Distinguishing closed 4-manifolds by slicing

Abstract

One approach to produce a pair of homeomorphic-but-not-diffeomophic closed 4-manifolds is to find a knot which is smoothly slice in one but not the other. This approach has never been run successfully. We give the first examples of a pair of closed 4-manifolds with the same integer cohomology ring where the diffeomorphism type is distinguished by this approach. Along the way, we produce the first examples of 4-manifolds with nonvanishing Seiberg-Witten invariants and the same integer cohomology as $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ which are not diffeomorphic to $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$. We also give a simple new construction of a 4-manifold which is homeomorphic-but-not-diffeomorphic to $\mathbb{C}{P}^2\#5\overline{\mathbb{C}{P}^2}$.

arXiv comment10 pages, 3 Figures