Unitary representations of Brieskorn spheres
type: Item
instance of: scholarly article
title: Unitary representations of Brieskorn spheres
zbMATH Open document ID: 0861.57035
DOI: 10.1215/S0012-7094-94-07506-6
author: Hans U. Boden
published in: Duke Mathematical Journal
publication date: 1997-05-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 706710
zbMATH Keywords: representations
zbMATH Keywords: homology 3-spheres
zbMATH Keywords: Brieskorn spheres
zbMATH Keywords: Casson invariant
reviewed by: Jerome Levine
MaRDI profile type: Publication
cites work: The Yang-Mills equations over Riemann surfaces
cites work: Spectral asymmetry and Riemannian Geometry. I
cites work: Spectral asymmetry and Riemannian geometry. II
cites work: Spectral asymmetry and Riemannian geometry. III
cites work: Topological methods to compute Chern-Simons invariants
cites work: Representations of orbifold groups and parabolic bundles
cites work: Variations of moduli of parabolic bundles
cites work: Q3961493
cites work: A symplectic geometry approach to generalized Casson’s invariants of 3-manifolds
cites work: Instanton Homology of Seifert Fibred Homology Three Spheres
cites work: Pseudofree orbifolds
cites work: Unitary representations of knot groups
cites work: Chern-Simons invariants of 3-manifolds and representation spaces of knot groups
cites work: Representation spaces of Seifert fibered homology spheres
cites work: Splitting the spectral flow and the Alexander matrix
cites work: Moduli of vector bundles on curves with parabolic structures
cites work: Cohomology of the moduli of parabolic vector bundles
cites work: Q4126536
cites work: Casson's invariant and gauge theory
description: scientific article; zbMATH DE number 706710