#### 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.

Notes on file formats
Daniel Ruberman, Nikolai Saveliev

Department of Mathematics, MS 050, Brandeis University

Waltham, MA 02454, USA

and

Department of Mathematics, University of Miami

PO Box 249085, Coral Gables, FL 33124, USA

Email: ruberman@brandeis.edu, saveliev@math.miami.edu

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