### Abstract

This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss a Casson-type invariant of a 3-manifold Y with the integral homology of the 3-torus, given by counting projectively flat U(2)-connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a certain sum of Rohlin invariants of Y. Our counting argument makes use of a natural action of H^{1}(Y; ℤ_{2}) on the moduli space of projectively flat connections; along the way we construct perturbations that are equivariant with respect to this action. Combined with the Floer exact triangle, this gives a purely gauge-theoretic proof that Casson's homology sphere invariant reduces mod 2 to the Rohlin invariant.

Original language | English (US) |
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Pages (from-to) | 618-646 |

Number of pages | 29 |

Journal | Commentarii Mathematici Helvetici |

Volume | 79 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2004 |

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### Keywords

- Casson invariant
- Flat moduli spaces
- Floer homology
- Rohlin invariant

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Commentarii Mathematici Helvetici*,

*79*(3), 618-646. https://doi.org/10.1007/s00014-004-0816-y