58 problems

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

Let $(M,g)$ be a closed orientable Riemannian three-manifold, with $g$ a metric of constant sectional curvature $\pm1$. Let $\operatorname{Isom}(M)$ be the isometry group of $M$ an…

Four-dimensional Smale conjecture. The group $\mathrm{Diff}_{\partial}(D^4)$ is contractible.

Recall that $\overline{\mathrm{B}\Gamma}^r_k$ classifies codimension $k$ $C^r$-Haefliger structures with trivial normal bundle. Haefliger–Thurston's conjecture. The space…

Let $M$ be a compact manifold of dimension $2$, and let the $L^2$-metric be the right-invariant metric on the group of volume-preserving diffeomorphisms induced by the $L^2$-length…

Almost action conjecture. If $\sigma:G\to H$ is a map such that

General distortion conjecture. The element $h$ is a distortion element in $\operatorname{Diff}^1(M)$ whose distortion function can be chosen to grow faster than any given function…

The Schatten-class embedding conjecture. Every countable subgroup of $Diff(M)$ admits a coarse uniform embedding into $C_p$ for some $p>1$.

Let $\Sigma$ be a spherical space form, and let $\operatorname{Isom}(\Sigma)$ and $\operatorname{Diff}(\Sigma)$ denote its isometry and diffeomorphism groups. Hatcher's conjecture.…

Let $G$ be the complexification of a simply connected compact Lie group, and let $L_{hyp}G$ be the hyperfunction completion of its loop group. Let $\mathcal D$ be the group of anal…

Let a swept surface carry sweeping functors arising from two-dimensional parallel transport, and consider their smooth limit. Let the group of diffeomorphisms of the swept surface…

Let $\hat {\Cal D}_{an}$ be the universal central extension of the group of analytic, orientation-preserving diffeomorphisms of $S^1$, acting on a completion of $\Cal D_{an}$, and…

Let $Z$ be a smooth connected closed $n$-dimensional $K(\Gamma,1)$-manifold, and let $\operatorname{Diff}(Z)$ be its group of diffeomorphisms with the natural smooth topology. Farr…

Let $X$ be a smooth closed manifold. Write $\mathrm{Diff}(X)$ for its diffeomorphism group, $B\mathrm{Diff}(X)$ for its classifying space, and $\pi_k$ and $H_k(-;\mathbb{Z})$ for h…

Let $\mathcal{X}$ be an unbounded metric space. Call $\mathcal{X}$ large if its asymptotic space in some (or every) nonstandard model $M$ contains a subspace isometric to…

Let $M$ be a manifold with a distribution $\xi$, and let $\Xi$ be the distribution on the group of contactomorphisms whose fiber at the identity is … A distribution is bracket gene…

Michor–Mumford conjecture. For every $k\geq 0$, the map

The paper considers controllability results for diffeomorphism groups of simple polytopes, including generation of the identity component by exponentials of suitable vector fields…

Let $n$ be a positive integer, let $\Gamma$ be a finite group, and let $\rho,\rho':\Gamma\to\operatorname{O}(n)$ be homomorphisms. Two such homomorphisms are $H$-equivalent if they…

Let $M$ be an $n$-manifold, and let $\mathrm{BDiff}_c^{r}(M)^{\delta}$ denote the classifying space of the discrete group of compactly supported $C^r$ diffeomorphisms, while…

Let $M$ be a connected smooth manifold with boundary of dimension $m\neq 3$. For each subset $\mathcal{X}\subseteq\pi_{0}(\partial M)$, let $\mathfrak{t}_{co}^{\mathcal{X}}$ be the…

Let $n\geq 3$ be odd, let $W_{g,1}$ denote the corresponding high-dimensional manifold, and consider the stabilization maps on rational homology … An optimal stability bound should…

Let $f(x,y)=-\cos(mx)\cos(ny)$, where $m$ and $n$ are positive integers with $m\geq n$. A flow has a conjugate point if a conjugate point occurs along its geodesic. Conjugate-point…

Chern–Simons nonlifting conjecture. Chern–Simons classes lie in the cokernel of this map.

Haefliger–Thurston conjecture. The map $\eta$ is a homology isomorphism in degrees at most $\operatorname{dim}(M)$ and is surjective on homology in degree $\operatorname{dim}(M)+1$…