### 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) |
---|---|

Pages (from-to) | 35-76 |

Number of pages | 42 |

Journal | Geometry and Topology |

Volume | 8 |

State | Published - 2004 |

### Fingerprint

### Keywords

- Casson invariant
- Floer homology
- Rohlin invariant

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Geometry and Topology*,

*8*, 35-76.

**Rohlin's invariant and gauge theory II. Mapping tori.** / Ruberman, Daniel; Saveliev, Nikolai.

Research output: Contribution to journal › Article

*Geometry and Topology*, vol. 8, pp. 35-76.

}

TY - JOUR

T1 - Rohlin's invariant and gauge theory II. Mapping tori

AU - Ruberman, Daniel

AU - Saveliev, Nikolai

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

KW - Casson invariant

KW - Floer homology

KW - Rohlin invariant

UR - http://www.scopus.com/inward/record.url?scp=4243059530&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4243059530&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:4243059530

VL - 8

SP - 35

EP - 76

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

ER -