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Andrew Casson's Z-valued invariant for Z-homology 3-spheres is shown to extend to a Q-valued invariant for Q-homology 3-spheres which is additive with respect to connected sums. We analyze conditions under which the set of abelian SL2(C) and SU(2) representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian SL2(C) or SU(2) representation is obtained. We also derive a sum theorem for Casson's invariant with respect to toroidal splittings of a Z-homology 3-sphere. Andrew Casson's lectures at MSRI in the spring of 1985 introduced an integer valued invariant of oriented integral homology 3-spheres. This invariant, constructed by means of representation spaces, yields interesting new results in low dimensional topology. In this paper we examine the extent to which Casson's procedure for defining his invariant can be used to obtain a rational valued invariant for oriented rational homology 3-spheres. Let 7r be a finitely generated group and G a Lie group. It is well known that the set R(7, G) of all homomorphisms of 7r into G can be given the structure of an analytic set in a natural manner. If G is an algebraic group, R(7r, G) becomes an algebraic set. The closed subspace of R(7r, G) consisting of representations 7r -* G with abelian image will be denoted by A(7r, G) . Let Rn(7,, G) be the union of those components of R(7r, G) which do not meet A(7r, G). When G is understood from the context, it will be dropped from the notation. If R is a commutative ring, an R-homology 3-sphere is a closed, orientable (over Z) 3-manifold with homology isomorphic to H*(S3; R). Let H(R) be the set of oriented homeomorphism types of oriented R-homology 3-spheres. For M E 11(Z) Casson defined an integer valued invariant A(M). We briefly recall his definition (see [AM] for a comprehensive exposition of Casson's MSRI lectures). Let M = WI UF W2 be a Heegard decomposition of M, where F = & W is of genus g and let F* be F punctured once. The diagram of I Received by the editors May 19, 1987 and, in revised form, December 9, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M99, 57M05, 20F99.

Varieties of group representations and Casson’s invariant for rational homology 3-spheres