Rohlin's invariant and gauge theory II. Mapping tori

Daniel Ruberman, Nikolai Saveliev

Research output: Contribution to journalArticle

10 Scopus citations

Abstract

This is the second in a series of papers studying the relationship between Rohlin's theorem and gauge theory. We discuss an invariant of a homology S 1 × S3 defined by Furuta and Ohta as an analogue of Casson's invariant for homology 3-spheres. Our main result is a calculation of the Furuta-Ohta invariant for the mapping torus of a finite-order diffeomorphism of a homology sphere. The answer is the equivariant Casson invariant (Collin-Saveliev 2001) if the action has fixed points, and a version of the Boyer-Nicas (1990) invariant if the action is free. We deduce, for finite-order mapping tori, the conjecture of Furuta and Ohta that their invariant reduces mod 2 to the Rohlin invariant of a manifold carrying a generator of the third homology group. Under some transversality assumptions, we show that the Furuta-Ohta invariant coincides with the Lefschetz number of the action on Floer homology. Comparing our two answers yields an example of a diffeomorphism acting trivially on the representation variety but non-trivially on Floer homology.

Original languageEnglish (US)
Pages (from-to)35-76
Number of pages42
JournalGeometry and Topology
Volume8
DOIs
StatePublished - 2004

Keywords

  • Casson invariant
  • Floer homology
  • Rohlin invariant

ASJC Scopus subject areas

  • Geometry and Topology

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