370 problems

Let $d$ be a nonnegative integer, and let $X$ be a nonempty finite free $(\mathbb{Z}/2)^d$-CW complex. Its homology is taken with coefficients in $\mathbb{F}_2$. Carlsson's conject…

Kropholler's conjecture. If there is a proper $H$-almost invariant set $A$ such that $A=AH$, then $G$ has a non-trivial action on a tree in which $H$ fixes a vertex $v$ and every e…

Fitting-height conjecture. If $A$ acts fixed-point-freely on $G$, then

Let $f\colon S^3\to S^3$ be a nontrivial orientation-preserving diffeomorphism of finite order, and let $\mathrm{Fix}(f)$ denote its fixed-point set. Smith conjecture. The set…

Webb's conjecture. The orbit space

Let $q\geq 5$ be a prime power, with the corresponding parity and residue value $\xi\in\{-1,0,1\}$ determined by $q\equiv\xi\pmod 3$. Let $G_q$ act on the EnΓ-lines in…

Let $G$ be a finite group acting on a contractible $2$-complex. Casacuberta–Dicks conjecture. The action has a fixed point. Serre's fixed-point theorem establishes the analogous st…

Brown's conjecture.

Kropholler–Mislin conjecture. The invariant $n_G$ is finite if and only if

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…

Positivity conjecture. The Novikov–Shubin invariants of $\bar X\to X$ are always positive; equivalently, the capacities are always finite.

Characteristic-invariant conjecture. The action $\alpha$ is classified up to cocycle conjugacy by the characteristic invariant $(H,\chi)$.

Let $q$ be a power of $3$, let $G=PGL_2(q)$ act on the lines of $PG(3,q)$, and let $C$ be the twisted cubic. A line is generic if it neither intersects $C$ nor lies in any osculati…

Homological finiteness conjecture. The homological version of the results in the cited work should hold; consequently, homological versions of the two stated theorems on flag-compl…

Let $G$ be a compact group acting continuously and transitively on a compact Hausdorff space $X$. A space of continuous functions on $X$ is called minimal if it is closed, invarian…

Let $n$ be a positive integer, let $H$ be a nonsolvable connected group of finite Morley rank, and suppose that $\operatorname{Alt}(n)$ acts definably and faithfully on $H$ by auto…

Let $G$ be a countable torsion-free amenable group, let $D$ be a Kirchberg algebra, and let $BG$ denote the classifying space of $G$. Consider outer actions of $G$ on $D\otimes K$,…

Let $\Gamma$ be a higher rank lattice acting continuously on the circle $S^1$. One-dimensional Zimmer rigidity conjecture. The action should factor through an action of a finite gr…

Let $G$ be a torsion-free group, and let ${\mathbb Z}(G)$ denote its integral group ring. Each element of ${\mathbb Z}(G)$ acts by left convolution on $\ell^{2}(G)$. Atiyah conject…

Let $(M,L)$ be a polarized manifold and let $\hat K$ be a compact subgroup of $\operatorname{Aut}(M)$. Suppose that $L$ is $\hat K$-linearized. Let $H_{\hat K}^0(M,L^m)$ be the max…

Let $p$ be a prime, let $n_1,\ldots,n_k$ be positive integers, and let $L_p^{2n_i-1}$ denote the corresponding lens spaces. Allday's conjecture. The free $p$-rank of their product…

Let $p$ be a prime, let $n_1,\ldots,n_k$ be positive integers, and let $(\mathbb{Z}_p)^r$ act freely on the product $S^{n_1}\times\cdots\times S^{n_k}$. Free-rank conjecture for pr…

A limit group is a finitely generated fully residually free group. Let $F^{\mathbb{Z}[t]}$ be Lyndon's free exponential group. Myasnikov–Remeslennikov embedding conjecture. Every l…

Let $G$ be a group acting freely on an $cmathbb{R}$-tree. Such a group is called finitely generated $cmathbb{R}$-free when it is finitely generated. Morgan–Shalen conjecture. Every…

Let $G$ be a torsion-free group that acts freely and properly discontinuously on some $S^n\times \mathbb{R}^k$. Talelli's conjecture. $G$ can act freely and properly discontinuousl…