| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

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.

    1. Is homology slice the same as slice?


      Add remark

      Problem 2.1.

      [Tye Lidman] Is there a non-smoothly slice knot $K$ in $S^3$ which is slice in a homology $B^4$? (i.e., is $\phi_{smooth}$ non-injective?)
        1. Remark. The possible obstructions to sliceness of a homology-slice $K$ 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


              Add remark

              Problem 2.2.

              [Clayton McDonald] Fix an integer homology sphere $Y$ which bounds a contractible 4-manifold $W$. 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 4-manifold?

                • Torsion in the cokernel


                  Add remark

                  Problem 2.3.

                  [Tye Lidman] Does $coker(\phi_{smooth})$ have torsion?

                    • Concordance of knots in homology spheres.


                      Add remark

                      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?)
                        1. 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


                              Add remark

                              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 null-homotopic in $W$. Must $K$ be homotopic in $Y$ to a knot $K'$ which is smoothly slice in $W$?
                                1. 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 3-handles, 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.