Rational knot concordance and homology cobordism
Tim D. Cochran, Bridget D. Franklin, Peter D. Horn

TL;DR
This paper demonstrates that zero-framed surgeries on knots can be rational homology cobordant without the knots being rationally concordant, providing counterexamples to a long-standing open question.
Contribution
It introduces counterexamples showing that rational homology cobordism of surgeries does not imply knot concordance, advancing understanding of knot concordance and homology cobordism.
Findings
Zero-framed surgeries on knots can be rational homology cobordant without the knots being rationally concordant.
For any positive integer p, the zero framed surgery on a knot is Z[1/p]-homology cobordant to that on its (p,1) cable.
Most knots are not rationally concordant to their (p,1) cables.
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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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Botulinum Toxin and Related Neurological Disorders
