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Varieties of group representations and Casson's invariant for rational homology $ 3$-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 $ 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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DOI: https://doi.org/10.1090/S0002-9947-1990-0972701-6
Keywords: Casson's invariant, $ \operatorname{SL} (2,{\mathbf{C}})$ representations, Zariski tangent space
Article copyright: © Copyright 1990 American Mathematical Society

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