\widetilde{H}_i(\Delta_{n,3}^2)=0$ for all $i>n-4$, and $\widetilde{H}_{n-4}(\Delta_{n,3}^2)$ is free. \label{snk3} \end{theorem} In the remaining cases the following can be deduced. \begin{theorem} Assume that $k>3$. \begin{itemize} \item[(i)] $\widetilde{H}_i(\Delta_{n,k}^2)=0$ if $i>n-k-1$ or $n-k-1>i>n-2k+2$. \item[(ii)] $\widetilde{H}_{n-k-1}(\Delta_{n,k}^2)$ is free of dimension ${{n-1} \choose {k-1}}$. \item[(iii)] If $n<2k-1$ then $\Delta_{n,k}^2$ has the homotopy type of a wedge of ${{n-1} \choose {k-1}}$ spheres of dimension $n-k-1$. \item[(iv)] If $n=2k-1$ then $\Delta_{n,k}^2$ has the homotopy type of a wedge of spheres. This wedge consists of ${{n-1} \choose {k-1}}$ $(n-k-1)$-spheres and $\frac{1}{2}n{{n-1} \choose {k-1}}$ $1$-spheres. \end{itemize} \label{snk4} \end{theorem} \begin{Proof} We use Theorem \ref{dnksnk} without reference throughout the proof. Claim (i) follows immediately from Theorem \ref{chsnk}(iii),(iv),(v). By Theorem \ref{chsnk}(v), the subposet of $\snk$ generated by chains of length $n-k+1$ is isomorphic to the poset obtained by removing all sets of sizes $1,2,\ldots,k-1$ from the Boolean algebra $B_n$. Claims (ii) and (iii) now follow immediately from the rank selection results in \cite{Bjo91,Sta82}, along with Theorem \ref{chsnk}(iii),(iv). If $n=2k-1$, let $W$ be the set of vertices in $\Delta(\ov{\snk})$ corresponding to graphs which consist of two $k$-cliques intersecting in a single vertex, and let $\Delta_0$ be the complex obtained by removing all simplices containing an element of $W$ from $\Delta(\ov{\snk})$. Then $\Delta_0$ is the order complex of the subposet of $\ov{\snk}$ generated by chains of length $n-k+1$, and is therefore homotopy equivalent to a wedge of ${{n-1} \choose {k-1}}$ $(n-k-1)$-spheres, as above. If $G \in \snk$ corresponds to an element $w \in W$, then by Lemma \ref{covsnk}, $(\ov{\snk})_{3$ when $n \leq 2k-1$, but not for $k=3$, nor for $k>3$ and $n>2k-1$. Indeed, other than the connectivity result given in Corollary \ref{skelcor}, in the case $k=3$ our only information on the topology of $\Da_{n,k}^2$ is given by Theorem \ref{snk3} (unless $n$ is very small), and in the case $k>3$ and $n>2k-1$ we were only able to determine the homology group $\widetilde{H}_{n-k-1}(\Da_{n,k}^2)$. In this section, we investigate the reduced Euler characteristic of $\Da_{n,k}^2$. We will determine a formula for the exponential generating function %$\tilde{\chi}^k(n):=\tilde{\chi}(\Da^{2}_{n,k})$ $$M_k(x):=\sum_{n=1}^\infty \tilde{\chi}(\Da^{2}_{n,k})\frac{x^n}{n!},$$ for all $k \geq 2$. That formula is stated in the following theorem. \begin{theorem}\label{th:mobius} For $k\ge 2$, we have \[M_k'\left(x\frac{p_{k-1}(x)}{p_k(x)}\right)= \ln\left(\frac{p_{k-1}(x)}{p_k(x)}\right), \] where $p_k(x):=1+x+\frac{x^2}{2!}+\cdots +\frac{x^{k-1}}{(k-1)!}$. \end{theorem} %\begin{corollary} Theorem \ref{th:mobius} gives another proof that $\tilde{\chi}(\Da^{2}_{n,2})=-(n-2)!$. It also implies \[M_3'(x)=\ln\left(\frac{-x(x-2)}{(x-1)+\sqrt{2-(x-1)^2}}\right), \] which gives the sequence $0,0,-1,3,-21,180,-2010,27090,-430290,\ldots$ for $\tilde \chi(\Da^2_{n,3})$, cf. Table 2. To obtain these corollaries set $y:=x\frac{p_{k-1}(x)}{p_k(x)}$ and solve for $x$ to get $x=\frac{1}{1-y}$ and $x=\frac{y-1+\sqrt{2-(y-1)^2}}{2-y},$ when $k=2$ and $3$ respectively. \medskip To prove Theorem \ref{th:mobius} we will use the posets $\snk$ defined in Section \ref{closed}. Note that $\tilde\chi(\Da^2_{n,k})=\mu_{\Sigma_{n,k}}(\hat 0,\hat 1)$. We will write $\mu_k(G)$ for $\mu_{\Sigma_{n,k}}(\hat 0,G)$ and $\mu_k(n)$ for $\mu_{\Sigma_{n,k}}(\hat 0,\hat 1)$. Let $\Pi_{n,k}$ be the $k$-equal lattice, which is the lattice of partitions of $\lb n \rb$ into subsets such that each subset has size one or at least $k$. Let $\tau_k(n):=\mu_{\Pi_{n,k}}(\hat 0,\hat 1)$ for $n\ge k$ and $\tau_k(2)=\cdots=\tau_k(k-1)=0$, but $\tau_k(1)=1$. The exponential generating function, $T_k(x):=\sum_{n=1}^\infty \tau_k(n)\frac{x^n}{n!}$, for the M\"obius function of $\Pi_{n,k}$ is known to be \[ T_k(x)=\ln(p_k(x)), \] where $p_k(x)$ is as above. It was first calculated in \cite{BL}. Let $\cC$ be the set of connected graphs in $\Sigma_{n,k}$. Now let \[ \sigma: \Sigma_{n,k}\setminus \cC \longrightarrow \Pi_{n,k} \] be the function which maps a disconnected graph in $\snk$ to the partition determined by its connected components. It is easily seen that for each $x \in \Pi_{n,k}$, $\sigma^{-1}_{\leq}(x)$ has a unique maximum element and therefore has a contractible order complex. Thus by Proposition \ref{quillen} and the definition of the M\"obius function, we have \[\tau_k(n)=-\sum_{G\in \Sigma_{n,k}\setminus \cC} \mu_k(G)= \sum_{G\in \cC} \mu_k(G). \] Thus it suffices to concentrate on the connected but not $2$-connected graphs. First we need a simple lemma. \begin{lemma} If $G \in \cC$ has blocks $W_1,\ldots,W_r$ with $|W_i|=w_i$, then \[\mu_k(G)=\prod_{i=1}^{r}\mu_k(w_i). \] \label{mult} \end{lemma} \begin{Proof} By Lemma \ref{interv} the interval $\lb \hat{0},G \rb$ is isomorphic to the product poset $\Sigma_{w_1,k} \times \ldots \times \Sigma_{w_r,k}$. The lemma now follows from the well known multiplicativity of the M\"obius function. \end{Proof} Now define $\displaystyle{\al_k(n):=\sum_{G\in\cC_1} \mu_k(G)}$, where $\cC_1:=\{G\in \cC: \mbox{$n$ is not a cutpoint of G}\}$. Also, set $\al_k(1)=\cdots=\al_k(k-1)=0$ and $A_k(x):=\displaystyle{\sum_{n=1}^\infty \al_k(n)\frac{x^n}{n!}}$. \begin{lemma}\label{lm:Dk} We have \[ A_k'(x)=\ln\left(\frac{p_{k-1}(x)}{p_k(x)}\right). \] \end{lemma} \begin{Proof} If $n$ is a cutpoint of $G \in \cC$, let $P_1,\dots,P_t$ be the connected components of $G-n$. Then for each $i \in \lb t \rb$, $n$ is not a cutpoint of the connected subgraph of $G$ induced on $P_i \cup \lp n \rp$. Using Lemma \ref{mult}, we get the recursive formula \[\tau_k(n)=\al_k(n)+\sum_{t=2}^{\left\lfloor\frac{n-1}{k-1}\right\rfloor} \sum_{P_1|\ldots|P_t \in \Pi_{n-1}}\al_k(|P_1|+1)\cdots\al_k(|P_t|+1), \] where each summand in the double sum on the right counts the M\"obius functions of all elements $G \in \cC$ such that $G-n$ has connected components $P_1,\ldots,P_t$. By the definition of $\al_k(n)$ we can rewrite this formula as \[ \tau_k(n)=\sum_{P_1|\ldots|P_t \in \Pi_{n-1}} \al_k(|P_1|+1)\cdots\al_k(|P_t|+1). \] The exponential formula (Proposition \ref{exp}) and easy power series manipulations then give \[T_k'(x)=e^{A_k'(x)}. \] \end{Proof} \noindent {\sf Proof of Theorem \ref{th:mobius}:} We will establish a recurrence relation for $\al_k$ involving $\mu_k$. Let $G \in \cC_1$ and let $W$ be the block of $G$ containing $n$. Then $W$ is the unique maximal clique in $G$ which contains $n$. Let $S=W \setminus \lp n \rp$. Let $B_G$ be the graph on the blocks of $G$ defined in Proposition \ref{sep}. Let $T_1,\ldots,T_t$ be the connected components of $B_G-W$ and for each $T_i$ let $V_i$ be the set of vertices of $G$ which are contained in a block that is contained in $T_i$. For each $T_i$ there is a unique $j_i \in S$ such that the subgraph of $G$ induced on $V_i \cup \lp j_i \rp$ is a connected union of blocks of $G$. Conversely, any $G \in \cC_1$ can be obtained by choosing $S$, $T_i$ and $j_i$ as above and then choosing graphs $H_i$ on $T_i \cup \lp j_i \rp$ such that each $H_i$ is either a clique of size at least $k$ or isomorphic to a connected element of $\ov{\Sigma}_{|T_i|+1,k}$. These choices can all be made independently, so we get the recurrence relation \[ \al_k(n)=\sum_{S\uplus T=[n-1]}\mu_k(|S|+1)\sum_{T_1\uplus\ldots\uplus T_t=T} (\al_k(|T_1|+1)|S|)\cdots(\al_k(|T_t|+1)|S|). \] We cannot apply the exponential formula directly at this point due to the factors $|S|$ which appear on the right hand side of the above equation. However, we get \[A_k'(x)=\sum_{n=1}^\infty \frac{\al_k(n)x^{n-1}}{(n-1)!} %\mathop{=}\limits^{{\tiny \{\mbox{take $i:=|S|$}\}}} \] \[=\sum_{n=1}^\infty\sum_{i=1}^{n-1}\binom{n-1}{i}\frac{\mu_k(i+1)x^i}{(n-1)!} \sum_{T_1|\ldots|T_t \in \Pi_{n-i-1}}(\al_k(|T_1|+1)i)\cdots(\al_k(|T_t|+1)i) x^{n-i-1} \] \[=\sum_{i=1}^\infty\frac{\mu_k(i+1)x^i}{i!}\sum_{n=i+1}^\infty \sum_{T_1|\ldots|T_t \in \Pi_{n-i-1}} (\al_k(|T_1|+1)i)\cdots(\al_k(|T_t|+1)i)\frac{x^{n-i-1}}{(n-i-1)!} \] Applying the exponential formula, for each $i$ we get \[ \sum_{n=i+1}^\infty \sum_{T_1|\ldots|T_t \in \Pi_{n-i-1}} (\al_k(|T_1|+1)i)\cdots(\al_k(|T_t|+1)i)\frac{x^{n-i-1}}{(n-i-1)!} =e^{iA_k'(x)}. \] Thus \[A_k'(x)=\sum_{i=1}^\infty\frac{\mu_k(i+1)x^i}{i!}e^{iA_k'(x)}= M_k'\left(xe^{A_k'(x)}\right). \] The theorem now follows from Lemma \ref{lm:Dk}. \qed %If $k > 4$ then homology no longer appears in one dimension only. % %$$ %\begin{array}{|c||c|c|c|c|c|c|} %\hline %n \backslash i & 0 & 1 & 2 & 3 \\ %\hline %\hline %4 & 0 & 0 & 0 & 0 \\ %\hline %5 & \ZZ^4 & 0 & 0 & 0 \\ %\hline %6 & \ZZ^6 & \ZZ^{13} & 0 & 0 \\ %\hline %7 & 0 & \ZZ^{70} & \ZZ^{20} & 0 \\ %\hline %\end{array} %$$ % %\centerline{{\sf Table 3:} Homology groups $\widetilde{H}_i(\Delta_{n,4}^2)$} % %\medskip \section{$(n-2)$-connected graphs and matching complexes} \label{match} Before we proceed to consider $(n-2)$-connected graphs, let us state some simple but useful facts about the general situation. What do maximal $(i-1)$-separable graphs on the $n$ element set $[n]$ look like? Is is clear that each such graph is described by an $(i-1)$-set $A$ and a partition $B \uplus C$ of $[n] \setminus A$ into two non-empty blocks $B$, $C$. The corresponding maximal $(i-1)$-separable graph is the complete graph on $[n]$ with all edges connecting $B$ and $C$ removed. Now let $G$ be an $(n-2)$-connected graph on $n$ vertices, so $G \not\in \Delta_{n}^{n-2}$. Then by the above description of maximal $(n-3)$-separable graphs the induced subgraph on any three vertices must contain at least two edges. Thus the complementary graph (i.e., the graph containing precisely the edges that are not in $G$) is a matching. The graphs on $n$ vertices that are matchings form a simplicial complex, that we denote by $M_n$. We conclude the following. \begin{proposition} The matching complex $M_n$ is Alexander dual (in the sense of Proposition \ref{alexander}) to the complex $\Delta_n^{n-2}$. In particular, there is an isomorphism $$\widetilde{H}_i(M_n) \cong \widetilde{H}^{\binom{n}{2}-i-3}(\Delta_n^{n-2}).$$ \end{proposition} The matching complexes $M_n$ have attracted attention for various reasons. In \cite[Theorem 4.1]{BLVZ92} the matching complex $M_n$ is shown to be topologically $( \lfloor \frac{n+1}{3} \rfloor-2)$-connected, which implies that $\widetilde {H}_i(M_n)=0$, for $i=0,\dots, \lfloor \frac{n+1}{3} \rfloor -2$. We thus get the following corollary. \begin{corollary} The cohomology of $\Delta_n^{n-2}$ vanishes in dimensions $i \geq \binom{n}{2}-\lfloor \frac{n+1}{3} \rfloor -1$. \end{corollary} The following table shows what we know about the homology groups $\widetilde{H}_i(M_n)$, based on the results of \cite{BLVZ92} for $n\leq 6$ and $n=8$, and our own computations. $$ \begin{array}{|c||c|c|c|c|c|c|} \hline n \backslash i & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \hline 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 3 & \ZZ^2 & 0 & 0 & 0 & 0 & 0 \\ \hline 4 & \ZZ^2 & 0 & 0 & 0 & 0 & 0 \\ \hline 5 & 0 & \ZZ^6 & 0 & 0 & 0 & 0 \\ \hline 6 & 0 & \ZZ^{16} & 0 & 0 & 0 & 0 \\ \hline 7 & 0 & \text{torsion}^1& \ZZ^{20} & 0 & 0 & 0 \\ \hline 8 & 0 & 0 & \ZZ^{132} & 0 & 0 & 0 \\ \hline 9 & 0 & 0 & \ZZ^{42} \oplus \text{torsion}^2 & \ZZ^{70} & 0 & 0 \\ \hline 10 & 0 & 0 & \text{torsion}^3 & \ZZ^{1216} & 0 & 0\\ \hline 11 & 0 & 0 & 0 & \ZZ^{1188} \oplus \text{torsion}^4 & \ZZ^{252} & 0\\ \hline 12 & 0 & 0 & 0 & \text{torsion}^5 & \ZZ^{12440} & 0\\ \hline \end{array} $$ \centerline{{\sf Table 3:} Homology groups $\widetilde{H}_i(M_n)$ of matching complexes} \medskip \footnotetext[1]{There is $\ZZ_3$-torsion of rank $1$. No $\ZZ_p$-torsion for $p=2$, $5 \leq p \leq 17$.} \footnotetext[2]{There is $\ZZ_3$-torsion of rank $8$. No $\ZZ_p$-torsion for $p=2$, $5 \leq p \leq 17$.} \footnotetext[3]{There is $\ZZ_3$-torsion of rank $1$. No $\ZZ_p$-torsion for $p=2, 5, 7$.} \footnotetext[4]{There is $\ZZ_3$-torsion of rank $35$. No $\ZZ_p$-torsion for $p=2, 5, 7$.} \footnotetext[5]{There is $\ZZ_3$-torsion of rank $56$. No $\ZZ_p$-torsion for $p=2, 5, 7$.} We see that the complexes $\Delta_n^{n-2}$ can have torsion, and that this phenomenon begins with $\Delta_{7}^{5}$. \section{The Euler characteristic of the complex $\Da_{n}^{n-3}$} \label{notn3} Consider the complex $(\Delta_n^{n-3})^*$ which is the Alexander dual of $\Delta_n^{n-3}$, and the exponential generating function of its reduced Euler characteristic $$F_n^{n-3} (x) := \displaystyle{\sum_{n \geq 0} \widetilde{\chi}((\Delta_n^{n-3})^*) \frac{x^n}{n!}}.$$ The values of $\widetilde{\chi}((\Delta_n^{n-3})^*)$ in the degenerate cases $n\leq 3$ will appear from an explicitly calculated expansion below. We will express the reduced Euler characteristic of $\Delta_n^{n-3}$ in terms of an expression for this series. \begin{theorem} We have that: $$F_n^{n-3}(x)= x-\frac{\exp(\frac{x}{2(1+x)})+x-\frac{1}{4}x^2-\frac{1}{8}x^4}{\sqrt{1+x}}.$$ The exponential generating function of the reduced Euler characteristic of $\Delta_n^{n-3}$ is then the sum of the real and imaginary parts of $-F_n^{n-3}(ix)$. \end{theorem} \Proof We will argue as we did for $\Delta_n^{n-2}$ in Section \ref{match}. If a graph $G$ is $(n-3)$-connected then the induced subgraph on any $4$ of its vertices contains either a vertex of degree $3$ or a path of length $3$. Thus in the complementary graph the induced subgraph on any $4$ vertices is either contained in a $3$-cycle or in a path of length $3$. In particular, there are no $4$-cycles and no vertices of degree $3$ in the complementary graph. Thus the connected components of the complementary graph are paths of any length and cycles of length different from $4$. Moreover, any graph in which every connected component is a cycle not of length $4$ or a path is the complement of an $(n-3)$-connected graph, so $(\Delta_{n}^{n-3})^*$ consists of all such graphs. There are exactly $n!/2$ different paths of length $n$ and $(n-1)!$ different $n$-cycles on an $n$-element vertex set. Now a direct application of the Exponential Formula \ref{exp} gives the result for the generating function of $(\Delta_n^{n-3})^*$. The remaining assertion follows from the fact that when passing from $\Delta_n^i$ to its Alexander dual the Euler characteristic changes by a factor of $-(-1)^{n(n-1)/2}$. \qed The complex $(\Delta_{n}^{n-3})^*$ has maximal simplexes of dimensions $n-1$ and $n-2$ only. It is easily collapsible to a pure complex of dimension $n-2$. A Maple computation (see below) shows that neither the Euler characteristic of $(\Delta_n^{n-3})^*$ nor the Euler characteristic of $\Delta_n^{n-3}$ alternate in sign, so the pure complexes are certainly not all Cohen-Macaulay. The calculation shows that %$$x - \frac{\exp(\frac{x}{2(1+x)})+x-\frac{1}{4}x^2-\frac{1}{8}x^4} %{\sqrt{1+x}} = $$ $$F_n^{n-3}(x)=-1-x+\frac{1}{4}x^4+\frac{1}{20}x^5+\frac{1}{20}x^6 -\frac{1}{27}x^7- \frac{1}{224}x^8-\frac{1}{480}x^9+O(x^{10}).$$ % 4 5 6 7 8 9 19 10 %- 1 - x + 1/4 x + 1/20 x + 1/20 x - 1/28 x - 1/224 x - 1/480 x - ---- x % 1200 % % 779 11 4601 12 2183 13 250979 14 1120979 15 % + ----- x - ------ x + ----- x - ------- x + -------- x % 30800 147840 58240 6115200 25872000 % % % 2621273 16 38362229 17 586244539 18 3106684571 19 20 % - -------- x + --------- x - ----------- x + ----------- x +O(x ) % 59136000 871270400 13722508800 76684608000$ % We have also studied the slightly larger complex of graphs which are the disjoint union of cycles and paths of any lengths (i.e., graphs with maximum vertex degree at most $2$). This is also a reasonable generalization of the matching complex, which is the complex of all graphs with maximum vertex degree at most $1$. The Euler characteristic of the corresponding Alexander dual has almost the same generating function as for $(\Delta_n^{n-3})^*$. That generating function is $$x - \frac{\exp(\frac{x}{2(1+x)})+x-\frac{1}{4}x^2}{\sqrt{1+x}} = $$ $$-1-x+\frac{1}{8}x^4-\frac{3}{40}x^5+\frac{1}{20}x^6-\frac{1}{28}x^7- \frac{17}{896}x^8-\frac{7}{1920}x^9-\frac{23}{2400}x^{10}+ O(x^{11}).$$ The maximal simplices in the cycles-and-paths complex have dimension $n-1$ or $n-2$, and the complex can be collapsed to a pure $(n-2)$-dimensional complex. The generating function for the Euler characteristic shows that these collapsed complexes are not all Cohen-Macaulay. % 4 5 6 7 17 8 9 23 10 % - 1 - x + 1/8 x - 3/40 x + 1/20 x - 1/28 x + --- x - 7/1920 x - ---- x % 896 2400 % % % 1283 11 35543 12 17373 13 347591 14 4782221 15 % + ----- x - ------- x + ------ x - ------- x + --------- x % 61600 1182720 465920 8153600 103488000 % % % 6512399 16 2036907647 17 5264945969 18 28290146389 19 % - --------- x + ----------- x - ------------ x + ------------ x % 135168000 41820979200 109780070400 613476864000 % % % % 20 % + O(x ) % \section{Final Remarks} \subsection{Homology and Topology of $\Delta_{n}^i$} The results and computations presented in this paper suggest that there is probably no uniform statement that covers the topology of all complexes $\Delta_n^i$. However, for $i \leq 2$ the homotopy type calculations for $\Delta_n^i$ give very nice answers. This is consistent with the graph theoretical study of not $i$-connected graphs, where there is a good structure theory only when $i \leq 3$ (see for example Chapter 6 of Lov\'asz' book \cite{Lov93}, or the survey article by Oxley \cite{Oxl96} and the references therein). As mentioned, there is a structure theory for $3$-connected graphs. The $3$-connected graphs on $n$ vertices for which neither the deletion nor the contraction of an edge leads to a $3$-connected graph were classified by Tutte (see Theorem 2.3 in \cite{Oxl96}) as ``wheels'' and ``whirls,'' both having $2n-2$ edges. Note that this does not provide a characterization of the deletion-minimally $3$-connected graphs (the minimal non-faces of $\Delta_n^3$), however it does show that no graph with less than $2n-2$ edges can be $3$-connected. Hence, $\Delta_n^3$ has a complete $2n-4$-skeleton, which shows that $\widetilde{H}_i(\Delta_n^3)=0$ for $i<2n-4$. This fact together with the following table leads to an interesting conjecture. $$ \begin{array}{|c||c|c|c|c|c|c|c|c|c|c|c|} \hline n \backslash i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \hline 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 4 & 0 & 0 & 0 & 0 & \ZZ^1 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 5 & 0 & 0 & 0 & 0 & 0 & 0 & \ZZ^6 & 0 & 0 & 0 & 0\\ \hline 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \ZZ^{36} & 0 & 0\\ \hline 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \ZZ^{240} \\ \hline \end{array} $$ \centerline{{\sf Table 4:} Homology groups $\widetilde{H}_i(\Delta_n^3)$} \medskip \noindent {\sf Conjecture 9.1:} $\Da_n^3$ has the homotopy type of a wedge of $\frac{(n-3)(n-2)!}{2}$ spheres of dimension $2n-4$. %The homology of $\Da_n^3$ is concentrated in %dimension $2(n-2)$ and free of rank $\frac{(n-3)(n-2)!}{2}$. \medskip For general $i$ the situation (concerning $\Da_n^i$) seems to be far more complicated. However, we would like to remark that for $i=2,3$ the known Betti numbers are (up to sign) the Lah-numbers $L_{n-2,1}$ and $L_{n-2,2}$ (see \cite[p.165-166]{Com70}). This coincidence unfortunately fails for $i=4$, which is easily seen by comparing $L_{n-2,3}$ with $\widetilde{\chi}(\Delta_n^4)$ for $n=6$. For $i>3$ no good structure theory for $i$-connected graphs is known, and the results of Section \ref{match} on $(n-2)$-connected graphs indicate that the topology of $\Delta_n^i$ will not behave nicely for all $i$. Nevertheless, the Alexander Duality with the matching complexes $M_n$ encourages a closer look at the complexes $\Delta_n^{n-2}$. Surprisingly, the prime $3$ seems to play a special role in the topology of these complexes. We have not detected $p$-torsion in the homology of $\Delta_n^{n-2}$ for any prime $p \neq 3$. \medskip \noindent {\sf Question 9.2:} Does the homology of $M_n$ have $p$-torsion for any prime $p \neq 3$? \medskip Even more surprising is the fact that the same prime $3$ seems to play an analogous role for the matching complexes $M_{n,n}$ on complete bipartite graphs $K_{n,n}$, also called chessboard complexes (see \cite{BLVZ92}). $$ \begin{array}{|c||c|c|c|c|c|c|c|} \hline n \backslash i & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \hline 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 2 & \ZZ & 0 & 0 & 0 & 0 & 0 & 0\\ \hline 3 & 0 & \ZZ^4 & 0 & 0 & 0 & 0 & 0 \\ \hline 4 & 0 & 0 & \ZZ^{15} & 0 & 0 & 0 & 0 \\ \hline 5 & 0 & 0 & \ZZ_3 & \ZZ^{56} & 0 & 0 & 0 \\ \hline 6 & 0 & 0 & 0 & \ZZ^{25}\oplus \text{torsion}^1 & \ZZ^{210} & 0 &0 \\ \hline 7 & 0 & 0 & 0 & 0 & \ZZ^{588} \oplus \text{torsion}^2 & \ZZ^{792} & 0 \\ \hline 8 & 0 & 0 & 0 & 0 & \text{torsion}^3 & ? & ? \\ \hline \end{array} $$ \footnotetext[1]{There is $\ZZ_3$-torsion of rank $10$. No $\ZZ_p$-torsion for $p=2,5,7$} \footnotetext[2]{There is $\ZZ_3$-torsion of rank $66$. No $\ZZ_p$-torsion for $p=2,5,7$} \footnotetext[3]{There is $\ZZ_3$-torsion of rank $1$. This group is finite according to \cite{FriHan96}.} \centerline{{\sf Table 5:} Homology groups $\widetilde{H}_i(M_{n,n})$ for the bipartite matching complexes} \medskip It is easy to see that $M_{n,n}$ collapses to an $(n-2)$-dimensional complex. Hence homology is free in dimension $n-2$ and vanishes in higher dimensions. Looking at the table and the footnotes the following question naturally occurs. \medskip \noindent {\sf Question 9.3:} Is the homology of $M_{n,n}$ free except for $3$-torsion ? \medskip \subsection{Homology and Topology of $\Delta_{n,k}^2$} For $k$-graph complexes the problem of determining the topology of the complex of separable $k$-graphs (the $i=2$ case) seems to be the most important. The complexes $\Delta_{n,k}^2$ play the same role in the study of spaces of ``knots'' for which $k$-fold self-intersections are forbidden as the complexes $\Delta_n^2$ play for ordinary knots. \medskip \noindent {\sf Question 9.4:} What is the homology and homotopy type of $\Delta_{n,k}^2$ ? \medskip The evidence from Section \ref{closed} leads us to anticipate the following answer for $k=3$. \medskip \noindent {\sf Conjecture 9.5:} $\Da_{n,3}^2$ is homotopy equivalent to a wedge of $(n-4)$-spheres. \medskip A natural approach to this question is through further combinatorial study of the lattices $\snk$ defined in Section \ref{closed}. \medskip \noindent {\sf Conjecture 9.6:} The lattice $\Sigma_{n,k}$ is shellable. \medskip This is open in all cases, except for the somewhat degenerate cases $n<2k-1$ when $\snk$ is a truncated Boolean algebra. If Conjecture 9.6 were verified for $k=2$ it would reprove Theorem \ref{main}, and it would via Lemma \ref{sn2sn3} imply the truth of Conjecture 9.5. If Conjecture 9.6 were verified for $k=3$ it would also imply the truth of Conjecture 9.5. If Conjecture 9.6 were verified for $k>3$ it would via Theorem \ref{chsnk}(iv) and the results of \cite{BjoWac96} imply the truth of the following. \medskip \noindent {\sf Conjecture 9.7:} If $k>3$ and $n \geq 2k-1$, then $\Da_{n,k}^2$ is homotopy equivalent to a wedge of spheres. Furthermore, the dimensions of the spheres are precisely $n-4-t(k-3)$ for $1\leq t\leq \lfloor \frac{n-1}{k-1} \rfloor$. \medskip \subsection{Generating series of Euler characteristics} Let $F_i(x) = \displaystyle{\sum_{n \geq 0} \widetilde{\chi}(\Delta_{n}^{i}) \frac{x^n}{n!}}$ and $G_i(x) = \displaystyle{\sum_{n \geq 0} \widetilde{\chi}((\Delta_{n}^{n-i})^*) \frac{x^n}{n!}}$. By the results presented in this paper we get the following table: \medskip \begin{center} $\begin{array}{|c||c|c|} \hline i & F_i(x) & G_i(x) \\ \hline \hline 1 & ln(1+x) & -1 \\ \hline 2 & (1-x)\log(1-x)+1+x & -\exp(x-\frac{x^2}{2}) \\ \hline 3 & & x - \frac{\exp(\frac{x}{2(1+x)})+x-\frac{1}{4}x^2-\frac{1}{8}x^4}{\sqrt{1+x}} \\ \hline \end{array}$ \end{center} \centerline{{\sf Table 6:} Generating functions of the Euler characteristics of $\Delta_n^i$ and $(\Delta_n^{n-i})^*$} \medskip We cannot formulate a conjecture about the entries in this table for $i>3$. Nevertheless, even though the actual homology computation may be too difficult, the generating series may be computable for a few more cases. Assuming a positive answer to Conjecture 9.1, we get $$F_3(x) = (x-\frac{3}{2})\log(1-x)+1-\frac{3}{2}x+\frac{1}{4}x^2.$$ \subsection{The representation of the symmetric group} \label{questionrep} All complexes $\Delta_{n,k}^i$ are invariant under the action of the symmetric group $S_n$. This action determines a linear representation of $S_n$ on each homology group $\widetilde{H}_j(\Delta_{n,k}^i)$. For fixed $n,k$ and $i$, the alternating sum of the characters of the given representations is a virtual character of $S_n$ %Since the character determines the %rank of the representation module, we cannot expect to %find the character in situations where Euler characteristic %computations seem to be hopeless. that we denote by $\omega_{n,k}^i$. %be the virtual character of $S_n$ on %$\widetilde{H}_*(\Delta_{n,k}^i)$ described above. For $k=2$ and $i=1,2$ this is an actual character (up to sign) and satisfies $$(-1)^{n+1}\omega_{n+1,k}^2 = (sign_n \omega_{n,k}^1) \uparrow_{S_n}^{S_{n+1}} + sign_{n+1} \omega_{n+1,k}^1.$$ %~\hskip1cm (1)$$ {From} looking at the dimensions of the homology modules it is clear that the analogous formula for $k \geq 3$ does not hold. We have also seen that homology of $\Delta_{n,k}^i$ is not torsion-free in general. On the other hand, \cite{ReiRob97} has demonstrated that for the closely related matching complexes it is possible to determine the representations on the rational homology. Thus it is reasonable to ask: \medskip \noindent {\sf Question 9.8:} What is the character of $S_n$ on each non-vanishing rational homology group of $\Delta_{n,k}^i$ ? \medskip The character $\omega_n^2 = \omega_{n,2}^2$ determined in Section \ref{homrep} has recently appeared in several different areas of mathematics. First in the work of C. A. Robinson \& S. Whitehouse \cite{RobWhi94} and S. Whitehouse \cite{Whi94-2} on gamma-homology of algebras and later in work of E. Getzler \& M. Kapranov \cite{GetKap95} on operads, O. Mathieu \cite{Mat96} on hyperplane arrangements and symplectic geometry, in the work of M. Kontsevich \cite{Kon93} on Lie algebras and symplectic geometry, and in the work of P. Hanlon \cite{Han96}, P. Hanlon \& R.P. Stanley \cite{HanSta95} and S. Sundaram \cite{Sun96} in a combinatorial and representation-theoretic context. It seems mysterious that the same character pops up in so many seemingly unrelated places. \medskip \noindent {\sf Question 9.9:} What are the deeper connections between the various contexts where the character $\omega_n^2$ appears ? \medskip The analogous question for the character $\omega_{n,2}^1= sign_n \cdot \lie_n$ has been studied quite extensively (see for example \cite{Bar90,BarBer90,Reu93}), and for that case much detailed information is known. An important aspect of the work in \cite{Bar90} and \cite{BarBer90} is the construction of explicit bases for the modules under consideration. Thus a first step towards an answer to Question 9.9 could be a solution of the following problem. \medskip \noindent {\sf Problem 9.10:} Describe a combinatorial basis for the homology of $\Delta_n^2$. \medskip A positive answer to the shellability conjecture 9.6 for $k=2$ could via the induced shelling basis (see \cite{BjoWac96}) lead to progress on Problem 9.10. \section{Notation and Tools}\label{notation} In this short section we will summarize the main tools that we use in the study of the complexes $\Delta_{n,k}^i$. We refer the reader to the survey paper \cite{Bjo91} for more details and references. Let $P$ be a finite partially ordered set -- {\it poset} for short. If $P$ has a unique minimum element $\hat{0}$ and a unique maximum element $\hat{1}$, we denote by $\ov{P}$ the {\it proper part} of $P$, that is the poset obtained by removing from $P$ the elements $\hat{0}$ and $\hat{1}$. By $\Delta(P)$ we denote the simplicial complex of all chains in ${P}$. The complex $\Delta(P)$ is called the {\it order complex} of $P$. By convention we include the empty set $\emptyset$ in every simplicial complex. For any simplicial complex $\de$, the {\it face lattice} $\Lat(\de)$ is the poset of faces of $\de$, ordered by inclusion and enlarged by an additional greatest element $\hat{1}$. Then the order complex $\Da(\ov{\Lat(\Da)})$ of the proper part of $\Lat(\Delta)$ is homeomorphic to $\Delta$. Indeed, $\Delta(\ov{\Lat(\Delta)})$ is the barycentric subdivision of $\Delta$. For a poset $P$ and $p \in P$ we denote by $P_{\leq p}$ the sub-poset $\{ p'~|~p' \in P;~p' \leq p~\}$. The posets $P_{\geq p}$, $P_{< p}$ and $P_{> p}$ are analogously defined. For $p \leq p'$ in $P$ we denote by $[p,p']$ the {\it closed interval} $P_{\geq p} \cap P_{\leq p'}$ in $P$, and by $(p,p')$ the {\it open interval} $P_{>p} \cap P_{x}) \Bigr).$$ \item[(ii)] Let $P$ be the proper part of a lattice and let $Co$ be the set of complements of some element $p \neq \hat{0}, \hat{1}$. Then $\Delta(P \setminus Co)$ is contractible. \end{itemize} \end{proposition} In the formulation of the proposition $\bigvee$ denotes the wedge product, $\Sigma$ denotes the suspension and $*$ denotes the join of topological spaces. Our next tool is the combinatorial version of a standard duality theorem from algebraic topology. \begin{proposition}[Combinatorial Alexander Duality] \label{alexander} Let $\Delta$ be a finite simplicial complex on vertex set $V$ and define $$\Delta^* = \{ B \subseteq V~|~V \setminus B \not\in \Delta \}.$$ Then $$\widetilde{H}_i(\Delta) \cong \widetilde{H}^{|V|-i-3}(\Delta^*).$$ \end{proposition} This is derived as follows. The usual Alexander duality theorem (see e.g. Munkres \cite{Mun84}) says that $$\widetilde{H}_i(A) \cong \widetilde{H}^{n-i-1} (S^n \setminus A)$$ for any compact subset $A$ of the $n$-sphere $S^n$. In our situation, let $P = 2^V \setminus \{ \emptyset, V \}$. This truncated Boolean algebra is the proper part of the face lattice of the boundary complex of a simplex, so $\Delta(P) \cong S^{|V|-2}$. Now let $A$ be the realization of $\Da(\ov{\Lat(\Da)})$ as a subspace of $\Delta(P)$. It is easy to see that $\Da(P\setminus \ov{\Lat(\Da)})$ is a strong deformation retract of $S^{|V|-2} \setminus A$, and since $P\setminus \ov{\Lat(\Da)} \cong \ov{\Lat(\Da^*)}$ the result follows. Finally, we recall a result from enumerative combinatorics. For a number sequence $(a_n)_{n \geq 0}$ the formal power series $\displaystyle{\sum_{n\geq 0} a_n \frac{x^n}{n!}}$ is called its {\it exponential generating function}. \begin{proposition}[Exponential formula] \label{exp} Suppose that two functions $a,b:\NN \longrightarrow \ZZ$ are given such that \[b(n)=\sum_{S_1|\cdots| S_t \in \Pi_n}a(|S_1|)\cdots a(|S_t|),\quad n\ge 1, \] where the sum ranges over all set partitions of $[n]$ and $a(0)=0$, $b(0)=1$. Then the exponential generating functions $A(x):=\sum_{n=0}^\infty \frac{a(n)x^n}{n!}$ and $B(x):=\sum_{n=0}^\infty \frac{b(n)x^n}{n!}$ satisfy \[B(x)=e^{A(x)}. \] \end{proposition} \noindent For the proof see \cite{Sta78,Sta98}. %\bibliography{group} %\bibliographystyle{plain} %\end{document} \begin{thebibliography}{BLVZ} \bibitem[Ba]{Bar90} H.~Barcelo. \newblock On the action of the symmetric group on the free {L}ie algebra and the partition lattice. \newblock {\em J. Comb. Theory, Ser. A} 55(1):93--129, 1990. \bibitem[BaBe]{BarBer90} H.~Barcelo and N.~Bergeron. \newblock The Orlik-Solomon algebra on the partition lattice and the free {L}ie algebra. \newblock {\em J. Comb. Theory, Ser. A} 55(1):80--92, 1990. \bibitem[B]{Bjo91} A.~Bj{\"o}rner. \newblock Topological methods. \newblock In: R.~Graham, M.~Gr{\"o}tschel, and L.~Lov{\'a}sz, editors, {\em Handbook of Combinatorics}, pages 1819--1872. North-Holland, Amsterdam, 1995. \bibitem[BL]{BL} A.~Bj{\"o}rner and L.~Lov{\'a}sz. \newblock Linear decision trees, subspace arrangements and M\"obius functions. \newblock {\em Journal of American Mathematical Society} 7(3):677--706, 1994. \bibitem[BLVZ]{BLVZ92} A.~Bj{\"o}rner, L.~Lov{\'a}sz, S.~T. Vre{\'c}ica, and R.~T. {\v Z}ivaljevi\'c. \newblock Chessboard complexes and matching complexes. \newblock {\em J. London Math. Soc. (2)}, 49(2):25--39, 1994. \bibitem[BW]{BjoWac96} A.~Bj{\"o}rner and M.~L. Wachs. \newblock Shellable nonpure complexes and posets, I. \newblock {\em Trans. Amer. Math. Soc.}, 348(4):1299--1327, 1996. \bibitem[BWa]{BjoWal83} A.~Bj{\"o}rner and J.~W. Walker. \newblock A homotopy complementation formula for partially ordered sets. \newblock {\em European J. Combin.}, 4(1):11--19, 1983. \bibitem[BWe]{BjoWel92} A.~Bj{\"o}rner and V.~Welker. \newblock Homology of the ``k-equal'' manifolds and related partition lattices. \newblock {\em Advances in Math.}, 110(2):277--306, 1995. \bibitem[Ch]{ch} M.~K.~Chari. \newblock On discrete Morse functions and combinatorial decompositions. \newblock Preprint, 1996. \bibitem[Co]{Com70} L.~Comtet. \newblock {\em Analyse Combinatoire}, Vol. I. \newblock Presses Universitaires de France, Paris, 1970. \newblock [English translation: {\em Advanced Combinatorics,} Reidel, Dordrecht, 1974.] \bibitem[F]{fo} R.~Forman. \newblock Morse theory for cell complexes. \newblock {\em Advances in Math.}, to appear. \bibitem[FH]{FriHan96} J.~Friedman and P.~Hanlon. \newblock On the Betti numbers of chessboard complexes. \newblock Preprint, 1996. %\bibitem[G]{Gla70} %L.C.~Glaser. %\newblock {\em Combinatorial Geometry Theory}. %\newblock van Nostrand, New York, 1970. \bibitem[GK]{GetKap95} E.~Getzler and M.~M.~Kapranov. \newblock Cyclic operads and cyclic homology. \newblock In: {\em Geometry, topology and physics}, pp 167-201. \newblock International Press, Cambridge, MA, 1995. \bibitem[H1]{Han82} P.~Hanlon. \newblock The fixed-point partition lattices. \newblock {\em Pacific J. Math.}, 96(2):319--341, 1981. \bibitem[H2]{Han96} P.~Hanlon. \newblock Otter's method and the homology of homeomorphically irreducible $k$-trees. \newblock {\em J. Combin. Theory Ser. A}, 74(2):301--320, 1996. \bibitem[HS]{HanSta95} P.~Hanlon and R.~P.~Stanley. \newblock A $q$-deformation of a trivial symmetric group action. \newblock {\em Trans. Amer. Math. Soc.}, to appear. \bibitem[K]{Kon93} M.~Kontsevich. \newblock Formal (non)-commutative symplectic geometry. \newblock In: L.~Corwin et al. (eds.) {\em The Gelfand Mathematical Seminar, 1990--1992}, pp. 173-187. \newblock Birkh\"auser, Boston, 1993. \bibitem[KSS]{KahSakStu84} J.~Kahn, M.~Saks and D. Sturtevant. \newblock A topological approach to evasiveness. \newblock {\em Combinatorica}, 4(4):279-306, 1984. \bibitem[L]{Lov93} L.~Lov{\'a}sz. \newblock {\em Combinatorial Problems and Exercises}, 2nd edition. \newblock North-Holland, Amsterdam, 1993. \bibitem[Ma]{Mat96} O. Mathieu. \newblock Hidden $S_{n+1}$-actions. \newblock {\em Comm. Math. Phys.}, 176(2):467--474, 1996. \bibitem[Mu]{Mun84} J.~R. Munkres. \newblock {\em Elements of Algebraic Topology}. \newblock Addison-Wesley, Menlo Park, CA, 1984. \bibitem[O]{Oxl96} J. Oxley. \newblock Structure theory and connectivity for matroids. \newblock in: J.E. Bonin et al (eds.) {\em Matroid Theory}, volume 197 of {\em Contemp. Math.}, pp. 129--170. \newblock Amer. Math. Soc., Providence, RI, 1996. \bibitem[Q]{Qui78} D.~Quillen. \newblock Homotopy properties of the poset of nontrivial $p$-subgroups of a group. \newblock {\em Advances in Math.}, 28(2):101--128, 1978. \bibitem[RR]{ReiRob97} V.~Reiner and J.~Roberts. \newblock Resolutions and the homology of matching and chessboard complexes. \newblock In preparation. \bibitem[Re]{Reu93} C. Reutenauer. \newblock {\em Free Lie Algebras}. \newblock Oxford University Press. \newblock Oxford, 1993. \bibitem[Ro]{Rob92} C.~A. Robinson. \newblock The space of fully grown trees. \newblock Technical report, Universit\"at Bielefeld, 1992. \bibitem[RW]{RobWhi94} C.~A. Robinson and S.~Whitehouse. \newblock The tree representation of ${\Sigma}_{n+1}$. \newblock {\em J. Pure Appl. Algebra}, 111(1-3):245--253, 1996. \bibitem[Si]{Si} R. Simion. Personal communication, September 1996. \bibitem[St1]{Sta78} R.~P.~Stanley. \newblock Exponential structures. \newblock {\em Stud. Appl. Math.}, 59(1):72-83, 1978. \bibitem[St2]{Sta82} R.~P. Stanley. \newblock Some aspects of groups acting on finite posets. \newblock {\em J. Combin. Theory Ser. A}, 32(2):132--161, 1982. \bibitem[St3]{Sta86} R.~P. Stanley. \newblock {\em Enumerative Combinatorics, Vol 1}. \newblock Wadsworth and Brooks/Cole. \newblock Monterey, CA, 1986. \bibitem[St4]{Sta98} R.~P. Stanley. \newblock {\em Enumerative Combinatorics, Vol 2}. \newblock Book manuscript. \bibitem[Su]{Sun96} S. Sundaram. \newblock Homotopy of non-modular partitions and the Whitehouse module. \newblock Preprint, 1996. \bibitem[SW]{SunWac94} S.~Sundaram and M.~Wachs. \newblock The homology representations of the k-equal partition lattice. \newblock {\em Trans. Amer. Math. Soc.}, 349(3):935-945, 1997. \bibitem[V1]{Vas93-1} V.~A. Vassiliev. \newblock Complexes of connected graphs. \newblock In: L.~Corwin et al. (eds.), {\em The Gelfand Mathematical Seminar, 1990-1992}, pp. 223--235. Birkh\"auser, Boston, 1993. \bibitem[V2]{Vas94} V.~A. Vassiliev. \newblock {\em Complements of Discriminants of Smooth Maps : Topology and Applications}, revised edition, volume~98 of {\em Transl. of Math. Monographs}. \newblock Amer. Math. Soc., Providence, RI, 1994. \bibitem[V3]{Vas97} V.~A. Vassiliev. \newblock Indices of complicated singular knots, homology of two-connected graphs, and calculation of finite order knot invariants. \newblock Preprint, 1997. \bibitem[We]{Wel90-1} V.~Welker. \newblock Equivariant homotopy of posets and some applications to subgroup lattices. \newblock {\em J. Combin. Theory Ser. A}, 69(1):61--86, 1995. \bibitem[Wh]{Whi94-2} S.~Whitehouse. \newblock The Eulerian representations of $\Sigma_n$ as restrictions of representations of $\Sigma_{n+1}$. \newblock {\em J. Pure Appl. Algebra}, 115(3):309--321, 1996. \end{thebibliography} \end{document}