Equivariant Casson invariants via gauge theory (Q2757782)

This paper defines an equivariant Casson invariant for an integral homology 3-sphere \(\Sigma\) equipped with a self-diffeomorphism \(\tau\) of finite order \(n\). Such an invariant had previously been defined by Cappell-Lee-Miller, but only when \(n\) is odd and prime. The definition is parallel to Taubes formulation of the usual Casson invariant. One considers the set of \(\text{SU}(2)\)-representations of \(\pi_1 (\Sigma)\) which are invariant under \(\tau_*\), up to conjugation by some \(u\in\text{SU}(2)\). Each representation is assigned a sign using spectral flow, as in Taubes work, and the invariant is the sum. In fact, by considering the trace of the element \(u\), the set of representations breaks up into subsets and one gets an invariant \(\lambda_k (\Sigma ,\tau)\) for each possible trace. NEWLINENEWLINENEWLINEThe main theorem is a formula which expresses \(\lambda_k (\Sigma ,\tau)\) in terms of \(\lambda (\Sigma')\) and the Tristram-Levine signature \(\sigma_{k/n}(K)\), where \(\Sigma'=\Sigma /\tau\) and \(K\) is a knot in \(\Sigma'\) which forms the branch set of the covering \(\Sigma\to\Sigma'\). It is required that \(\Sigma'\) be an integral homology sphere. The proof relies on a formula of C. Herald which expresses a signed count of the \(\text{SU}(2)\) representations of \(\Sigma' -K\), having a given trace on the meridian element, in terms of \(\lambda (\Sigma')\) and \(\sigma_{k/n}(K)\). To relate these two signed counts, the proof uses a third count, a signed sum of the \(\text{SO}(3)\) representations of the orbifold fundamental group of \((\Sigma' ,K,n)\). NEWLINENEWLINENEWLINEThere are some applications to branched covers over certain families of knots and giving alternative proofs of a formula of Fintushel-Stern for the Casson invariant of certain Brieskorn spheres and a formula of Neumann-Wahl.