10 problems

Let $cmathcal{M}_P(K)$ denote the pure complex of discrete Morse matchings of a simplicial complex $K$. Let $cmathrm{cDelta}^n$ be the $n$-simplex and…

For fixed $m>0$, let $\mathcal{G}_m^n$, $\Delta_m^{n,r}$, and $\Gamma_n^{\alpha,r}$ be the complexes defined in the paper. Let $V(\Delta)$ denote the vertex set of an induced subco…

Greedy matching conjecture. For any tangle diagram $T$, the greedy matching $M_{\operatorname{gr}}$ is a Morse matching on the delooped Khovanov complex…

Greedy matching conjecture. The greedy matching is a Morse matching for all torus braids, that is, the graphs

Orientable pseudomanifold characterization. $K$ is an orientable pseudomanifold if and only if

Živaljević's conjecture. Every simply connected smooth $d$-manifold admits some LC triangulations.

Let $K$ be a knot with bridge index $b$. Say that a $d$-complex has discrete Morse vector $(c_0,\ldots,c_d)$ if it admits a discrete Morse function with exactly $c_i$ critical face…

Heegaard–discrete Morse conjecture. Some closed $3$-manifold with Heegaard genus $g$ admits discrete Morse functions with less than $g$ critical edges.

Mazur-manifold LC triangulation conjecture. Mazur's $4$-manifold admits an LC triangulation.

Kirby's conjecture. Every simply connected closed smooth $4$-manifold is geometrically $1$-connected.