# Knot concordance and homology cobordism

We consider the question: &quot;If the zero-framed surgeries on two oriented knots in $$S^3$$ are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?&quot; 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 $$\mathbb{Z}$$-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 $$\tau$$- and $$s$$-invariants of $$K$$ and $$P(K)$$ differ. Consequently neither $$\tau$$ 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.