19 problems

Grünbaum's conjecture. The function $\theta_k(d+s,d)$ gives the minimum number of $k$-faces of a $d$-polytope with $d+s$ vertices.

Let $\Delta$ be a simplicial $(d-1)$-sphere. The $g$-numbers are certain alternating weighted sums of the $f$-numbers. McMullen–Walkup conjecture. For any $k\leq d/2$, one has … Th…

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…

Let $Q$ be a cubical $d$-polytope with $n$ vertices, and write $f_k(Q)$ for its number of $k$-dimensional faces. Jockusch's cubical lower bound conjecture. For $1\leq k\leq d-2$, ……

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

Let $d$-polytopes have at least $d+3$ facets and between $3d$ and $4d-4$ vertices. For $d=4$, the condition $4d-4=3d$ specifies the endpoint with $12$ vertices. Four-dimensional fa…

Let $d\geqslant 5$, and consider $d$-polytopes with at least $d+3$ facets and between $3d$ and $4d-4$ vertices. A simple edge is an edge whose endpoints are both simple vertices. A…

Let $d\geqslant 2$ and $1\leqslant\ell\leqslant d-1$. Let $J(\ell+1,d)$ be the $d$-polytope obtained by truncating a simple vertex from the $(\ell+1,d-\ell-1)$-triplex, and let the…

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 $M$ 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 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 ${\cal K}_k(d)$ be the class of triangulated closed $d$-manifolds whose vertex links belong to the corresponding generalised Walkup class. For a member $M\in {\cal K}_k(d)$ wit…

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 $d\geq 2k+1$, let $S$ be a triangulated $d$-sphere with face numbers $(f_0,\dots,f_d)$, and let $f_j$ denote its number of $j$-faces. Generalized Lower Bound Conjecture. For ev…