18 problems

Let ${\mathsf {BNM}}_k(r,s)$ be the bipartite matching complex on two parts of sizes $r$ and $s$, with the natural action of $S_r\times S_s$. Let $H(n,l)$ denote the irreducible…

Let $n=2k$ be even. The complexes ${\mathsf {NPM}}_n$ and the poset ${\mathcal L}_n^{(2)}$ have reduced homology groups carrying actions of $S_n$, where $\varepsilon_n$ denotes the…

The $D_n$ conjecture. There exists an integer $n_0\geq 8$ such that, if $n\geq n_0$, then $\widetilde H_{\nu_{2n}}(M(D_n))$ is an elementary $3$-group. Moreover, if $n\geq n_0$ and…

The $n=14$ conjecture. The result holds for $n=14$ as well.

Let $G$ be a planar graph, and let $\mathcal{M}(G)$ denote its matching complex, whose vertices are the edges of $G$ and whose simplices are sets of pairwise disjoint edges. Planar…

Optimality conjecture for $\mathcal{M}^{**}$. The augmented gradient vector field $\mathcal{M}^{**}$ is optimal; equivalently, there is a gradient vector field on $M_7$ with 22 cri…

Björner–Lovász–Vrećica–Živaljević connectivity conjecture. The connectivity bounds in this statement are sharp. The conjecture concerns the precise homotopical connectivity of matc…

Let $M_{n-2}(K_n)$ be the matching complex of $(n-2)$-element matchings in the complete graph $K_n$, and let $M_{n-1}(K_{n,n})$ be the matching complex of $(n-1)$-element matchings…

Matching-complex homology conjecture. The only non-zero homology group of $C_{nd}^d$ for $n=1,\ldots,3d-1$ is

Let $p=2q-1$ be an odd prime, and let $n=2k+1+3r$ and $d=k-1+r$. Then the relevant reduced homology group is $\tilde{H}_{k-1+r}(\mathsf{M}_{2k+1+3r};\mathbb{Z})$. The general odd-p…

Let $n=2k+1+3r$ and $d=k-1+r$, so that the relevant reduced homology group is $\tilde{H}_{k-1+r}(\mathsf{M}_{2k+1+3r};\mathbb{Z})$. The 7-torsion conjecture. The group contains ele…

Let $n=2k+1+3r$ and $d=k-1+r$, so that the relevant reduced homology group is $\tilde{H}_{k-1+r}(\mathsf{M}_{2k+1+3r};\mathbb{Z})$. The 5-torsion conjecture. The group contains ele…

For $k\geq 0$, consider the reduced homology group $\tilde{H}_{k+1}(\mathsf{M}_{2k+7};\mathbb{Z})$, where $\mathsf{M}_n$ denotes the matching complex on $n$ vertices. The 3-rank co…

Let $n\geq 1$ and write … Thus the relevant reduced homology group is $\tilde{H}_{k-1+r}(\mathsf{M}_{2k+1+3r};\mathbb{Z})$. Jonsson's 3-torsion conjecture. The group contains eleme…

Split exactness conjecture. This sequence, obtained by cutting the long exact $0$-$e$-$2$ sequence, is split exact for every $n\geq3$ and every $d$.

Let $n,d,k,r$ be related by … and let $\mathsf{M}_n$ denote the matching complex of the complete graph on $n$ vertices. 5-torsion conjecture. The group…

Let $n,d,k,r$ be related by … and write $\tilde{H}_d(\mathsf{M}_n;\mathbb{Z})$ for the reduced homology of the matching complex of the complete graph on $n$ vertices. 3-torsion con…

Rank conjecture. Based on calculations for $k\leq 5$, $r_k$ equals the coefficient of $x^k$ in $(1+x+x^2)^k$.