18 problems

Let $R$ be a commutative ring with unit, and let $G$ be a discrete group without $R$-torsion. The group $G$ has jump cohomology of height $k$ over $R$ if there is an integer…

Resolution conjecture. There exist a compactum $Z$ with $\dim Z\leq n$ and a surjective map $r:Z\longrightarrow X$ such that

Moore's conjecture. Assume that for every nontrivial element $x$ in $\Gamma$, at least one of the following conditions holds: $M1)$ $\langle x\rangle\cap H\neq\\{e\\}$; or $M2)$…

Lyubeznik's conjecture.

Bhatt–Lurie’s cohomological dimension conjecture. The functor

Kato–Kuzumaki's conjecture. A field $k$ satisfies $C_i^q$ if and only if

Let $\phi:\Gamma\to\pi$ be a homomorphism of discrete groups. The LS-category $\operatorname{cat}\phi$ is the least integer $k$ such that a classifying-space map $B\Gamma\to B\pi$…

Let $R$ be a Noetherian domain, let $S$ be a solid $R$-algebra, and let $I$ be an ideal of $R$. Solid-algebra persistence conjecture. One has … The conjecture asks whether cohomolo…

Restricted cohomological-dimension-one structure conjecture. Any Lie $p$-algebra $L$ with $\operatorname{cd}_*(L)=1$ is of the form

Let $\mathcal A_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$, and let $\operatorname{cd}(\mathcal A_g)$ be its cohomological dimension. Co…

Vanishing conjecture. For every free $\mathbb QG$-module $F$,

Cohomological-dimension conjecture. The cohomological dimension $\operatorname{cd}_{k}(G)$ is finite if and only if

Let $G$ be a group and let $\frakF$ be a semi-full family of subgroups of $G$. Write $\cd_{\frakF}G$ for the Bredon cohomological dimension and $\gd_{\frakF}G$ for the Bredon geome…

Let $G$) be a group without $R$-torsion, where $R$ is a commutative ring with a unit, and let $k$ be a nonnegative integer. Jump cohomology conjecture. $G$ has jump cohomology of h…

Period-index conjecture.

Let $\Gamma$ be a torsion-free group acting smoothly and properly discontinuously on $S^n\times\mathbb{R}^k$. Talelli's reformulated conjecture. If $\Gamma$ is a torsion-free group…

Let $\Gamma$ be a group. Its cohomological dimension is … The group $\Gamma$ has periodic cohomology after $k$ steps if there exists an integer $q>0$ such that $H^i(\Gamma,-)$ and…

Let $S_g$ be a closed oriented surface of genus $g$, let $\Mod(S_g)$ be its mapping class group, and let $\I^k(S_g)$ be the subgroup consisting of elements that act trivially on a…