Geometry & Topology, Vol. 8 (2004) Paper no. 2, pages 35--76.

Rohlin's invariant and gauge theory II. Mapping tori

Daniel Ruberman, Nikolai Saveliev

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 cross 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

AMS subject classification. Primary: 57R57. Secondary: 57R58.

DOI: 10.2140/gt.2004.8.35

E-print: arXiv:math.GT/0306188

Submitted to GT on 30 June 2003. (Revised 18 December 2003.) Paper accepted 16 January 2004. Paper published 21 January 2004.

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Daniel Ruberman, Nikolai Saveliev
Department of Mathematics, MS 050, Brandeis University
Waltham, MA 02454, USA
Department of Mathematics, University of Miami
PO Box 249085, Coral Gables, FL 33124, USA


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