Orientation-reversing involutions on homology 3-spheres

Abstract

An important problem in geometric topology is to find a homology 3-sphere M with Rohlin invariant μ(M) = ½ such that M # M bounds an acyclic 4-manifold. If such an M exists then, for instance, all closed topological manifolds of dimension ≥ 5 support a polyhedral structure(3, 4). One way of producing such examples is to find a homology 3-sphere M with μ(M) = ½ such that M admits an orientation-reversing diffeomorphism h, for then M # M would be diffeomorphic to M # − M which bounds an acyclic manifold. In this paper we observe by elementary means that if h is further assumed to be an involution, then this is not possible, namely:Theorem. Let M be a homology 3-sphere which admits an orientation reversing involution. Then M bounds a parallelizable rational 4-ball. In particular, μ(M) = 0.