Rohlin’s invariant and gauge theory II. Mapping tori

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 $S^{1} \times S^{3}$ 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.

Keywords

Casson invariant, Rohlin invariant, Floer homology

Mathematical Subject Classification 2000

Primary: 57R57

Secondary: 57R58

References

Forward citations

Publication

Received: 30 June 2003

Revised: 18 December 2003

Accepted: 16 January 2004

Published: 21 January 2004

Proposed: Ronald Stern

Seconded: Robion Kirby, Tomasz Mrowka

Authors

Daniel Ruberman
Department of Mathematics MS 050 Brandeis University Waltham MA 02454 USA

Nikolai Saveliev
Department of Mathematics University of Miami PO Box 249085 Coral Gables FL 33124 USA

Volume 8, issue 1 (2004)

Daniel Ruberman and Nikolai Saveliev

Abstract