264 problems

Church–Farb–Putman vanishing conjecture. If $i\leq n-2$, then

Eilenberg–Ganea conjecture. For every group $G$,

Church–Farb representation stability conjecture. The sequence induced in cohomology and homology by this stabilization sequence satisfies representation stability.

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…

Friedlander–Milnor reformulation. The Friedlander–Milnor conjecture for $\operatorname{SL}_2$ over $\overline{\mathbb{Q}}$ can be reformulated as

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…

Let $G$ be a torsion-free group. Say that $G$ has periodic cohomology after $k$ steps over $R$ if there is an integer $q\geq 0$ such that the functors $H^i_R(G,-)$ and…

Let $\Gamma$ be any lattice in $\mathop{\mathrm{SO}}\nolimits(1,n)$. Thurston's conjecture. There are a finite-index subgroup $\Gamma'$ of $\Gamma$ and a surjective homomorphism ……

Habiro–Katada conjecture. The isomorphism

Sylow-subgroup generation conjecture. The bounded derived category $\mathsf{D}^{b}(C^*BG)$ is generated by the object

Church–Farb invariant-cohomology conjecture. The $\operatorname{GL}(n,\mathbb{Z})$-invariant part of $H^*(\operatorname{IA}_n,\mathbb{Q})$ vanishes stably.

Let $q=p^r$, let $GL_n(\mathbb F_q)$ be the general linear group over the finite field with $q$ elements, and let $k$ be the coefficient field. Consider the cohomology groups…

Let $G$ be the relevant symmetry group, and consider symmetry protected topological phases in $u$-dimensional quantum spin systems. An SPT phase is a gapped, short-range-entangled…

Guralnick's conjecture. There exists an absolute constant $c$ such that

Lee–Szczarba conjecture. For a prime $p$ and $n\geq 2$, the induced map

Let $H$ be a finite group and let $V$ be an absolutely irreducible faithful representation of $H$. The first cohomology group is denoted by $\operatorname{H^1}(H,V)$. Guralnick's c…

Friedlander's generalized isomorphism conjecture. This map should be an isomorphism. The conjecture is used in the source to relate étale cohomology to the cohomology of discrete g…

Let $G$ be a finite simple group, and let $B_0(G)\subset H^2(G,\mathbb{Q}/\mathbb{Z})$ be the subgroup consisting of cohomology classes that restrict trivially to every abelian sub…

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

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 $R={\mathbb Z}[t]$ or $R={\mathbb Z}[t,t^{-1}]$, and let $SL_n(R)$ be the corresponding special linear group. Cohomology vanishing conjecture. If $n\ge 3$, then … The paper giv…

Let $\Gamma$ be a group, let $\lambda$ be a parameter, and let $m$ be an integer, with $W_{\lambda,m}$ the coefficient space used in the paper's cohomology construction. Finite-dim…

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…