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Fundamental Group of Moduli Spaces of Representations

Let S be a surface of genus g with n points removed, G a connected Lie group, and X(G) the moduli space of representations of the fundamental group of S into G. We compute the fundamental group of X(G) when n>0 and G is a real or complex reductive algebraic group, or a compact Lie group; and when n=0 and G=GL(m,C), SL(m,C), U(m), or SU(m).


Introduction
Consider G a connected reductive algebraic group over C. We will say a Zariski dense subgroup G ⊂ G is real reductive if G(R) 0 ⊂ G ⊂ G(R). Let Γ be a finitely generated discrete group. Then G acts on the analytic space Hom(Γ, G) by conjugation. Let Hom(Γ, G) * ⊂ Hom(Γ, G) be the subspace with closed orbits. Then the quotient space X Γ (G) := Hom(Γ, G) * /G for the adjoint action is called the G-character variety of Γ. By [25], the space X Γ (G) is Hausdorff and when G is real algebraic it is moreover semi-algebraic (and thus deformation retracts to compact space, which implies its fundamental group is finitely generated). Two cases of interest in this paper are when G is taken to be G itself and then X Γ (G) is the GIT quotient Hom(Γ, G)/ /G; and when G is a maximal compact subgroup of G, in which case X Γ (G) is the usual orbit space Hom(Γ, G)/G.
Moduli spaces of representations arise naturally throughout mathematics and physics. In particular, when Γ is the fundamental group of a complex manifold M, then X Γ (G) is called the Betti moduli space and is biholomorphic to theČech moduli space of flat principal G-bundles over M. This is in turn biholomorphic to the moduli space of flat Gconnections on M, called the de Rham moduli space. Lastly, there is the Dolbeault moduli space of Higgs bundles over M, which although remains diffeomorphic to the others, is equipped with a different complex structure which depends on M. See [29] for more about these four moduli spaces.
The topological study of these moduli spaces has a long history; beginning with the papers [2,16] and [13]. More recent developments concerning their homotopy groups include [5] for closed surface groups, and [10] for open surface groups.
For open surface groups Γ and connected G, it is easy to see that the spaces X Γ (G) are path-connected. For closed surface groups Γ and G connected and compact, the components of the moduli spaces X Γ (G) were described in [17,18]. When G is complex reductive and Γ is a Riemann surface group, the components have been described in [21], [20], and [11]. For example, when the derived subgroup of G (in either the compact or complex case) is simply connected, the moduli spaces are path-connected.
In [20], the systematic study of the fundamental group of these moduli spaces was initiated. In this paper, we complete part of that project. In particular, we consider Riemann surfaces M = Σ g,n of genus g with n points removed. We compute the fundamental groups of the above moduli spaces when n > 0 for any complex reductive G which by [9] gives the same result for compact groups K. In particular, we prove: Theorem A. Let G be either a connected reductive algebraic group over C, or a connected compact Lie group. Then π 1 (X π 1 Σg,n (G)) = π 1 (G/[G, G]) 2g+n−1 , whenever n > 0. In particular, X π 1 Σg,n (G) is simply connected if G is semisimple and n > 0.
After addressing moduli spaces over open Riemann surfaces, we turn to closed Riemann surfaces.

Acknowledgments
The second author thanks the Tata Institute of Fundamental Research for hosting him while this work was completed, and thanks Dan Ramras for comments. We also thank an anonymous referee for helpful suggestions.

Open surfaces
Let G be a connected reductive affine algebraic group over C. In this section we assume that Σ g,n is open, so n ≥ 1 and π 1 (Σ g,n ) is a free group F r of rank r = 2g + n − 1. Therefore, Hom(π 1 (Σ g,n ), G) ∼ = G r is a smooth complex variety, and where G acts by simultaneous conjugation given by g · (g 1 , · · · , g r ) = (gg 1 g −1 , · · · , gg r g −1 ). Thus X g,n (G) is irreducible and normal because G r is irreducible and smooth (hence normal), and these properties are inherited in GIT (see [8], for example). This provides some known topological properties worth pointing out that will be used in this section and in Section 3.
First, irreducible varieties are connected, and moreover any Zariski open subset is also connected. Normal varieties have their singular locus in codimension at least two. If S is a connected subvariety, then there is an analytic neighborhood U of S such that S is a deformation retraction of U (by Proposition A.5 in [15]), and the complement U \ S is connected (by [23]). This latter property implies that given any non-empty Zariski open subset U ⊂ X g,n , the homomorphism π 1 (U) −→ π 1 (X g,n ) induced by the inclusion map is surjective (see [1], for example).
Let DG := [G, G] be the derived subgroup of G, and Z(G) the center of G. We will denote by T = Z 0 (G) the connected component of Z(G) containing the identity element, and F = T ∩ DG is a central finite subgroup.
We have an exact sequence: where e is the identity in G, and G/DG ∼ = (C * ) d is a complex torus whose dimension coincides with the dimension of Z(G). Moreover, If G is Abelian, π 1 (X g,n (G)) = π 1 (Hom(F r , G)) ∼ = π 1 (G) r , so we assume that G is not Abelian.
For r = 1, the GIT quotient G/ /G deformation retracts to K/K by [9], and by [7,Corollary 17], we have π 1 (K/K) = π 1 (K/DK); here K/DK is the quotient for the right translation action and not for the conjugation action. Therefore, we conclude that π 1 (G/ /G) = π 1 (G/DG) since G/DG ∼ = (C * ) d and K/DK ∼ = (S 1 ) d is contained in G/DG (the quotient G/DG is for the right translation action). So we assume that r ≥ 2.
Remark 2.1. Also, Corollary 18 in [7] states K/K is simply-connected if and only if K is semi-simple.
The following discussion builds on results in [28,19,12]. A representation ρ ∈ Hom(F r , G) is irreducible if ρ(F r ) is not contained in any proper parabolic subgroup of G, and ρ is good if it is irreducible and the centralizer of ρ(F r ) in G is equal to the center Z(G) of G.
For any group π 1 (Σ g,n ), the set of good representations Hom g (π 1 (Σ g,n ), G) form a Zariski open G-stable subset of the polystable locus (points with closed orbits). Moreover, when n > 0 they are smooth as they are open subsets in a smooth manifold, and when n = 0 they are also smooth by [26,Proposition 37]. The action of P G on the set of irreducible representations is proper, and so we conclude that X g g,n (G) = Hom g (π 1 (Σ g,n ), G)/G ⊂ X g,n (G) is a smooth manifold since the P G-stabilizers are trivial. For open surfaces these sets are non-empty when r ≥ 2, and for closed surfaces we need only assume g ≥ 2 for these sets to be non-empty. The dimension is computed to be (2g + n − 2) dim G + dim Z(G) in the open case, and (2g − 2) dim G + 2 dim Z(G) in the closed case.
Lemma 2.2. Let G be a connected reductive affine algebraic group over C. Then Hom g (F r , G) → X g g,n (G) is a P G-bundle, and Hom i (F r , G) → X i g,n (G) is a P G-orbibundle. Proof. This follows from the Slice Theorem for proper actions (see Appendix of [14]), and the fact that P G acts effectively and properly on the space of irreducible representations Hom i (F r , G) by [19], and locally freely by [26]. On Hom g (F r , G) the action is free by definition.
A consequence of the above lemma, is that the projection mapping Hom g (F r , G) → X g g,n (G) is a Serre fibration, but by [24] Hom i (F r , G) → X i g,n (G) is a Serre fibration if and only if it has only one orbit type (that is, exactly when the good representations are the irreducible representations). Theorem 2.3. Let n ≥ 1 and let G be either a connected reductive affine algebraic group over C, or a connected compact Lie group. Then π 1 (X g,n (G)) ∼ = π 1 (G/DG) r .
Proof. First assume that G is a connected reductive affine algebraic group over C. A continuous map will be called π 1 -surjective if the corresponding homomorphism of fundamental groups is surjective. We will first show that p : Hom(F r , G) → X g,n (G) is π 1 -surjective.
Normality shows that the map j g is π 1 -surjective. Commutativity then implies that p is π 1 -surjective.
Next, the fibration DG → T × DG → T naturally gives a fibration Hom(F r , DG) → Hom(F r , T × DG) → Hom(F r , T ), and consequently, a fibration X g,n (DG) → X g,n (T ×DG) → X g,n (T ), where X g,n (F ) ∼ = F r equivariantly acts freely on X g,n (T × DG) ∼ = X g,n (T ) × X g,n (DG). This latter fibration descends to a fibration: since T × F DG ∼ = G and T /F ∼ = G/DG (see [4] for a generalization). Thus, we have a long exact sequence in homotopy: · · · → π 1 (X g,n (DG)) → π 1 (X g,n (G)) → π 1 (X g,n (G/DG)) → π 0 (X g,n (DG)) = 0 , since X g,n (DG) is connected. But because G/DG is Abelian, we have X g,n (G/DG) ∼ = (G/DG) r . So it suffices to prove that X g,n (DG) is simply-connected. Indeed, DG is semi-simple, so for any 1 ≤ k ≤ r consider this commutative diagram: where the top map is the k-th factor inclusion g → (e, ...e, g, e, ...e), and the bottom map is well-defined since the k-th factor inclusion is DG-equivariant. By Remark 2.1, DG/ /DG is simply-connected, and thus the generators of π 1 ((DG) r ) all map to 0 in π 1 (X g,n (DG)) by commutativity. However, we have shown p is π 1 -surjective. Together, this implies that X g,n (DG) is simply-connected.
In [9] is it shown that X g,n (K G ) is a deformation retraction of X g,n (G), where K G is a maximal compact subgroup of G. Therefore, the compact case follows from the complex reductive case.
Proof. By [6], X g,n (G) is homotopic to X g,n (K) where K is a maximal compact subgroup of G. By Theorem 2.3, we conclude that π 1 (X g,n (K)) ∼ = π 1 (K/DK) r .
Remark 2.5. We note that we cannot replace K/DK in the above theorem with G/DG (as in the complex reductive case); for example, G = U(p, q) gives a counter-example. However, we do know in general that K/DK is a geometric torus as it is a compact connected Abelian Lie group.

Closed surfaces
Let X be a compact connected Riemann surface of genus g, with g ≥ 2. Take any n ≥ 2 such that n = 2 if g = 2. The holomorphic cotangent bundle of X will be denoted by K X .
A Higgs field on a holomorphic vector bundle E on X is a holomorphic section of End(E) ⊗ K X = E ⊗ E * ⊗ K X . A Higgs bundle is a pair of the form (E , θ), where E is a holomorphic vector bundle on X and θ is a Higgs field on E. A Higgs bundle (E , θ) is called semistable if degree(F )/rank(F ) ≤ degree(E)/rank(E) for any nonzero holomor- Let M H (X, n) be the moduli space of semistable Higgs bundles on X of rank n and degree zero. Let  The morphism M H (X, n) −→ Pic 0 (X) that sends any (E , θ) to the line bundle n E induces an isomorphism of fundamental groups. In particular, π 1 (M H (X, n)) = Z 2g .
Proof. We will first show that N H (X, n) is irreducible and normal. In [27,p. 70,Theorem 11.1] it is proved that M H (X, n) is irreducible and normal (hence connected). The same proof works for N H (X, n). But normality of N H (X, n) can be deduced from the normality of M H (X, n) as follows: let be morphism defined by (E , θ) −→ ( n E , trace(θ)). Then φ is an algebraic fiber bundle with fiber N H (X, n). Since M H (X, 1) = Pic 0 (X) × H 0 (X, K X ) is smooth, it follows that N H (X, n) is normal if and only if M H (X, n) is normal. As mentioned before, M H (X, n) is normal by a theorem of Simpson. Let N s X (n) be the moduli space of stable vector bundles on X of rank n and trivial determinant. The total space of the cotangent bundle T * N s X (n) of the variety N s X (n) is naturally contained in N H (X, n). From the openness of the stability condition (see [22, p. 635, Theorem 2.8(B)]) it follows that T * N s X (n) is a Zariski open subset of N H (X, n). It is known that N s X (n) is simply connected (see [3, p. 266, Proposition 1.2(b)]). Therefore, T * N s X (n) is simply connected. Since N H (X, n) is normal and T * N s X (n) is a simply connected Zariski open subset of it, we conclude that N H (X, n) is simply connected (see the discussion at the beginning of Section 2). Now consider the variety M H (X, n). The morphism φ in (1) is an algebraic fiber bundle with fiber N H (X, n). Since N H (X, n) is connected and simply connected, we conclude that the corresponding homomorphism φ # : π 1 (M H (X, n)) −→ π 1 (M H (X, 1)) is an isomorphism. The projection q : M H (X, 1) = Pic 0 (X) × H 0 (X, K X ) −→ Pic 0 (X) produces an isomorphism of fundamental groups because H 0 (X, K X ) is contractible. Therefore, the map q • φ : M H (X, n) −→ Pic 0 (X) induces an isomorphism of fundamental groups.