Rational knot concordance and homology cobordism

Details Rational knot concordance and homology cobordism Rational knot concordance and homology cobordism

2010-11-22

arxiv.org/abs/1011.4901v2

Mathematics

Geometric Topology

9 pages, 6 figures; Reference added and 2 errors corrected in introduction

Scientific paper

The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvious rational version of this question has a negative answer. Namely, we give examples of knots whose zero-framed surgeries are rational homology cobordant 3-manifolds, wherein the knots are not rationally concordant (that is not concordant in any rational homology S^3 x [0,1]). Specifically, we prove that, for any positive integer p and any knot K, the zero framed surgery on K is Z[1/p]-homology cobordant to the zero framed surgery on its (p,1) cable. Then we observe that most knots are not rationally concordant to their (p,1) cables.

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