A Fake Compact Contractible 4-manifold

Here we construct a fake smooth structure on a compact contractible 4-manifold W , where W is a well-known Mazur manifold obtained by attaching in two-handle to S x B along its boundary as in Figure 1. * Here we use the conventions of [2]. The results of this paper imply: Theorem 1. There is a smooth contractible 4-manifold V with dV = d W, such that V is homeomorphic but not diffeomorphic to W relative to the boundary. Let a be the loop in dW given by S x p c S x S = d{S x B) as in Figure 2. Zeeman raised the question whether a is slice in W [12], i.e., if a bounds an imbedded smooth D in W. Even though it turned out that a is slice in another smooth contractible manifold with the same boundary [2], the original question has remained open. Let / : d W —• d W be the diffeomorphism, obtained by first surgering S x B to B xS in the interior of W, then surgering the other imbedded B x S back to S x B (i.e., replacing the dots in Figure 2.) Clearly this diffeomorphism extends to a self-homotopy equivalence of W. In fact, by [9], / extends to a homeomorphism F: W —• W. In [2, p. 279] the question of whether / extends to a diffeomorphism of W was posed. If it did, a would be slice in W since f(a) is clearly slice in W. Here we answer these questions negatively: Theorem 2. α is not slice in W, in particular f does not extend to a self-diffeomorphism of W. Theorem 1 follows from Theorem 2 as follows: Let F: W —• W be a homeomorphism extending / . Let V be the smooth structure on W obtained by pulling back the smooth structure of W by F . This gives a diffeomorphism F: V —> W extending / on the boundary. If G: W -> V