On the vanishing of the Rokhlin invariant

Tetsuhiro Moriyama (1)

1 IF - Institut Fourier

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-action 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.

Mots clés: Rokhlin invariant; integral homology 3-sphere; signature; cobordism; Casson invariant; Kontsevich-Kuperberg-Thurston invariant; configuration space

Domaines: Topologie géométrique [math.GT]

Tetsuhiro Moriyama. On the vanishing of the Rokhlin invariant. 2008. ⟨hal-00296614⟩