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# Varieties of Group Representations and Casson's Invariant for Rational Homology 3-Spheres

S. Boyer and A. Nicas
Transactions of the American Mathematical Society
Vol. 322, No. 2 (Dec., 1990), pp. 507-522
DOI: 10.2307/2001712
Stable URL: http://www.jstor.org/stable/2001712
Page Count: 16
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## Abstract

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 $\mathrm{SL}_2(C)$ 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 $\mathrm{SL}_2(C)$ or $\mathrm{SU}(2)$ representation is obtained. We also derive a sum theorem for Casson's invariant with respect to toroidal splittings of a $Z$-homology 3-sphere.

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