10 problems

Let $\mathcal H_{g,1}$ and $\overline{\mathcal H}_{g,1}$ be the homology-cobordism groups in the source, let $\sigma$ be the homomorphism to the inverse limit of automorphism group…

Let $\Sigma$ and $\Sigma'$ be homology spheres. A simply connected homology cobordism between them is a homology cobordism whose fundamental group is trivial. Simply connected homo…

For an integer $n$, the triangulation conjecture in dimension $n$ is the assertion that every closed topological $n$-manifold is homeomorphic to a simplicial complex. Triangulation…

Let $K\subseteq Y$ be a link in a homology sphere. A homology cobordism $W$ from $Y$ to $S^3$ is a cobordism inducing homology isomorphisms from its boundary components. Two links…

Kim–Livingston conjecture. This homomorphism is an isomorphism.

Levine's conjecture. Every link with vanishing $cbar\mu$-invariants is an $chF$-link in the sense of Levine.

Saveliev's conjecture. The quantity $\nu(Y)$ is an invariant of homology cobordism.

Neumann's conjecture. The invariant $\bar\mu$ is a homology cobordism invariant.

Let $\mathcal{H}_{g,1}$ be the homology cobordism group of homology cylinders over the surface $\Sigma_{g,1}$. Perfectness conjecture. The group $\mathcal{H}_{g,1}$ is perfect. Thi…

The $bar\mu$-invariant is an integral lift of the Rohlin invariant for plumbed homology $3$-spheres. Let $Y=\Sigma(a_1,\ldots,a_n)$ be a Seifert fibered homology sphere, and let…