Knot Concordance and Homology Cobordism
Cochran, Tim D.; Franklin, Bridget D.; Hedden, Matthew; Horn, Peter D.
We consider the question: “If the zero-framed surgeries on two oriented knots in S3 are 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 is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the τ and s-invariants of K and P(K) differ. Consequently neither τ nor 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 and its (p, 1)-cables.