Knot concordance and homology cobordism

Abstract

We consider the question: “If the zero-framed surgeries on two oriented knots in S 3 S^3 are Z \mathbb {Z} -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 K is Z \mathbb {Z} -homology cobordant to the zero-framed surgery on many of its winding number one satellites P ( K ) P(K) . Then we prove that in many cases the τ \tau and s s -invariants of K K and P ( K ) P(K) differ. Consequently neither τ \tau nor s 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 K and its ( p , 1 ) (p,1) -cables.