5 problems

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…

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…

Sparla's conjecture. The inequality