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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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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Keywords: Casson’s invariant, <!– MATH $\operatorname {SL} (2,{\mathbf {C}})$ –> <IMG WIDTH="80" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$\operatorname {SL} (2,{\mathbf {C}})$"> representations, Zariski tangent space
Article copyright: © Copyright 1990 American Mathematical Society