On the Vanishing of the Rokhlin Invariant
Vol. 18 (2011), No. 2, Page 239--268.
Moriyama, Tetsuhiro
On the Vanishing of the Rokhlin Invariant
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]
Abstract:
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--action 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.
Mathematics Subject Classification (2010): Primary 57M27, Secondary 57N70, 57R20, 55R80.
Received: 2011-05-19
