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Let $K$ be a simplicial complex on vertex set $V$. $K$ is called $d$-Leray if the homology groups of any induced subcomplex of $K$ are trivial in dimensions $d$ and higher. $K$ is called $d$-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most $d$ that is contained in a unique maximal face. We define the $t$-tolerance complex of $K$, $mathcal{T}_t(K)$, as the simplicial complex on vertex set $V$ whose simplices are formed as the union of a simplex in $K$ and a set of size at most $t$. We prove that for any $d$ and $t$ there exists a positive integer $h(t,d)$ such that, for every $d$-collapsible complex $K$, the $t$-tolerance complex $mathcal{T}_t(K)$ is $h(t,d)$-Leray. The definition of the complex $mathcal{T}_t(K)$ is motivated by results of Montejano and Oliveros on tolerant
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Given a graph $G$ on the vertex set $V$, the {em non-matching complex} of $G$, $NM_k(G)$, is the family of subgraphs $G subset G$ whose matching number $ u(G)$ is strictly less than $k$. As an attempt to generalize the result by Linusson, Shareshian and Welker on the homotopy types of $NM_k(K_n)$ and $NM_k(K_{r,s})$ to arbitrary graphs $G$, we show that (i) $NM_k(G)$ is $(3k-3)$-Leray, and (ii) if $G$ is bipartite, then $NM_k(G)$ is $(2k-2)$-Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex $NM_k(G)$, which vanishes in all dimensions $dgeq 3k-4$, and all dimensions $d geq 2k-3$ when $G$ is bipartite. As a corollary, we have the following rainbow matching theorem which generalizes the result by Aharoni et. al. and Driskos theorem: Let $E_1, dots, E_{3k-2}$ be non-empty edge subsets of a graph and suppose that $ u(E_icup E_j)geq k$ for every $i e j$. Then $E=bigcup E_i$ has a rainbow matching of size $k$. Furthermore, the number of edge sets $E_i$ can be reduced to $2k-1$ when $E$ is the edge set of a bipartite graph.
Let $mathcal{H}$ be a hypergraph on a finite set $V$. A {em cover} of $mathcal{H}$ is a set of vertices that meets all edges of $mathcal{H}$. If $W$ is not a cover of $mathcal{H}$, then $W$ is said to be a {em noncover} of $mathcal{H}$. The {em noncover complex} of $mathcal{H}$ is the abstract simplicial complex whose faces are the noncovers of $mathcal{H}$. In this paper, we study homological properties of noncover complexes of hypergraphs. In particular, we obtain an upper bound on their Leray numbers. The bound is in terms of hypergraph domination numbers. Also, our proof idea is applied to compute the homotopy type of the noncover complexes of certain uniform hypergraphs, called {em tight paths} and {em tight cycles}. This extends to hypergraphs known results on graphs.
Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any $n geq 1$, there exists a constant $epsilon(n)>0$ such that for any $0 leq k <n$ the $k$-th coboundary expansion constant of any $n$-dimensional spherical building is at least $epsilon(n)$.
The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex. It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders. Furthermore, comparing the description of its homotopy type induced from the two shellings re-interprets a known convolution formula counting bases of the matroid. The representation of the automorphism group of the matroid on the homology of the augmented Bergman complex turns out to have a surprisingly simple description. This last fact is generalized to closures beyond those coming from a matroid.
It is shown that if T is a connected nontrivial graph and X is an arbitrary finite simplicial complex, then there is a graph G such that the complex Hom(T,G) is homotopy equivalent to X. The proof is constructive, and uses a nerve lemma. Along the way several results regarding Hom complexes, exponentials, and subdivision are established that may be of independent interest.
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