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 -spheres

Authors: S. Boyer and A. Nicas
Journal: Trans. Amer. Math. Soc. 322 (1990), 507-522
MSC: Primary 57N10
DOI: https://doi.org/10.1090/S0002-9947-1990-0972701-6
MathSciNet review: 972701
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Abstract: Andrew Casson's ${\mathbf{Z}}$ -valued invariant for ${\mathbf{Z}}$ -homology -spheres is shown to extend to a ${\mathbf{Q}}$ -valued invariant for ${\mathbf{Q}}$ -homology -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 -sphere.

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