On the vanishing of the rokhlin invariant

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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 II M II (-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.

Original languageEnglish
Pages (from-to)239-268
Number of pages30
JournalJournal of Mathematical Sciences
Volume18
Issue number2
Publication statusPublished - 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Mathematics(all)
  • Applied Mathematics

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