Unitary Representations of Brieskorn Spheres

In this article, we commence an investigation of the SU(N) representation space of Seifert bered homology spheres (a 1 ; : : : ; a n): Under mild assumptions (e.g. if N is prime), then Theorem 3.1 implies that any closed connected component of irreducible SU(N) representations of (a 1 ; : : : ; a n) is homeomorphic to a component of SU(N) representations of an associated genus zero Fuchsian group. The latter representation spaces can be studied using the general correspondence between representations of Fuchsian groups and the moduli of parabolic bundles given by Mehta and Seshadri. For example, the inductive procedure of Atiyah-Bott-Nitsure determines the cohomology of this mod-uli space and it follows that the odd dimensional cohomology groups of any component of irreducible SU(N) representations of (a 1 ; : : : ; a n) vanish. In particular, any irre-ducible component of the SU(3) representation space of a Brieskorn spheres (p; q; r) is either a point or a two sphere. By repeated application of the inductive procedure, the precise number of points and two spheres in this representation space is determined. Speciic results for the Brieskorn spheres with p = 2 are given, where the representation space is a collection of points. In the last section, the SU(N) spectral ow of irreducible representations of Seifert bered homology spheres is shown to be even. This gives a calculation of the leading term in a gauge-theoretic deenition of the generalized Casson invariants.