Unitary representations of Brieskorn spheres
The aim of this paper is to study the \(\text{SU}(N)\)-representation space of Seifert fibred homology 3-spheres; in particular the Brieskorn spheres \(\Sigma(2,p,q)\). The approach used is comparison with the moduli space of semistable parabolic bundles and use of the inductive procedure of Atiyah-Bott-Nitsure to compute the \({\mathcal P}\)-equivariant cohomology of the strata (\({\mathcal P}\) is the gauge group of parabolic automorphisms). In the case of \(\text{SU}(3)\)-representations of \(\Sigma(2,p,q)\), the representation space turns out to be a discrete set of points and the results give explicit answers for the number of irreducible representations for many values of \(p\) and \(q\). With the aim of computing a generalized \(\text{SU}(N)\) Casson invariant, it is shown that the spectral flow along a path of irreducible \(\text{SU}(N)\)-representations of a Seifert fibred homology sphere is always even (generalizing the case \(N=2\) proved by Fintushel-Stern). Thus the number of irreducible representations gives the leading term of what is expected to be the generalized Casson invariant.
scientific article; zbMATH DE number 706710
scientific article