On the vanishing of the Rokhlin invariant

TL;DR

This paper provides a new direct proof that the Rokhlin invariant of an amphichiral integral homology 3-sphere vanishes, using cobordism and spin manifold techniques inspired by the Kontsevich-Kuperberg-Thurston invariant.

Contribution

It introduces a novel construction of a manifold pair and employs cobordism theory to prove the vanishing of the Rokhlin invariant for amphichiral integral homology 3-spheres.

Findings

The Rokhlin invariant of amphichiral integral homology 3-spheres is zero.

A new proof method based on cobordism and spin manifolds is established.

The construction relates to the Kontsevich-Kuperberg-Thurston invariant.

Abstract

It is a natural consequence of fundamental properties of the Casson invariant that the Rokhlin invariant 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 \cup M \cup (-M) and (Y,Q) \cong (-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 minus one point. For any such (Z,X), the signature of X vanishes, and this implies the vanishing of the Rokhlin invariant. The idea of the construction of (Y,Q) comes from the definition of the Kontsevich-Kuperberg-Thurston invariant for rational homology 3-spheres.