Rational knot concordance and homology cobordism

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.