On the vanishing of the Rokhlin invariant

It is a natural consequence of fundamental properties of the Casson invariant that the Rokhlin invariant μ(M) of an amphichiral integral homology 3–sphere M vanishes. In this paper, we give a new direct proof of this vanishing property. For such an M , we construct a manifold pair (Y,Q) of dimensions 6 and 3 equipped with some additional structure (6–dimensional spin e-manifold), such that Q ∼= M ∐ M ∐ (−M) , and (Y,Q) ∼= (−Y,−Q) . We prove that (Y,Q) bounds a 7–dimensional spin e–manifold (Z,X) by studying the cobordism group of 6– dimensional spin e-manifolds and the Z/2–actions on the two–point configuration space of M\{pt} . For any such (Z,X) , the signature of X vanishes, and this implies μ(M) = 0. The idea of the construction of (Y,Q) comes from the definition of the Kontsevich–Kuperberg–Thurston invariant for rational homology 3–spheres.