Rational knot concordance and homology cobordism

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

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arXiv · 1011.4901

Rational knot concordance and homology cobordism

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

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Abstract

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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BibTeX RIS

Tim D. Cochran, Bridget D. Franklin, Peter D. Horn. 2010-11-22. Rational knot concordance and homology cobordism. https://arxiv.org/abs/1011.4901

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