Knot concordance and homology cobordism

Details Knot concordance and homology cobordism Knot concordance and homology cobordism

2011-02-28

arxiv.org/abs/1102.5730v1

Mathematics

Geometric Topology

15 pages, 8 figures

Scientific paper

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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