An exotic orientable 4-manifold (Q1062316)

We construct two compact, orientable 4-manifolds with boundary which are homeomorphic but not diffeomorphic. These are distinguished by Rohlin's theorem, i.e., they realize the obstruction \(H^ 3(M,\partial M;{\mathbb{Z}}_ 2)={\mathbb{Z}}_ 2\) predicted by Kirby-Siebenmann theory. In this sense, they are analogous to the Cappel-Shaneson fake \({\mathbb{R}}P^ 4\) and related examples, although this is the first such example in the oriented category. Our example contrasts with Donaldson's recent discovery of two closed simply connected 4-manifolds which are homeomorphic but not diffeomorphic. Donaldson's example is much more subtle and difficult, requiring extensive use of gauge theory and algebraic geometry, since no classical obstruction exists to distinguish the smoothings. Our construction proceeds as follows: We begin with Scharlemann's candidate for a fake \(S^ 3\times S^ 1\#S^ 2\times S^ 2\), which has a Poincaré homology sphere \(\Sigma\) representing \(H_ 3\). Using Freedman theory, we cut away the troublesome \(S^ 2\times S^ 2\), leaving a manifold-with-boundary M containing \(\Sigma\). We compare this with a model M' lying in the standard \(S^ 3\times S^ 1\#S^ 2\times S^ 2\) and containing \(S^ 3\) in place of \(\Sigma\). We arrange for M and M' to be homeomorphic, but show that they are not diffeomorphic, due to the nontrivial Rohlin invariant of \(\Sigma\).