### Abstract

Let Σ be an integral homology 3-sphere with a finite cyclic group action making it into a branched cover of another integral homology sphere with branch set a knot K. We define a family of equivariant Casson invariants of Σ by essentially counting equivariant SU(2)-representations of its fundamental group. We relate these invariants to the Euler characteristic of 3-orbifold Floer homologies, and identify them with certain equivariant knot signatures of K. We give explicit formulae for the equivariant Casson invariants of the branched covers over the graph knots and Montesinos knots. Applying this to various natural cyclic group actions on the links of singularities, we obtain a geometric proof of the Fintushel-Stern and Neumann-Wahl formulae and their extensions, and give a closed form formula for the Floer homology of Brieskorn homology spheres.

Original language | English (US) |
---|---|

Pages (from-to) | 143-169 |

Number of pages | 27 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 541 |

State | Published - 2001 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal fur die Reine und Angewandte Mathematik*,

*541*, 143-169.

**Equivariant Casson invariants via gauge theory.** / Collin, Olivier; Saveliev, Nikolai.

Research output: Contribution to journal › Article

*Journal fur die Reine und Angewandte Mathematik*, vol. 541, pp. 143-169.

}

TY - JOUR

T1 - Equivariant Casson invariants via gauge theory

AU - Collin, Olivier

AU - Saveliev, Nikolai

PY - 2001

Y1 - 2001

N2 - Let Σ be an integral homology 3-sphere with a finite cyclic group action making it into a branched cover of another integral homology sphere with branch set a knot K. We define a family of equivariant Casson invariants of Σ by essentially counting equivariant SU(2)-representations of its fundamental group. We relate these invariants to the Euler characteristic of 3-orbifold Floer homologies, and identify them with certain equivariant knot signatures of K. We give explicit formulae for the equivariant Casson invariants of the branched covers over the graph knots and Montesinos knots. Applying this to various natural cyclic group actions on the links of singularities, we obtain a geometric proof of the Fintushel-Stern and Neumann-Wahl formulae and their extensions, and give a closed form formula for the Floer homology of Brieskorn homology spheres.

AB - Let Σ be an integral homology 3-sphere with a finite cyclic group action making it into a branched cover of another integral homology sphere with branch set a knot K. We define a family of equivariant Casson invariants of Σ by essentially counting equivariant SU(2)-representations of its fundamental group. We relate these invariants to the Euler characteristic of 3-orbifold Floer homologies, and identify them with certain equivariant knot signatures of K. We give explicit formulae for the equivariant Casson invariants of the branched covers over the graph knots and Montesinos knots. Applying this to various natural cyclic group actions on the links of singularities, we obtain a geometric proof of the Fintushel-Stern and Neumann-Wahl formulae and their extensions, and give a closed form formula for the Floer homology of Brieskorn homology spheres.

UR - http://www.scopus.com/inward/record.url?scp=0035648714&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035648714&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0035648714

VL - 541

SP - 143

EP - 169

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

ER -