18 problems

Let $\Delta$ be an orientable $(d-1)$-dimensional $\mathbf{k}$-homology manifold without boundary. Define the modified $h$-numbers $h_i”$ as in the source by … where…

Let $\Delta$ be a $(d-1)$-dimensional orientable homology manifold over $k$, let $k[\Delta]$ be its face ring, and let $\Theta\subseteq k[\Delta]$ be a linear system of parameters.…

Extremal homology-manifold conjecture. 1. Suppose that $K$ is a $15$-vertex $Z$-homology $8$-manifold such that

Combinatorial-manifold conjecture. Any $(3d/2+3)$-vertex $\mathbb{Z}$-homology $d$-manifold is a combinatorial manifold.

Let $\Delta$ be a connected $3$-dimensional balanced $\mathbb{F}$-homology manifold. The Klee–Novik conjecture. Equality … holds if and only if $\Delta$ belongs to the balanced Wal…

Let $G$ be a finitely presented Poincaré duality group. ENR homology-manifold realization conjecture. $G$ is the fundamental group of a closed aspherical $\operatorname{ENR}$ homol…

Let $\Delta$ be an orientable $\mathbbm{k}$-homology manifold of dimension $d-1$ admitting a free action by the cyclic group $\mathbb{Z}/p\mathbb{Z}$. For the finer…

Let $(\Delta,\nu)$ be a PL realization of an orientable simplicial $\mathbb{R}$-homology $d$-manifold without boundary in $\mathbb{R}^d$, and let $\Psi(\Delta,\nu)$ be its stress a…

Let $\Delta$ be an orientable simplicial $\mathbb{R}$-homology $d$-manifold without boundary, with $H_1(\Delta;\mathbb{Z}/2\mathbb{Z})=0$ whenever $d\geq2$. Let $\Upsilon$ be a sub…

Let $\Delta$ be an orientable simplicial $\Bbbk$-homology $d$-manifold without boundary, and let…

Let $\Delta$ be a homology manifold of dimension $d\geq 4$ with boundary, and let $g_2(\Delta,\partial\Delta)$ and $\tilde b_i(\Delta,\partial\Delta)$ denote its relative second…

Let $\mathbb{F}$ be a field, and let $M$ be an $\mathbb{F}$-orientable $k$-stacked and $(k+1)$-neighbourly $\mathbb{F}$-homology manifold without boundary of dimension $2k+1$. Conj…

Let $M$ be a connected, closed, triangulated homology manifold of dimension $d$. For $1 \leq \ell \leq \frac{d-1}{2}$ and any field $\mathbb{F}$, let $g_{\ell+1}(M)$ denote the cor…

Let $\Delta$ be a connected orientable homology $(d-1)$-manifold without boundary whose vertex links have the weak Lefschetz property. Let $\Theta$ and $w$ be as in Theorem 5.4, se…

Socle characterization conjecture. The equality

Let $\Delta$ be a homology manifold, let $h'_i$ denote its h-prime vector, and let $\beta_{i-1}$ denote its $(i-1)$st Betti number. Manifold g-conjecture. For every…

Let $\Delta$ be a homology manifold, and let $L^i_s(\Delta)$ denote the set of pairs consisting of a linear system of parameters and a one-form satisfying the partial Lefschetz con…

Let $Z$ be a rational homology manifold. The class ${T_1}_*(Z)$ is the specialization at $y=1$ of the Hirzebruch class transformation $T_{y*}$, while the total Goresky–MacPherson h…