12 problems

Local model conjecture. The pair $(\mathcal H(M),\mathcal H_c(M))$ is locally homeomorphic to $(\square^\omega l_2,\boxdot^\omega l_2)$ at the identity $\mathrm{id}_M$ of $M$. In p…

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 $N\in \mathbb{N}$ 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…

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…

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…

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…