23 problems

Let $\operatorname{Homeo}_{+}(\mathbb{S}^{n})$ denote the group of orientation-preserving homeomorphisms of the sphere $\mathbb{S}^{n}$. Simplicity conjecture. A simple application…

A homeomorphism $h:X\to X$ is stable if it can be written as a finite composition of homeomorphisms, each of which is the identity on some open set. A group of homeomorphisms is st…

Let $f\taut$ be a map, where $\taut$ denotes the boundary of the space of embeddings, and let $\tP(f)$ denote the corresponding pseudo-isotopy space. Local weak contractibility con…

Let $M$ be a finite-dimensional closed connected manifold, meaning a compact connected manifold without boundary. A subgroup is periodic if every one of its elements has finite ord…

Let $\Sigma$ be one of the four exceptional closed surfaces … A subgroup is periodic if every one of its elements has finite order. Burnside conjecture for exceptional surfaces. Fo…

Let $M$ be a closed orientable surface of positive genus, and let $H^2_{(1)}({\rm Homeo}_0(M),{\mathbf R})$ denote the $1$-bounded cohomology of the identity component of the homeo…

Rubin's constructibility conjecture. Assuming Gödel's axiom of constructibility $V=L$, if the homeomorphism groups of $M$ and $N$ are elementarily equivalent, then $M$ and $N$ are…

Let $P$ be the pseudoarc, and let $\operatorname{Homeo}(P)$ denote its autohomeomorphism group with the uniform topology. Kwiatkowska's conjecture. The group…

Non-singular mixed identities conjecture. There do not exist non-singular mixed identities for

Let $N\in \mathbb{N}$ and let $G\leq {\mathrm{Homeo}}(\mathbb{R})$ be a subgroup with at most $N$ fixed points. Let $I\subset \mathbb{R}$ be a maximal interval without global fixed…

Vanishing conjecture. The second bounded cohomology vanishes for the groups of piecewise linear homeomorphisms of flows:

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…

Hoehn's small-case conjecture. The statement $(\ast)_2$ is true.

Let $M$ be a compact manifold, and let $\operatorname{Homeo}_0(M)$ denote the identity component of the group of homeomorphisms of $M$. A group morphism is nontrivial if it is not…

Let $E$ be a closed subset of the Cantor set, and let $\operatorname{Homeo}(E)$ denote its homeomorphism group. A basic counterexample to recovering the homeomorphism type of $E$ f…

Let $F$ be Thompson's group, and let $H$ be a finitely generated subgroup of $F$. A finite generating set of a subgroup of $\operatorname{Homeo}_+(I)$ is geometrically fast if it h…

Let $D^2$ be the two-disc, with $[?]$ its area form, and let $Hameo(D^2,\partial D^2)$ be the subgroup of Hamiltonian homeomorphisms inside $Homeo^\Omega(D^2,\partial D^2)$. The Ca…

Let $\mu_0$ be Werner's measure on self-avoiding loops, let $\nu_0$ be its induced measure on welding homeomorphisms $\sigma$, and let $W$ be the welding map. A quasicircle is a Jo…

Let $\mathbb{S}^{1}$ be the circle, $\mathbb{S}^{2}$ the sphere, and $\mathbb{D}^{2}$ the closed disc. Consider non-trivial actions of the group…

Let $S$ be a closed surface, let $D$ be an open disc in $S$, and write $S-D$ for the surface obtained by removing $D$. Let $\mathrm{Homeo}_{0}(S-D)$ be the identity component of th…

Let $M$ be a compact manifold, and let $\mathrm{Homeo}_{0}(M)$ denote the identity component of its homeomorphism group. A group morphism is trivial if it maps every element to the…

Let $PSL(2,\mathbb{R})$ be the projective special linear group of $2\times 2$ real matrices. De La Harpe's conjecture. $PSL(2,\mathbb{R})$ is a maximal closed subgroup. The conject…

Small-homeomorphism compact-subgroup conjecture. There exists $\epsilon>0$ such that this subset of $\operatorname{Homeo}(M)$ contains no nontrivial compact subgroup. For any speci…