32 problems

Abelian rank conjecture. The group $G$ must be abelian of rank at most $2$.

Let $(\Gamma,\alpha)$ be a finite connected regular $(n+1)$-valent graph whose edges are colored by $(\mathbb Z_2)^{n+1}$, with $\alpha$ a “good” coloring. A moment graph is the gr…

Let $G$ be a finite group, and let $r(G)$ denote its rank, namely the maximum rank of an elementary abelian $p$-subgroup of $G$ over all primes $p$ dividing $|G|$. Minimal-sphere-p…

Let $G=(\mathbb Z/2)^r$ act freely on … and let $N=n_1+\dots+n_m$. Write $\pi_G:X/G\to BG$ for the classifying map of the action. Top-dimensional surjectivity conjecture. The induc…

Let $G$ be a finite group, and let $G$ act freely on … The rank $r(G)$ is the maximum of the $p$-ranks $r_p(G)$ over primes $p$ dividing $|G|$, where $r_p(G)$ is the largest $r$ su…

Let $M$ be a compact, simply connected manifold with an isometric action by a compact group $G$. The orbit space $M/G$ is assumed to have dimension two and to be covered by the hyp…

Let $M$ be a compact, simply connected manifold with a polar action by a compact group $G$, meaning an isometric action admitting a section $S$ with $\dim S=\dim M/G$ that meets al…

Let $p$ be a prime, let $G=({\Bbb Z}/p)^k$, and let $X$ be a finite free $G$-complex. Carlsson's toral rank conjecture. One has … This conjecture gives a lower bound on the total m…

Stable Carlsson conjecture. There exists a constant $C$ such that, for every prime $p>C$,

Toral rank and Carlsson conjectures.

Let $G$ be a finite subgroup of $\operatorname{O}(n)$. Say that $G$ has the amphichiral linear centralizer property (LCP) if its centralizer in $\operatorname{O}(n)$ contains an el…

Let $X$ be a connected manifold, let $\cl G$ be a closed subgroup of $\operatorname{Homeo}(X)$, and let $f\in\cl G$ be almost periodic, meaning that the subgroup…

Let ${\mathbb{Z}}_p$ denote the topological group of $p$-adic integers, and let $X$ be a connected $m$-manifold. Hilbert–Smith conjecture, $p$-adic formulation. The group…

Equivariant Mahowald-invariant conjecture. If

Let $T^k$ act isometrically and effectively on $M^n$, where $M^n$ is a closed, simply connected, non-negatively curved Riemannian manifold. Define … where $n_i\geq 2$,…

For a prime $p$, let $\widehat{\mathbb{Z}}_p$ denote the $p$-adic group. The p-adic Hilbert–Smith conjecture. For every prime $p$, there is no faithful action of…

Let $M$ be a $2n$-dimensional compact oriented manifold homotopy equivalent to the complex projective space $4\mathbb{CP}^n4$. Suppose that $M$ admits a non-trivial $S^1$-action. P…

Let $S_g$ be a closed orientable surface of genus $g\geq 3$, let $\Mod(S_g)$ denote its mapping class group, and let $\Homeo^+(\mathbb{S}^n)$ denote the group of orientation-preser…

Let $M$ and $N$ be smooth closed manifolds of dimensions $p$ and $q$, respectively. Write … A $(\mathbb{Z}/2)^2$-equivariant map is understood with respect to the involutions excha…

Let $M$ be a compact manifold with a rigid geometric structure $\omega$, and let $H$ be a sufficiently large transformation group. Gromov–D’Ambra conjecture. All triples…

Let $M$ be a closed orientable $4$-manifold with Euler characteristic $(M) \neq 0$, and let $G$ be a finite group acting effectively, locally linearly, and homologically trivially…

Let $M$ be a simply connected manifold that is not homeomorphic to $S^n$, and let $N$ be a manifold of dimension $n+k<2n$. The group $\operatorname{Homeo}_0(M)$ acts nontrivially o…

Weak Hilbert–Smith conjecture. $G$ is a Lie group.

Let a compact Lie group act smoothly and isometrically on a simply connected compact manifold of nonnegative curvature. The action is irreducible when its associated reflection gro…

Free-rank conjecture. One has