Equivariant Casson invariants via gauge theory

Olivier Collin, Nikolai Saveliev

Research output: Contribution to journalArticle

17 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)143-169
Number of pages27
JournalJournal fur die Reine und Angewandte Mathematik
Volume541
StatePublished - 2001
Externally publishedYes

Fingerprint

Casson Invariant
Invariant Theory
Gauge Theory
Equivariant
Knot
Gages
Branched Cover
Homology Spheres
Floer Homology
Cyclic group
Group Action
Geometric proof
Euler Characteristic
Orbifold
Fundamental Group
Homology
Explicit Formula
Counting
Closed-form
Finite Group

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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 journalArticle

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