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Topological components of spaces of representations. (English) Zbl 0655.57019
From the introduction: Let $$S$$ be a closed oriented surface of genus $$g>1$$ and let $$\pi$$ denote its fundamental group. Let $$G$$ be a semisimple Lie group. The purpose of this paper is to investigate the global properties of the space $$\operatorname{Hom}(\pi,G)$$ of all representations $$\pi\to G$$, when $$G$$ is locally isomorphic to either PSL(2,$${\mathbb{C}})$$ or PSL(2,$${\mathbb{R}})$$. The main results of this paper may be summarized as follows:
Theorem A. (i) Let $$G$$ be the $$n$$-fold covering group of PSL(2,$${\mathbb{R}})$$. Then the number of connected components of $$Hom(\pi,G)$$ is given by the following formula: $2n^{2g}+(4g-4)/n-1\quad\text{if} \quad n| 2g-2;\quad 2[(2g-2)/n]+1\quad\text{if}\quad n\nmid 2g-2.$ For example $$\operatorname{Hom}(\pi,\text{SL}(2,\mathbb{R}))$$ has $$2^{2g+1}+2g-3$$ components. (ii) Let $$G=SO(3)$$ or PSL(2,$${\mathbb{C}})$$. Then $$\operatorname{Hom}(\pi,G)$$ has exactly two connected components. If $$G=SU(2)$$ or SL(2,$${\mathbb{C}})$$ then $$\operatorname{Hom}(\pi,G)$$ is connected.

##### MSC:
 57R22 Topology of vector bundles and fiber bundles 57R30 Foliations in differential topology; geometric theory 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 57M05 Fundamental group, presentations, free differential calculus 20F99 Special aspects of infinite or finite groups 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 22E99 Lie groups
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