Knot concordance and homology cobordism (Q2838978)

If oriented knots \(K\) and \(K'\) in \(S^3\) are CAT concordant then the knot manifolds \(M(K)\) and \(M(K')\) obtained by longitudinal (0-framed) surgery on these knots are meridian-preserving CAT \(\mathbb{Z}\)-homology cobordant. In other words, there is an orientable 4-manifold \(W\) such that \(\partial{W}=M(K)\amalg{-M(K')}\) and the inclusions of the boundary components induce isomorphisms on homology, and carry the preferred meridians of the knots to the same generator of \(H_1(W;\mathbb{Z})\). The Ozsváth-Szabó-Rasmussen \(\tau\)-invariant is used to show that the converse fails in the smooth case CAT = DIFF. There are TOP slice knots \(K\) such that \(M(K)\) is smoothly \(\mathbb{Z}\)-homology cobordant to \(M(U)=S^2\times{S^1}\), although \(K\) is not smoothly slice. There are related notions of rational concordance and \(\mathbb{Q}\)-homology cobordism, and it is shown that meridian-preserving \(\mathbb{Q}\)-homology cobordism of knot manifolds does not imply rational concordance of the knots, even in the TOP case.