11 problems

Let $d\geqslant 3$ and $1\leqslant\ell\leqslant d-1$. Let $P$ be a $d$-polytope with $2d+\ell$ vertices, and let $T(r)$ denote the $r$-simplex. Pineda-Villavicencio's conjecture. 1…

The centrally symmetric lower bound conjecture. If $\Delta$ is a simplicial sphere, a connected simplicial manifold, or a normal pseudomanifold, then

Klee–Nevo–Novik–Zheng's conjecture. There exists a unique polytopal complex ${\mathcal C}$ in ${\mathbb R}^d$ such that: (i) one face of ${\mathcal C}$ is the cross-polytope…

Let $\Delta$ be a balanced connected ${\bf k}$-homology manifold that is orientable over ${\bf k}$ of dimension $d-1 \geq 3$. Let $\mathcal{BH}^d$ denote the class of balanced conn…

Let $\Delta$ be a balanced connected simplicial complex of dimension $d-1 \geq 3$. If $\Delta$ is a ${\bf k}$-homology manifold that is orientable over ${\bf k}$, then the balanced…

Let $\Delta$ be a connected triangulated $(d-1)$-manifold without boundary, and let $h_r(\Delta)$ denote its $h$-numbers and $\beta_{j-1}(\Delta)$ its Betti numbers. Bagchi–Datta's…

Let $M$ be an $m$-vertex connected triangulated closed $d$-manifold with Betti numbers $\beta_i$, and let ${\cal K}_l^{\ast}(d)$ denote the class of $(l+1)$-neighbourly members of…

Generalised lower bound conjecture for triangulated manifolds. The displayed inequality holds, and equality for some $l<\frac{d-1}{2}$ holds if and only if $M\in {\cal K}_l(d)$. Th…

Let $M$ be an $m$-vertex connected closed triangulated $d$-manifold, and let $\beta_i$ be its Betti numbers over some field. Let ${\cal K}_l^{\ast}(d)$ denote the class used in the…

Let $M$ be a triangulation of a connected closed $d$-manifold. Write $g_j(M)$ for its $g$-numbers, $\beta_i(M;\mathbb F)$ for Betti numbers over a field $\mathbb F$, and let…

Let $d$ and $k$ be integers with $d\geq 2k+1$, and let $X$ be a triangulated $d$-sphere with face-vector $(f_0,\dots,f_d)$. Generalized lower bound conjecture. For every $j$, the f…