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

Andrew Casson’s $\mathbf{Z}$ -valued invariant for $\mathbf{Z}$ -homology $3$ -spheres is shown to extend to a $\mathbf{Q}$ -valued invariant for $\mathbf{Q}$ -homology $3$ -spheres which is additive with respect to connected sums. We analyze conditions under which the set of abelian ${\mathrm{SL}}_2(\mathbf{C})$ and $\mathrm{SU}(2)$ representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian ${\mathrm{SL}}_2(\mathbf{C})$ or $\mathrm{SU}(2)$ representation is obtained. We also derive a sum theorem for Casson’s invariant with respect to toroidal splittings of a $\mathbf{Z}$ -homology $3$ -sphere.