q}})$ and $\Free(\mathcal{C}_{Q_{q\\ q' \quad &\textrm{if } x = q\\ s \quad &\textrm{if } x < q\\ i \quad &\textrm{if } x \textrm{ is incomparable to } q. \end{cases} \] induces a map from $\del_{\Free(\mathcal{C}_Q)}(q)$ to $\del_{\Free(\mathcal{C}_{Q'})}(q')$. One can check the relevant fibers $f^{-1}(Q'_{\leq y})$ to see that they are all contractible, and thus $\del_{\Free(\mathcal{C}_Q)}(q)$ has the homotopy type of $\del_{\Free(\mathcal C_{Q'})}(q')$, and hence is contractible. \end{proof} \begin{corollary} Let $Q$ be a poset and $\mathcal{C}_Q$ be the collection of order convex sets of $Q$. Then $\beta(\mathcal{C}_Q)$ is equal to the number of bottlenecks in $Q$. \end{corollary} \begin{proof} From the previous lemma we see that \[ \widetilde{\chi}(\del_{\Free}(q))= \begin{cases} 1 \quad &\textrm{ if $q$ is a bottleneck}\\ 0 \quad &\textrm{ otherwise}. \end{cases} \] Apply this formula to (\ref{e:betaconvex}). \end{proof} \subsection{Chordal graphs}\label{ss:graph} Gordon has previously interpreted $\beta(\Li(G))$ for a chordal graph $G$ in \cite{Go}. He did this by means of a deletion-contraction argument. Our topological approach gives a more detailed result. \begin{lemma} Let $G$ be a connected chordal graph. A subset $K$ of $V$ is free in $\Li(G)$ if and only if it induces a clique in $G$. \end{lemma} \begin{proof} See \cite[Lemma 5.1]{Go}. \end{proof} For each vertex $v$ in $G$ let $c(v)$ be the number of connected components of $G-v$. \begin{lemma}\label{l:graphtop} Let $G$ be a connected chordal graph and $\Free=\Free(\Li(G))$. For each vertex $v$ in $G$, the complex $\del_{\Free}(v)$ has the homotopy type of $c(v)$ disjoint points. \end{lemma} \begin{proof}[Sketch of Proof] It is easy to show that $\del_{\Free}(v)$ is just the disjoint union of $\Free(\Li(G'))$ for each connected component $G'$ of $G-v$. By Lemma~\ref{l:convexfree} these are all contractible so the lemma follows. \end{proof} Recall that a {\em block} of a graph is a maximal subgraph which contains no cut-vertex. We denote by $b(G)$ the number of blocks in the graph $G$. \begin{corollary} \cite[Theorem 5.1]{Go} Let $G$ be a connected chordal graph. Then $$ \beta(\Li(G))=b(G)-1. $$ \end{corollary} \begin{proof} From the previous lemma we see that \[ \widetilde{\chi}(\del_{\Free}(v))=c(v)-1, \] and so \[ \beta(\Li(G))=\sum_{v \in G} (c(v)-1). \] It is then an elementary graph-theoretic fact \cite[Problem 4.1.28]{We} that \[ \sum_{v \in G} (c(v)-1)=b(G)-1. \] \end{proof} \section{Open problems}\label{s:open} In this section we discuss some open problems aimed toward generalizing Theorem~\ref{t:main} to all convex geometries. As a first step in this direction we formulate a conjecture that generalizes Lemma~\ref{l:boundary} for all convex geometries. Let $\Li$ be a convex geometry on the ground set $X$. A subset $A$ of $X$ will be called {\em independent} if $a \not\in \Li(A-a)$ for all $a \in A$. We will say that {\em $x$ depends on $y$} if there exists an independent set $A$ such that $y \in A$, $ x \in \Li(A)$ but $x \not\in \Li(A-y)$. Let $Dep(x)$ be the set of all points $y$ such that $x$ depends on $y$. The situation in which $Dep(x)=X$ includes the following as special cases \begin{itemize} \item $x \in \inte(\A)$ for a point configuration, \item $x$ is a bottleneck of a poset, \item $x$ is a vertex for which $c(v)=1$ in a chordal graph. \end{itemize} \begin{conjecture}[cf. Lemmata~\ref{l:boundary}, \ref{l:ordertop}, and \ref{l:graphtop}] For a convex geometry $\Li$ on the ground set $X$, the complex $\del_{\Free(\Li)}(x)$ is contractible unless $Dep(x) = X$. \end{conjecture} The question of what happens if $Dep(x)=X$ is much less clear, as the diversity of outcomes in Lemmata~\ref{l:interior}, \ref{l:ordertop}, and \ref{l:graphtop} indicate. In this direction, we pose the following question. \begin{question} For any convex geometry $\Li$ on the ground set $X$, and any point $x$ with $Dep(x)= X$, does the complex $\del_{\Free(\Li)}(x)$ have same integral homology as a bouquet of equidimensional spheres? 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