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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?


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      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


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              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


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                  Problem 2.3.

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

                    • Concordance of knots in homology spheres.


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                      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


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                              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.