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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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