67 problems

Hilbert–Smith conjecture. Every faithful action of a locally compact group on a connected manifold has a Lie group as its acting group.

Let $M$ be a manifold homotopy equivalent to $\mathbb{C}P^n$ and admitting a nontrivial circle action. Petrie's conjecture. Any homotopy equivalence between $M$ and $\mathbb{C}P^n$…

Ghys's Jordan conjecture. The diffeomorphism group $\operatorname{Diff}(X)$ is Jordan for every compact smooth manifold $X$.

Lipsman's conjecture. The CI condition is equivalent to properness of the pair $(L,H)$.

Palais–Terng conjecture. If $\mathcal H$ is integrable, then its integral manifolds can be extended to sections.

Laitinen's conjecture. If $G$ is an Oliver group with $\operatorname{prim}(G)\geq 2$, then there exist non-isomorphic $\mathbb{R}G$-modules $U$ and $V$ which are Smith equivalent,…

Morimoto's conjecture. The group $G$ is isomorphic to one of

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 $n_i\geq 2$,…

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

Let $M$ be a connected $n$-manifold. A homeomorphism $f$ of $M$ is almost periodic when the subgroup of $\operatorname{Homeo}(M)$ that it generates has compact closure. Almost peri…

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

Smith conjecture. The isotropy representation of $G$ at $p$ is isomorphic, as a representation over $\mathbb{R}$, to the isotropy representation of $G$ at $q$.

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…

A compact Lie group acts smoothly on the 3-sphere $S^3$. Linearity conjecture. Every such action is conjugate to a linear action. This asserts that, despite the existence of non-li…

Let $G$ be a compact Lie group acting on Euclidean $n$-space or on the closed $n$-disk. Conner's conjecture. The orbit space is contractible. Conner's conjecture was proved by Oliv…

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…

Conjecture. Either $\frak G$ is a Moufang quadrangle, or $\frak G$ is of Clifford type with topological parameters $(3,4k)$ or $(7,8)$.

Löffler–Raußen conjecture. Non-trivial $\mathbb{Z}_p$-actions should exist on any simply-connected manifold.

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…

Simplicity conjecture. This group is not a simple group.

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

Toral rank and Carlsson conjectures.