2. Knots in homology spheres
For $*$ either smooth or topological, let $\hat{\mathcal{C}}_*$ denote the group of knots in integer homology spheres modulo $*$concordance in homology cobordisms. There is a natural map $\phi_* \colon \mathcal{C}_* \to \hat{\mathcal{C}}_*$. Many of the problems in this section center on understanding this map.
Is homology slice the same as slice?
Problem 2.1.
[Tye Lidman] Is there a nonsmoothly slice knot $K$ in $S^3$ which is slice in a homology $B^4$? (i.e., is $\phi_{smooth}$ noninjective?)
Remark. The possible obstructions to sliceness of a homologyslice $K$ are Rasmussen’s $s$invariant, the $s$type invariant coming from singular instanton Floer homology, the other $s$style invariants coming from Khovanov homotopy type.


Homology slice vs. contractibly slice
Problem 2.2.
[Clayton McDonald] Fix an integer homology sphere $Y$ which bounds a contractible 4manifold $W$. Is there a knot $K$ in $Y$ such that $K$ bounds a disc in some integer homology ball $W'$, but $K$ does not bound a disc in any contractible 4manifold? 
Torsion in the cokernel
Problem 2.3.
[Tye Lidman] Does $coker(\phi_{smooth})$ have torsion? 
Concordance of knots in homology spheres.
Problem 2.4.
[Christopher Davis] Is every knot in every integer homology sphere topologically concordant in some integer homology cobordism to a knot in $S^3$? (i.e. is $\phi_{top}$ surjective?)
Remark. In the smooth category, the answer is ‘No’ due to work of Levine [MR3589337] The answer is ’Yes’ modulo every term of the $n$solvable filtration, due to work of Davis [arXiv:1803.01086].


Homotopy to slice knots
Problem 2.5.
[Christopher Davis] Let $K$ be a knot in an integer homology sphere $Y$, and suppose that $Y$ bounds an integer homology ball $W$ such that $K$ is nullhomotopic in $W$. Must $K$ be homotopic in $Y$ to a knot $K'$ which is smoothly slice in $W$?
Remark. [Patrick Orson] It is known that $K$ is homotopic to a knot $J$ with Alexander polynomial 1, which by Freedman is topologically slice in some integer homology ball $V$. (In fact, $V$ is contractible.)

Remark. [Christopher Davis] If $W$ has a handle description without 3handles, then the answer to the question is ’Yes’.

Cite this as: AimPL: Smooth concordance classes of topologically slice knots, available at http://aimpl.org/concordsliceknot.