18 problems

Let $M$ be a manifold, let $nl^{0}(M)$ denote the space of null Lagrangians of order zero on $M$, and let $J_{c}(M)$ denote the relevant compactly supported jet space. A classical…

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…

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 $n$ 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…

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 $(M,g)$ 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…

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

The 3-torus diffeomorphism conjecture. There are identifications among continuous groups:

Let $f\colon S^4\to S^4$ be an orientation-preserving self-diffeomorphism. Smale conjecture, special case. Every orientation-preserving self-diffeomorphism of $S^4$ is smoothly iso…

Let $M$ and $N$ be compact connected $C^p$ manifolds, and let $\Diff^p_c(M)_0$ and $\Diff^p_c(N)_0$ denote their groups of compactly supported $C^p$ diffeomorphisms diffeotopic to…

Let $p':= p-b$ and $q':= q-a$. For $\xi_1\in\pi_pSO_{q'}$ and $\xi_2\in\pi_qSO_{p'}$, let $h\colon\pi_iSO_j\to\pi_i(S^{j-1})$ be induced by $SO_j\to S^{j-1}$, let…

Let $\operatorname{Emb}(S^1,S^1\times S^3;S^1_0)$ be the based embedding space, and let $\operatorname{Diff}(S^1\times B^3\operatorname{fix}\partial)$ and…

For a group of diffeomorphisms, define the distance threshold by … the smoothness threshold by … and the Fredholm threshold by … Here $d_s$ is the $H^s$ geodesic distance,…

Let $M$ be a connected $C^\infty$ manifold without boundary, let $r\geq 1$, and denote by ${\rm Diff}^r_c(M)_0$ the identity component of the group of compactly supported $C^r$ dif…

Let $M$ be the closed manifold under consideration, let $Z^1(M)$ denote the space of closed $1$-forms on $M$, and let $\mu$ be an endomorphism of the relevant space of $1$-forms. C…

Let $M$ be a compact manifold and let $\Gamma$ be a finitely generated group of smooth diffeomorphisms of $M$. Ghys's alternative conjecture. Either $\Gamma$ contains a free group…

Let $M=S^2$ be a two-dimensional sphere, let $\mathcal{D}_\mu$ be the group of area-preserving diffeomorphisms, and identify $T^*\mathcal{D}_\mu$ with…