4 problems

Let $g\ge 1$. A handlebody of genus $g$ is a compact 3-manifold homeomorphic to a regular neighborhood of a finite connected graph with first Betti number $g$, embedded in $S^4$. B…

Let ${\cal S}$ be an $m$-dimensional sheet smoothly and isometrically embedded in a $d$-dimensional Euclidean space, with $d<2m$, and let $p$ be any point of ${\cal S}$. A subsheet…

Smoothing conjecture. Every such $N$ is smoothable. More precisely, $N$ admits a transverse field of 2-planes and therefore there is an arbitrarily small topological isotopy carryi…

Properness conjecture. For every smooth embedding $\imath:T^p\hookrightarrow\mathbb{R}^{p+2}$, the subgroup ${\mathscr{E}}_{\mathtt{Top}}(\imath)$ is proper in…