Knot concordance and homology cobordism

Tim D. Cochran; Bridget D. Franklin; Matthew Hedden; Peter D. Horn

doi:10.1090/s0002-9939-2013-11471-1

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arXiv · 1102.5730

Knot concordance and homology cobordism

Tim D. Cochran·Bridget D. Franklin·Matthew Hedden·Peter D. Horn

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Abstract

We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the tau and s-invariants of K and P(K) differ. Consequently neither tau nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show, that a natural rational version of this question has a negative answer in both the topological and smooth categories, by proving similar results for K and its (p,1)-cables.

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BibTeX RIS

Tim D. Cochran, Bridget D. Franklin, Matthew Hedden, Peter D. Horn. 2011-02-28. Knot concordance and homology cobordism. https://doi.org/10.1090/s0002-9939-2013-11471-1

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