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

Boyer, S.; Nicas, A.

doi:10.1090/S0002-9947-1990-0972701-6

Varieties of group representations and Casson’s invariant for rational homology $3$-spheres
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Trans. Amer. Math. Soc. 322 (1990), 507-522 Request permission

Abstract:

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 ${\operatorname {SL} _2}({\mathbf {C}})$ and $\operatorname {SU} (2)$ representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian ${\operatorname {SL} _2}({\mathbf {C}})$ or $\operatorname {SU} (2)$ 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.

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