Unsolved Problems

Problem Let $M$ be a $3$-dimensional smooth submanifold of $S^4$, $M$ diffeomorphic to $S^3$. By the Jordan-Brouwer separation theorem, $M$ separates ...

Conjecture If a $4$-manifold has the homotopy type of the $4$-sphere $S^4$, is it diffeomorphic to $S^4$?...

Conjecture Given a knot in $S^3$ which is slice, is it a ribbon knot?...

Problem Is there a complete and computable set of invariants that can determine which (rational) homology $3$-spheres bound (rational) homology $4$-ba...

Problem $Diff(S^4)$ has the homotopy-type of a product space $Diff(S^4) \simeq \mathbb O_5 \times Diff(D^4)$ where $Diff(D^4)$ is the group of diffeom...

Conjecture Let $f\in Diff^{r}(M)$ and $p\in\omega_{f}$. Then for any neighborhood $V_{f}\subset Diff^{r}(M)$ there is $g\in V_{f}$ such that $p$ is pe...

Conjecture Let $Diff^{r}(M)$ be the space of $C^{r}$ Diffeomorphisms on the connected, compact and boundaryles manifold M and $\chi^{r}(M)$ the space ...