An intersection theory count of the [formula omitted]-representations of the fundamental group of a 3-manifold

Abstract

We define an invariant of closed 3-manifolds counting the signed equivalence classes of representations of the fundamental group in SL 2( C ) . The invariant is an SL 2( C ) -analog of the Casson-Walker invariant for SU(2). We reinterpret the invariant algebro-geometrically and show that it is non-negative. We relate the invariant to a generalization of the norm of Culler, Gordon, Luecke and Shalen. We show that an analog of the Casson-Walker knot invariant exists in this setting. We obtain a Dehn surgery formula for the invariant for manifolds which are the result of Dehn surgery on knots in integral homology spheres, where the surgery coefficients obey certain technical conditions.