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Connected Components of The Space of Surface Group Representations

Nan-Kuo Ho·Chiu-Chu Melissa Liu
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Abstract

Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface $Σ$ of genus $l >1$, the order of the group $H^2(Σ,π_1(G))$ is equal to the number of connected components of the space $Hom(π_1(Σ),G)/G$ which can also be identified with the moduli space of gauge equivalence classes of flat G-bundles over $Σ$. We show that the same statement for a closed compact nonorientable surface which is homeomorphic to the connected sum of k copies of the real projective plane, where $k\neq 1,2,4$, can be easily derived from a result in A. Alekseev, A.Malkin and E. Meinrenken's recent work on Lie group valued moment maps.

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Nan-Kuo Ho, Chiu-Chu Melissa Liu. 2003-03-20. Connected Components of The Space of Surface Group Representations. https://arxiv.org/abs/math/0303255

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