Connected Components of The Space of Surface Group Representations

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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 H²(Σ,π₁(G)) is equal to the number of connected components of the space Hom(π₁(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≠ 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.