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

Abstract

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