137 problems

For every finite group $G$ and every field $K$, if $\operatorname{Ext}^1_{KG}(K,K)\neq 0$, where $K$ is regarded as the trivial $KG$-module, then…

Church–Farb–Putman conjecture. The codimension-$i$ cohomology of $\operatorname{SL}_n(\mathbb Z)$ vanishes for $n\geq i+2$; equivalently,

Let $S_g^n$ be a compact oriented surface of genus $g$ with $n$ boundary components, and let $\mathrm{Mod}(S_g^n)$ be its mapping class group. Let…

Let $G$ be an infinite Kähler group, meaning the fundamental group of a smooth complex Kähler manifold. Carlson–Toledo conjecture. There exists a finite-index subgroup $G^\circ$ of…

Let $\ell$ be a prime number. Let $K$ be a number field with $\zeta_\ell\in K$, and let $S$ be a finite set of places containing the infinite places and the places over $\ell$. The…

Benson's strong regularity conjecture. The cohomology ring $H^*(G,k)$ always contains a very strongly quasi-regular system of parameters.

Katada's conjecture. For $n\gg *$,

Let $G$ be a finite $p$-group. Define $\omega_a(G)$ as the minimum of $\dim \operatorname{H}^{*}(G;\mathbb{F}_p)/\mathfrak{p}$ over associated primes $\mathfrak{p}$ of…

Let $\operatorname{SL}_n\mathbb{Z}$ be the special linear group over the integers, and let $H^*(-;\mathbb{Q})$ denote rational group cohomology. Vanishing conjecture.…

Schmid's cohomological conjecture. For every integer $n$, one has

Let $\ul\la$ be a bipartition and let $n\ge l(\ul\la)$ be an integer. For $i=1,\dots,n$, define … where $\mu_i$ is given in the source, and let…

Let ${\cal K}_g$ be the Johnson kernel, let $d_i\in H^{4i-3}({\cal K}_g;{\mathbb Q})$ be the higher secondary classes, and let ${\cal M}_g$ act on $H^*({\cal K}_g;{\mathbb Q})$. Se…

Throughout, let $H^i(S)=H^i(S;\mathbb{Z})$ denote the integral cohomology of a finite group $S$. For a finite $p$-group $P$, let $e(P)$ be the exponent of…

Let $F$ be a field with $|\cdot F/\cdot F^2|=2^n$, so that $E=E_n\cong (\mathbb Z/2)^n$, and let ${\mathcal G}_F$ be the associated $W$-group. Let…

Let $P$ be a pro-$p$ group, and write … Let $\varphi^2:H_{\operatorname{cont}}^{2}(P)\to H_{\operatorname{disc}}^{2}(P)$ be the comparison map. Prasad's conjecture. For every $p$-a…

Let $\Gamma\subset \operatorname{Mob}(\mathbb S^n)$ be a lattice. Virtual first Betti number conjecture. There exists a finite-index subgroup $\Gamma'\subset\Gamma$ such that … Equ…

Strict inequality conjecture. If $G$ is not $p$-central, then

Let $G$ be a finite $p$-group, let $k$ be a positive integer, and let $d$ and $a_1,\dots,a_d$ be integers. A regular sequence has degree sequence $a_1,\dots,a_d$ when its elements…

Let $G$ be a finite $p$-group. Its cohomology ring $H^*(G,\mathbb F_p)$ is a graded-commutative, noetherian, local $\mathbb F_p$-algebra. Write $\operatorname{rank}_p Z(G)$ for the…

For each $n$, let $QA_n$ be the quadratic algebra generated by the cohomology classes associated with the generators of the quasitriangular group, and let…

Let $\mathcal H_{g,1}$ and $\overline{\mathcal H}_{g,1}$ be the homology-cobordism groups in the source, let $\sigma$ be the homomorphism to the inverse limit of automorphism group…

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…

Let $G$ be a group, let $H$ be a subgroup of finite index, and let $R$ be an arbitrary ring. Assume that for every nontrivial element $x$ of $G$, at least one of the following cond…

Let $G$ be an algebraic group over an algebraically closed field $k$, and let $n$ be a positive integer such that the characteristic of $k$ does not divide $n$. The simplicial sche…

Let $G$ be an infinite, finitely generated group. Let $M$ be a Banach space with norm $||\ ||_M$ that is also a left $\C G$-module, and suppose that $G$ acts on $M$ by continuous…