### 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} × 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.

Original language | English (US) |
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Pages (from-to) | 35-76 |

Number of pages | 42 |

Journal | Geometry and Topology |

Volume | 8 |

DOIs | |

State | Published - 2004 |

### Keywords

- Casson invariant
- Floer homology
- Rohlin invariant

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Geometry and Topology*,

*8*, 35-76. https://doi.org/10.2140/gt.2004.8.35