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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Equivariant Casson invariants via gauge theory. / Collin, Olivier; Saveliev, Nikolai.
In: Journal fur die Reine und Angewandte Mathematik, Vol. 541, 2001, p. 143-169.Research output: Contribution to journal › Article
}
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 -