10 problems

Lutz's conjecture. The only manifolds with $\beta_1(M,\mathbb{F})=3$ admitting a $13$-vertex triangulation are $(S^2\times S^1)^{\# 3}$ and $(S^2\uplus S^1)^{\# 3}$.

Sparla's conjecture. The inequality

Let M=(S^3\text{timeshspace{-1.62ex}hspace{-.4ex}hspace{.7ex}}S^1)\#({\mathbb C}{\bf P}^{\,2})^{\# 5}, and let $\text{}^4\hspace{.3pt}15^{\,4}_{\,1}$ denote the specified combi…

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

A $d$-sphere is understood in the graph-theoretic sense used in the paper. Its chromatic number is denoted by $\chi$. Chromatic-number conjecture. Every $d$-sphere has chromatic nu…

Let $M$ be a triangulated manifold, and suppose that $M$ is $\mathbb{F}$-tight, meaning tight over a field $\mathbb{F}$. Say that a triangulation is strongly minimal if it has the…

Let $X$ be a simplicial complex. Call $X$ $\mathbb{F}$-tight when it has the tightness property over a field $\mathbb{F}$, and call it strongly minimal when it achieves the compone…

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…