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Representations of surface groups in the projective general linear group

Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,R), for n>=4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in R_SL(3,R) is homotopically equivalent to R_SO(3).


Introduction
Let X be a closed, oriented surface of genus g 2 and π 1 X be its fundamental group. Let R PGL(n,R) = Hom red (π 1 X, PGL(n, R))/PGL(n, R) be the quotient space of reductive representations of π 1 X in the projective general linear group PGL(n, R) = GL(n, R)/R * , where PGL(n, R) acts by conjugation. In this paper we determine the number of connected components of R PGL(n,R) , for n 4 even, applying the general theory of G-Higgs bundles to the PGL(n, R) case.
For a semisimple Lie group G, this general theory, created among others by Hitchin [14], Simpson [29,30,31], Corlette [8] and Donaldson [9], supplies a strong relation between different subjects such as topology, holomorphic and differential geometry and analysis. On the one hand, we have the moduli space R G of reductive representations of π 1 X in G, also known as a character variety. An element in R G is topologically classified by certain invariants of the isomorphism class of the associated flat principal G-bundle over X. If c is a topological class of principal G-bundles, we denote by R G (c) the subspace of R G consisting of classes of representations which belong to the class c. On the other hand we fix a complex structure on X turning it into a Riemann surface, and consider G-Higgs bundles over it. A G-Higgs bundle is a pair consisting of a holomorphic bundle, whose structure group depends on G, and a section of a certain associated bundle (see below for precise definitions). Topologically, a G-Higgs bundle is also classified by invariants taking values in the same set as the representations in R G . Again, if c is one topological class, we denote by M G (c) the moduli space of polystable G-Higgs bundles in the class c. a correspondence similar to the previous one, but replacing π 1 X by its universal central extension Γ, defined in (2.2) below. We denote the space of such representations, with fixed topological class c, by R Γ,G (c). Related to these two moduli spaces, and essential in the proof of the existence of the homeomorphism, is a third moduli space: the moduli space of solutions to the so-called Hitchin's equations on a fixed C ∞ principal G-bundle over X.
For G compact and connected, the spaces R G and M G have been studied in the seminal papers of Narasimhan and Seshadri [18] and of Ramanathan [21] from an algebraic viewpoint, and by Atiyah and Bott [1] from a gauge theoretic point of view. In this case, the answer about the number of components is known: for each topological type c, each subspace of R G (c) is connected. Since then much has been done to study the geometry and topology of these spaces. When G is complex, connected and reductive, the answer to the problem of counting connected components is the same as in the compact case by the works of Hitchin [14], Donaldson [9], Corlette [8] and Simpson [29,30,31]. When G is a non-compact real form of a complex semisimple Lie group, the study of the topology of R G started with the seminal papers of Goldman [12] and Hitchin [15] and, although much work has been done since then by several people (see, in particular, the paper [13] of Gothen, the works [2,3,4] of Bradlow, García-Prada and Gothen and also [10] by García-Prada, Gothen and Mundet i Riera), it is still far from finished.
In this paper we are interested in studying the components of R PGL(n,R) , for n 4 even (when n 3 is odd, PGL(n, R) ∼ = SL(n, R) hence the components of R PGL(n,R) are known to be 3 by the work of Hitchin in [15]; the n = 2 case was studied by Xia in [33]). Following the ideas of Hitchin [14,15], the main tool to reach our goal should be the L 2 -norm of the Higgs field in M PGL(n,R) (c), but in our case another group naturally appears. We will work with the space M EGL(n,R) of EGL(n, R)-Higgs bundles (EGL(n, R) = GL(n, R) × U(1)/ ∼, where (A, z) ∼ (−A, −z)). This is done mainly for two related reasons. One is that with this new group we can work with holomorphic vector bundles, rather than just principal or projective bundles. The other is that we can realize space of EGL(n, R)-Higgs bundles as closed subspace of M GL(n,C) × Jac d (X), where M GL(n,C) is the moduli space of Higgs bundles (see [14]). In general, when M G (c) is smooth, the function f given by the L 2 -norm of the Higgs field is a non-degenerate Morse-Bott function which is also a proper map and, in some cases, the critical submanifolds are well enough understood to allow the extraction of topological information such as the Poincaré polynomial. However, even when M G (c) has singularities, the properness of f allows us to draw conclusions about the connected components, although one cannot directly apply Morse theory.
The study of the local minima of f is sufficient to obtain the number of connected components of the space of EGL(n, R)-Higgs bundles and thus of R Γ,EGL(n,R) , for n 4. There is a projection from this space to R PGL(n,R) and using this we compute the components of R PGL(n,R) , obtaining the first of our two main results (see Theorem 10.2): Theorem 1.1. Let n 4 be even. Then the space R PGL(n,R) has 2 2g+1 + 2 connected components.
Essential in the count of components of R PGL(n,R) is the topological classification of real projective bundles over X. This is done in the first part of the paper, where we have found explicit discrete invariants which classify continuous principal PGL(n, R)-principal bundles over any closed oriented surface. This classification shows for example that, in contrast to the complex case, there are real projective bundles which are not projectivization of real vector bundles. It shows also that, in most cases, there is a collapse of the second Stiefel-Whitney class, when we pass from real vector bundles to projective bundles.
Combining the results of Xia [33] for n = 2 and of Hitchin [15] for n 3 odd with our Theorem 1.1, we have the number of connected components of R PGL(n,R) , for arbitrary n, as follows: Theorem 1.2. The number of connected components of R PGL(n,R) is: if n 3 is odd; if n 4 is even.
Using the results of Hitchin in [15], we are able to obtain more topological information of R PGL(3,R) (observe that this is the same as R SL(3,R) since PGL(3, R) ∼ = SL(3, R)), because in this case there are no critical submanifolds of f besides the local minima and these are of a very special type. The result we obtain is the following (see Theorem 11.1): (3,R) has one contractible component and the space consisting of the other two components is homotopically equivalent to R SO (3) .
Actually, using a computation of the Poincaré polynomials of R SO(3) recently done by Ho and Liu in [16], this theorem gives the Poincaré polynomials of R SL(3,R) almost for free.

2.
Representations of π 1 X in G and G-Higgs bundles 2.1. Representations of π 1 X in G. Let X be a closed oriented surface of genus g 2 and let G be a semisimple Lie group.
Consider the space Hom(π 1 X, G) of all homomorphisms from the fundamental group of X to G. Such a homomorphism ρ : π 1 X → G is also called a representation of π 1 X in G. Considering the presentation of π 1 X given by the usual 2g generators (2.1) π 1 X = a 1 , b 1 , . . . , a g , b g | g i=1 [a i , b i ] = 1 one sees that a representation ρ ∈ Hom(π 1 X, G) is determined by its values on the set of generators a 1 , b 1 , . . . , a g , b g . The set Hom(π 1 X, G) can thus be embedded in G 2g via ρ → (ρ(a 1 ), . . . , ρ(b g )), becoming the subset of 2g-tuples (A 1 , B 1 , . . . , A g , B g ) of G 2g satisfying the algebraic equation g i=1 [A i , B i ] = 1, and we consider the induced topology on Hom(π 1 X, G). Letting G act on Hom(π 1 X, G) by conjugation g · ρ = gρg −1 we obtain the quotient space Hom(π 1 X, G)/G.
This space may not be Hausdorff because there may exist different orbits with non-disjoint closures, so we consider only reductive representations of π 1 X in G, meaning the ones that, when composed with the adjoint representation of G on its Lie algebra, become a sum of irreducible representations. Denote the space of such representations by Hom red (π 1 X, G).
The corresponding quotient is the space we are interested in: Definition 2.1. The moduli space of representations of π 1 X in G is the quotient space R G = Hom red (π 1 X, G)/G.
The space R G is also known as the G-character variety of X.
If G acts on G 2g through the diagonal adjoint action, the inclusion j : Hom(π 1 X, G) ֒→ G 2g becomes G-equivariant and, from Theorem 11.4 in [24], a representation ρ ∈ Hom(π 1 X, G) is reductive if and only if the orbit of j(ρ) in G 2g is closed, hence it follows that R G is indeed Hausdorff.
If we allow G to be reductive and not just semisimple, then we consider a universal central extension Γ of π 1 X given by the short exact sequence It is generated by 2g generators a 1 , . . . , b g (which are mapped to the corresponding ones of π 1 X) and by a central element J, subject to the relation g i=1 [a i , b i ] = J: Let H ⊆ G be a maximal compact subgroup of G. Analogously to the case of π 1 X, let us consider the reductive representations ρ of Γ in G such that ρ(J) ∈ (Z(G) ∩ H) 0 , the identity component of the centre of G intersected with H: This definition does not depend on the choice of H. Indeed, any other maximal compact To give a representation ρ ∈ Hom red ρ(J)∈(Z(G)∩H) 0 (Γ, G) is equivalent to give a representation of π 1 (X \ {x 0 }), the fundamental group of the punctured surface, in G such that the image of the homotopy class of the loop around the puncture is ρ(J) ∈ (Z(G) ∩ H) 0 .
Of course, if G is semisimple, R Γ,G = R G . The main result of this paper is the computation of the number of connected components of R PGL(n,R) , for n 4 even.
As is well-known, there is a bijection between isomorphism classes of representations of π 1 X in G and isomorphism classes of flat G-bundles over X. There is as well a one-to-one correspondence between isomorphism classes of representations of Γ in G and isomorphism classes of projectively flat G-bundles over X, i.e., G-bundles equipped with connections with constant central curvature in Z(g) = Lie(Z(G) 0 ). Taking these correspondences into account, we make the following definition: Let ρ be a representation of Γ in G. A topological invariant of ρ is a topological invariant of the associated projectively flat G-bundle.
If two representations ρ 1 , ρ 2 ∈ Hom red (π 1 X, G) are equivalent, then the associated principal flat G-bundles E ρ 1 and E ρ 2 are isomorphic and vice-versa. Hence the topological invariants of ρ 1 and of ρ 2 are the same. Thus it makes sense to define a topological invariant of an equivalence class of representations. Given a topological class c of G-bundles over X, denote by R G (c) the subspace of R G whose representations belong to the class c. Analogously, define R Γ,G (c).
2.2. G-Higgs bundles. In this section we introduce the main objects which we shall work with. These are called Higgs bundles and roughly are pairs consisting of a holomorphic bundle and a section of an associated bundle (see Definition 2.4 below). Higgs bundles were introduced by Hitchin [14] on compact Riemann surfaces and by Simpson [29] on any compact Kähler manifold.
Let H ⊆ G be a maximal compact subgroup of G and H C ⊆ G C their complexifications. There is a Cartan decomposition of g, where m is the complement of h with respect to the non-degenerate Ad(G)-invariant bilinear B form on g. If θ : g → g is the corresponding Cartan involution then h and m are its +1-eigenspace and −1-eigenspace, respectively. Complexifying, we have the decomposition and m C is a representation of H C through the so-called isotropy representation (2.4) Ad with fibre m C , associated to the isotropy representation.
Let K = T * X 1,0 be the canonical line bundle of X.
Definition 2.4. A G-Higgs bundle over a Riemann surface X is a pair (E H C , Φ) where E H C is a principal holomorphic H C -bundle over X and Φ is a global holomorphic section of E H C (m C ) ⊗ K, called the Higgs field.
Any continuous G-bundle has certain discrete invariants which distinguish bundles which are not isomorphic as continuous (or equivalently C ∞ ) G-bundles. On Riemann surfaces, if G is connected, these invariants take values in π 1 G. For example, complex vector bundles of rank n are classified by their degree d ∈ Z = π 1 U(n). For G not necessarily connected, these topological invariants may take values in more complicated sets which depend only on the homotopy type of G. If H is a maximal compact subgroup of G, then the inclusion H ⊂ G is a homotopy equivalence so the classification of G-bundles is equivalent to that of H-bundles. Now, a G-Higgs bundle (E H C , Φ) is topologically classified by the topological invariant of the corresponding H C -bundle E H C and, as the maximal compact subgroup of H C is H, the topological classification of G-Higgs bundles is the same as the one of H-principal bundles. Now we consider the moduli space of G-Higgs bundles. The notion of (poly)stability, for general G is subtle (see [6,10,25,27]) but for GL(n, C) it is easy. Consider a (GL(n, C)-)Higgs bundle (V, Φ) and let Definition 2.5. A Higgs bundle (V, Φ) is: ⊗K is the map induced from f and from the isotropy representation H C → Aut(m C ).
In order to construct moduli spaces, we need to consider S-equivalence classes of semistable G-Higgs bundles (cf. [27]). For a stable G-Higgs bundle, its S-equivalence class coincides with its isomorphism class and for a strictly semistable G-Higgs bundle, its S-equivalence contains precisely one (up to isomorphism) representative which is polystable so this class can be thought as the isomorphism class of the unique polystable G-Higgs bundle which is S-equivalent to the given strictly semistable one.
These moduli spaces have been constructed by Schmitt in [25,26,27], using methods of Geometric Invariant Theory, showing that they carry a natural structure of algebraic/complex variety.
Definition 2.7. For a reductive Lie group G, the moduli space of G-Higgs bundles over a Riemann surface X is the algebraic/complex variety of isomorphism classes of polystable G-Higgs bundles. We denote it by M G : The relation between G-Higgs bundles over X and representations π 1 X → G is given by the following fundamental theorem.
Theorem 2.8. Let G be a semisimple Lie group. A G-Higgs bundle is polystable if and only if it arises from a reductive representation of π 1 X in G. Moreover, this correspondence induces a homeomorphism between the spaces R G (c) and M G (c).
If G is reductive, there is a similar correspondence which induces a homeomorphism between the spaces R Γ,G (c) and M G (c).
Strictly speaking, this theorem has been proved for G = GL(n, C) and G = SL(n, C) by Hitchin in [14] and Simpson in [29] (see also the papers [8] of Corlette and [9] of Donaldson). The general definition of polystability and the proof of the Hitchin-Kobayashi correspondence for arbitrary G-Higgs bundles appears in the preprint [10] of García-Prada, Gothen and Mundet i Riera.

Topological invariants for PGL(n, R)-bundles over closed oriented surfaces
In this section we obtain a topological classification of continuous principal PGL(n, R)bundles over X, with n 4 even. We shall, however, start by obtaining a general topological classification for any principal G-bundles, with π 0 G abelian.
3.1. The case of any topological group G with π 0 G abelian. Let G be a topological group. Denote by C(G) the sheaf of continuous G-valued functions on X and by G 0 the identity component of G. We have the short exact sequence of groups and, associated to the corresponding short exact sequence of sheaves of continuous functions with values in the corresponding groups, we have the sequence of cohomology sets: Recall that the cohomology set H 1 (X, C(G)) is in natural bijection with the set of isomorphism classes of continuous G-principal bundles over X. So, from the previous sequence, we define the first topological invariant of a continuous G-bundle E.
Definition 3.1. The topological invariant µ 1 of E is defined by Of course, this invariant yields the obstruction to reducing the structure group of E to G 0 . Notice that, if π 0 G is abelian, H 1 (X, π 0 G) ∼ = Hom(π 1 X, π 0 G) ∼ = π 0 G 2g . From now on we assume that we are on this case: π 0 G is an abelian group.
Our initial classification of G-bundles with µ 1 fixed was much more complicated and was splitted into two assymetric parts: µ 1 = 0 and µ 1 = 0. I am gretly indebted to an anonymous referee for providing a much simpler argument for the case µ 1 = 0 and which allows to study both cases µ 1 = 0 and µ 1 = 0 simultaneously. The argument is as follows.
The surface X is homeomorphic to the result of identifying (using orientation reversing homeomorphisms) the sides of a regular 4g-gon P according to the rule g . Let π : P → X be the natural projection. Let c be the centre of P , B(c, ǫ) ⊂ P be a small disc centred at c of radius ǫ, disjoint from the boundary of P , and let U = π(P \ {c}) and V = π(B(c, ǫ)).
A G-principal bundle E on X can be described by its restrictions E U = E| U and E V = E| V and by the gluing data As V is contractible, E V is isomorphic to the trivial bundle. On the other hand, the fact that the bundle E U can be extended to X implies that π * E U → P \{c} can be trivialized. The invariant µ 1 (E) describes the isomorphism type of E U and can be thought of as specifying, up to homotopy, how to glue the restrictions of π * E U → P \ {c} to the sides of P . Choosing a trivialization of π * E U , this is the same as associating, for each j, connected components of G to A j and to B j . This is how µ 1 can be seen as a homomorphism . Let G(E U ) and G(E V ) be the gauge groups of E U and E V . Then the relevant gluing information to recover the bundle E, up to isomorphism, is the class of ρ : Choosing adequate trivializations of π * (E U | U ∩V ) and of E V | U ∩V , the map ρ is given by a map ). Since U ∩ V ∼ S 1 and π 1 (G 0 ) is abelian, we can identify ρ 0 with an element, still denoted by ρ 0 , of π 1 (G 0 ) = π 1 G (we define the fundamental group of a topological group as the fundamental group of its identity component): Now, recall that π 0 G acts on π 1 G via the adjoint action (hence [α 1 ] = [α 2 ] in π 1 G/π 0 G if and only if there is a ∈ G such that α 1 and aα 2 a −1 are homotopic). Since π 1 G is an abelian group, we denote the group structure additively. Given µ 1 : π 1 X → π 0 G, define Γ µ 1 ⊂ π 1 G as the subgroup of π 1 G generated by the elements of the form γ 2 − γ 1 · γ 2 , where γ 2 ∈ π 1 G and γ 1 lies in the image of µ 1 : Then, since π 0 G is abelian, the action of π 0 G on π 1 G descends to the quotient π 1 G/Γ µ 1 .
It should be noticed that the values which the invariant µ 2 can take depend on the invariant µ 1 .
Similar arguments to the ones used in Proposition 5.1 of [21] show that the pair (µ 1 , µ 2 ) is well-defined and that uniquely characterizes the bundle E. In terms of (3.1) this can understood as follows: • π 1 G corresponds to π 0 (Isom(E U | U ∩V , E V | U ∩V )).
• the action of π 0 G on π 1 G corresponds to the action of π 0 (G(E V )) on π 0 (Isom(E U | U ∩V , E V | U ∩V )).
We have therefore the following topological classification of G-principal bundles over closed oriented surfaces. Proposition 3.3. Let X be a closed, oriented surface and let G be a topological group such that π 0 G is abelian. Given µ 1 ∈ H 1 (X, π 0 G), there is a bijection between the set of isomorphism classes of continuous G-principal bundles E over X, with µ 1 (E) = µ 1 , and (π 1 G/Γ µ 1 )/π 0 G. Remark 3.4. In case G is connected, this classification coincides with the well-known topological classification of G bundles over X, given π 1 G (cf. [21]).
Remark 3.5. For the case of the sphere S 2 (in fact for S n ) this was already known (cf. [32], Section 18).
Remark 3.6. The same result is valid not only for closed, oriented surfaces, but also for any 2-dimensional connected CW-complex. A proof of this fact, using different methods, can be found in [20].
3.2. The case of PGL(n, R). Now we shall apply the result obtained in the previous section to obtain invariants which classify continuous PGL(n, R)-principal bundles over our surface X.
As PGL(n, R) is homotopically equivalent to PO(n, R) = O(n, R)/Z 2 , its maximal compact subgroup, this is equivalent to classify PO(n, R)-bundles. From now on, we will write PO(n) instead of PO(n, R) for the real projective orthogonal group, as well as O(n) instead of O(n, R) for the real orthogonal group.
For PO(n), we have that This class is the obstruction to reduce the structure group to PSO(n).
For n 4 even, if n = 2 mod 4. More precisely, the universal cover of PO(n) is Pin(n) and, if p : Pin(n) → PO(n) is the covering projection, then, as a set, ker(p) = {1, −1, ω n , −ω n } where ω n = e 1 · · · e n is the oriented volume element of Pin(n) in the standard construction of this group via the Clifford algebra Cl(n) (see, for example, [17]).
For PO(n)-bundles with µ 1 = 0, then it is easy to see that Γ µ 1 = {0, 1} ∼ = Z 2 , therefore and π 0 PO(n) acts trivially on this quotient: Hence, we have the invariant µ 2 defined in general in Definition 3.2, which, for PO(n)-principal bundles over X, is such that: Remark 3.8. When µ 1 = 0, we also write the possible elements of µ 2 ∈ Z 2 by 0 and by ω n , instead of [0] and [ω n ]. This requires a little attention because, for example, µ 2 = 0 has different meanings whenever µ 1 = 0 or µ 1 = 0. However, it should always be clear in which situation we are.
Remark 3.9. When µ 1 = 0, we are reduced to the topological classification of PSO(n)-bundles over X which, for PSO(n)-equivalence, is given by the elements in {0, 1, ω n , −ω n } = π 1 PSO(n). However, since we are interested in PO(n)-equivalence, the bundles with invariants ω n and −ω n become identified.
The next proposition gives the interpretation of the class µ 2 in terms of obstructions. The case of µ 1 (E) = 0 is proved in a similar way, noticing also that over the 1-skeleton X 1 of X there are no obstructions to lifting the bundle because there the bundle is trivialized on contractible open sets.
Another way to see that a PO(n)-bundle lifts to a Pin(n)-bundle if it lifts to an O(n)bundle is as follows. Suppose that E is a real projective bundle, with µ 1 (E) = 0, and which is the projectivization of a real vector bundle W . Since the projection from O(n) onto PO(n) preserves components of the groups (because n is even), w 1 (W ) = µ 1 (E) = 0 where w 1 (W ) is the first Stiefel-Whitney class of W . So the first Stiefel-Whitney class of all lifts of E to O(n) is the same (another way to see this is to note that w 1 (W ⊗ F ) = w 1 (W ), for any real line bundle F , whenever rk(W ) is even). Nevertheless, different lifts of E can have different second Stiefel-Whitney class because their first Stiefel-Whitney class is non-zero. In fact, given a real vector bundle W of rank n on X with w 1 (W ) = 0, it is easy to see that there exists a real line bundle F such that . This is the reason why the second Stiefel-Whitney class "disappears" on projective bundles with µ 1 = 0. Hence either W or W ⊗ F has w 2 = 0 and therefore lifts to a Pin(n)-bundle. Choosing this lift of E, we see that E lifts to a Pin(n)-bundle.
Remark 3.12. If E is a real projective bundle with µ 1 (E) = 0 and µ 2 (E) ∈ {0, 1}, then also the second Stiefel-Whitney class of the lifts is well defined and is equal to µ 2 .
From Proposition 3.3, we obtain a full topological classification of real projective bundles over X.
Theorem 3.13. Let n 4 be even, and let X be a closed oriented surface of genus g 2. Then continuous PO(n)-bundles over X are classified by

Representations and topological classification
4.1. Representations of π 1 X in PGL(n, R). In this section we begin our analysis of the space R PGL(n,R) . The first thing to do is to define a topological invariant of a representation ρ : π 1 X → PGL(n, R). From Definition 2.3, we already know that is done via the correspondence between representations and flat bundles.
For fixed topological invariants,
Let PGL(n, R) 0 denote the identity component of PGL(n, R) and let PGL(n, R) − denote the component of PGL(n, R) which does not contain the identity. Definition 4.2. Given a representation ρ : π 1 X → PGL(n, R), let A 1 , B 1 , . . . , B g ∈ PGL(n, R) be the images of the generators of π 1 X by ρ. The invariant δ 1 of ρ is defined as to be such that: Obviously, δ 1 (ρ) is the obstruction to reducing the representation to PGL(n, R) 0 . So we have In order to obtain something similar for the invariant µ 2 , we will consider representations in the maximal compact PO(n). In terms of topological invariants, there is no loss of generality in doing this and has the advantage that these representations are automatically reductive due to the compactness of PO(n).
Since ker(p ′ ) ⊆ Z(O(n)), the value of this product does not depend on the choice of the lifts A ′ i , B ′ i and it is the obstruction to lifting ρ : π 1 X → PO(n) to a representation ρ ′ : π 1 X → O(n). Definition 4.3. Let ρ : π 1 X → PO(n) be a representation and let A 1 , B 1 , . . . , B g ∈ PO(n) be the images of the generators of π 1 X by ρ. The invariant δ 2 of ρ is defined as In this remark (and only here) we will not use the additive notation of Notation 3.7, since here we are going to work on the Pin(n) and Spin(n) group (which are not abelian). If δ 2 (ρ) = I n , one can ask whether ρ ′ : π 1 X → O(n) lifts to a representation ρ ′′ : π 1 X → Pin(n) under the projection p ′′ : Pin(n) → O(n) and the way to measure the obstruction to the existence of this lift is exactly the same as in the previous case: . . , g}, and consider the value Again this is well-defined because ker(p ′′ ) ⊆ Z(Pin(n)) and it is the obstruction to lifting ρ ′ to a representation ρ ′′ : π 1 X → Pin(n).
Ifp : Pin(n) → PO(n) is the universal cover (p = p ′ • p ′′ ) then, in the case δ 1 (ρ) = 0, we could not use the same procedure as in the previous cases to measure directly the obstruction to lifting ρ to a representationρ : π 1 X → Pin(n) because ker(p) = {±1, ±ω n } ⊂ Z(Pin(n)) = {±1}. In principle, the above procedure only gives partial information about the possible lifts of ρ to Pin(n): if δ 2 (ρ) = −I n then clearly ρ does not lift to Pin(n); if δ 2 (ρ) = I n and the lift ρ ′ of ρ to O(n) lifts to Pin(n) then ρ lifts to Pin(n); if δ 2 (ρ) = I n but the lift ρ ′ of ρ to O(n) does not lift to Pin(n), we cannot conclude that ρ does not lift to Pin(n) because if we change the lift of ρ to O(n) (or, equivalently, if we change the lifts of some of the generators A i and B i ) then this new representation of π 1 X on O(n) might lift to Pin(n). In fact, this is always possible, if µ 1 (ρ) = 0 (i.e., if δ 1 (ρ) = 0). To see this, suppose ρ is such that δ 1 (ρ) = 0 and i 0 to Pin(n) we obtain ±ω nBi 0 and now, sinceÃ −1 i 0 ∈ Pin(n) − and sinceB i 0 andB −1 i 0 belong to the same component of Pin(n), we havẽ Thus, if δ 1 (ρ) = 0, the value of (4.2) does not give any new information.
Recall Definition 4.1. From Proposition 3.10 and from what we have seen, we have the following lemma: Lemma 4.6. Let n 4 be even. The following equivalences hold: Proposition 4.7. Let n 4 even be given. Then, the space Proof. Let us start by seeing that R PGL(n,R) (µ 1 , ω n ) is non-empty for each µ 1 ∈ (Z 2 ) 2g . To do so we will find an explicit representation of π 1 X in PO(n) (hence in PGL(n, R)) with these invariants. From (4.1) and Lemma 4.6, in order to show that R PGL(n,R) (µ 1 , ω n ) is non-empty we only need to find a reductive representation ρ : π 1 X → PO(n) ⊂ PGL(n, R) with In other words, from the definition of δ 1 (ρ), we need to find n × n invertible matrices A ′ i and and only if the 2i-th coordinate of δ 1 (ρ) is 1. and, from the definition of δ 2 (ρ), which satisfy the equality As we are using the compact group PO(n), the reductiveness condition on the representation is automatically satisfied.
Let us start with the following orthogonal matrices: Note that X 2 , X ′ 2 and Z 2 are pairwise anti-commuting and that Y 2 and Y ′ 2 commute. For n 4 even, define We have the following facts: n anti-commute for all n even; • Y n and Y ′ n commute for all n even; • Z n anti-commutes with X n for all n even; • W n and W ′ n anti-commute for all n 4 even. Using these orthogonal matrices and the identity I n it is possible to construct the required representation. The important thing to note is that for each n we always have a pair of commuting and anti-commuting matrices both in SO(n) or both in O(n) − or one in SO(n) and the other in O(n) − .
The case of commuting matrices is easy: if one of the matrices is to be in SO(n), use the identity I n ; if both must be in O(n) − , use Y n and Y ′ n : The case of anti-commuting matrices is also easy: This shows that given in SO(n) or in O(n) − depending on x 1 and on x 2 and such that [A ′ 1 , B ′ 1 ] = −I n . Then, for i 2, we can also choose A ′ i and B ′ i accordingly to x 2i−1 and to x 2i respectively, and gives a representation ρ : π 1 X → PGL(n, R) with the given µ 1 and µ 2 (ρ) = ω n . For the other cases, the proof is similar but easier. For R PGL(n,R) (µ 1 , 0) with µ 1 = 0, we have, from (4.1) and Lemma 4.6, to find a representation ρ with δ 1 (ρ) = µ 1 and δ 2 (ρ) = I n and this is done in same way as above, using the first table.
The cases R PGL(n,R) (0, µ 2 ) with µ 2 = 0, 1 should be dealt with similarly, but now we would need to consider the invariantδ of Definition 4.5 and, hence, elements on the Pin(n) group. Instead, note that, since µ 1 = 0, we are looking for representations on the connected group PGL(n, R) 0 . Hence, the non-emptiness of R PGL(n,R) (0, µ 2 ) follows from Proposition 7.7 of [21].
The map in R PGL(n,R) which takes a class ρ to (µ 1 (ρ), µ 2 (ρ)) is continuous hence, if classes lie in the same connected component of R PGL(n,R) , they must have the same topological invariants. From this and from Theorem 3.13 and Proposition 4.7, we conclude that, for n 4 even, R PGL(n,R) has at least 2 2g+1 + 1 connected components this being the number of topological invariants.
It remains to see whether, for each pair (µ 1 , µ 2 ), R PGL(n,R) (µ 1 , µ 2 ) is connected or not, and it is now that the theory of Higgs bundles comes into play.

PGL(n, R)-Higgs bundles and EGL(n, R)-Higgs bundles
In this section we begin the study of PGL(n, R)-Higgs bundles and explain why and how one wants to work with another group instead of PGL(n, R).
Definition 5.1. A PGL(n, R)-Higgs bundle over X is a pair (E, Φ), where E is a holomorphic principal PO(n, C)-bundle and Φ ∈ H 0 (X, E × PO(n,C) so(n, C) ⊥ ⊗ K) where so(n, C) ⊥ is the vector space of n × n symmetric and traceless complex matrices.
We would like to work naturally with holomorphic vector bundles associated to the corresponding PGL(n, R)-Higgs bundles. However, this will not be done directly because PO(n, C) does not have a standard action on C n , and to fix this we use a standard procedure as follows. Enlarge the complex orthogonal group O(n, C) so that it still has a canonical action on C n and such that it has a non-discrete centre, and consider the sheaf of holomorphic functions with values in this centre. Then, the second cohomology of X of this sheaf vanishes, so there is no obstruction to lifting a holomorphic PO(n, C)-bundle to a holomorphic bundle with this new structure group, and hence to do the same to Higgs bundles with the corresponding groups.
Applying again Definition 2.4, we give now a concrete definition of EGL(n, R)-Higgs bundle. Notice that, if G = EGL(n, R), then a maximal compact subgroup of G is H = EO(n), so Proof. We have the following short exact sequence of groups Consider the sheaf EO(n, O) of holomorphic functions on X with values in EO(n, C). The above short exact sequence induces the following exact sequence hence, we see that there is no obstruction to lifting E to a principal EO(n, C)-bundle O)).
Write G = PGL(n, R), H = PO(n), G = EGL(n, R) and H = EO(n). We have We have the bundle E, which is a lift of E and we have a map π : E → E induced by the projection , the frame bundle associated to K), and E × X E K is the fibred product over X of E and E K .
induces, in a natural way, a Higgs field Φ ∈ H 0 (X, E × H C m C ⊗ K) given by (5.4).
Consider the actions of EO(n, C) on C n and on C induced, respectively, by the group homomorphisms and by the corresponding standard actions of GL(n, C) and C * .  With these two bundles we have a H C -equivariant map where H C acts on C n ⊗ C n by [(w, λ)] → λw ⊗ λw and on C as above. In other words The outcome of these results is that one can work with EGL(n, R)-Higgs bundles instead of PGL(n, R)-Higgs bundles with the advantage that in the former case we work with the objects (V, L, Q, Φ), involving holomorphic vector bundles. That is what we will do from now on.
We shall also call EGL(n, R)-Higgs bundles the quadruples (V, L, Q, Φ) mentioned in the previous proposition.
Given an EGL(n, R)-Higgs bundle (V, L, Q, Φ), we associate a PGL(n, R)-Higgs bundle (E, Φ 0 ), where E is given by the projection in sequence (5.1) and Φ 0 is obtained by projecting Φ to its traceless part.
is also a lift of (E, Φ 0 ) and the degree of the line bundle L ⊗ F 2 is 1.
From [3], a GL(n, R)-Higgs bundle is a triple (V, Q, Φ) where V is a rank n holomorphic vector bundle, equipped with a nowhere degenerate quadratic form, and Φ is symmetric endomorphism of V .
commute, where q and q ′ are the isomorphisms associated to Q and Q ′ , respectively.
Now we consider twisted orthogonal bundles, i.e., triples (V, L, Q) where V is a holomorphic rank n vector bundle equipped with a nowhere degenerate L-valued quadratic form Q. Of course, two twisted quadratic pairs (V, L, Q) and (V ′ , L ′ , Q ′ ) are isomorphic if there is a pair (f, g) of isomorphisms f : V → V ′ and g : L → L ′ such that ((f t ) −1 ⊗ g)q = q ′ f . Let E and E ′ be two principal EO(n, C)-bundles over X and let (V, L, Q) and (V ′ , L ′ , Q ′ ) be the corresponding twisted orthogonal bundles through the actions (5.5) and (5.6). It is easy to see that E and E ′ are isomorphic if and only if (V, L, Q) and (V ′ , L ′ , Q ′ ) are isomorphic. Now, consider the notion of isomorphism between two EGL(n, R)-Higgs bundles, by applying Definition 2.6. Consider also Definition 5.8 of isomorphism of EGL(n, R)-Higgs bundles written in terms of vector bundles, through Proposition 5.5. We have then that, when applied to EGL(n, R)-Higgs bundles, the isomorphism notion of Definition 2.6 is equivalent to the one of Definition 5.8, and that Proposition 5.5 gives a bijection between isomorphism classes of EGL(n, R)-Higgs bundles and of the objects (V, L, Q, Φ) which, because of this bijection, we also called EGL(n, R)-Higgs bundles.
Furthermore, the construction of Proposition 5.5 can be naturally applied to families of EGL(n, R)-Higgs bundles parametrized by varieties. Hence, the bijection between isomorphism classes of EGL(n, R)-Higgs bundles and of the objects (V, L, Q, Φ), naturally extends to families. The comparison of stability and strict polystability of EGL(n, R)-Higgs bundles and that of the associated Higgs bundles is a delicate question, since the conditions might not correspond. Nevertheless, this problem does not occur for polystability due to the correspondence with the solutions to Hitchin's equations.
For a Lie group G, and a maximal compact subgroup H, the Hitchin's equations are equations for a pair (A, Φ) where A is a H-connection on a C ∞ H C -principal bundle E H C and Φ ∈ Ω 1,0 (X, E H C (m C )). The equations are is the curvature of A, τ is the involution on G which defines H, λ ∈ Z(h) and ω is the normalized volume form on X so that vol(X) = 2π. Furthermore,∂ A is the unique∂ operator on E H C , corresponding to the H-connection A (∂ A is then the unique holomorphic structure on E H C induced from the (0, 1)-form A 0,1 ) and the second equation on (6.1) says that Φ is holomorphic with respect to this holomorphic structure. The details of this theory can be found in [14,11,6]. Now, given a G-Higgs bundle, (E H C , Φ) one associates to it a pair (A, Φ) given by a Hconnection A on the C ∞ H C -principal bundle E H C and Φ ∈ Ω 1,0 (X, E H C (m C )) (see, for instance, [14,11]). The Hitchin-Kobayashi correspondence says that a G-Higgs bundle (E, Φ) is polystable if and only if the associated pair (A, Φ) is a solution to the G-Hitchin's equations (see [11] for the details).
Let us check that, the homomorphism and β ∈ Ω 1 (E H , u(1))), we obtain the connection β Proof. The previous discussion shows that j * A ′ = β +A where j is the homomorphism in (6. .

From this we obtain
which are the GL(n, C)-Hitchin's equations. Furthermore, α + M is the curvature of β + A, and notice again that∂ β+A Φ =∂ A Φ due to the definition of m C and to the map m C → (m C ) ′ defined above. So, the equations in E EO(n,C) and in the GL(n, C) bundle E GL(n,C) obtained from E EO(n,C) by extending the structure group through the homomorphism j, are equivalent, and this proves the result.
Hence, the previous proposition and the Hitchin-Kobayashi correspondence show that Definition 6.1 is consistent with the notion of polystability for EGL(n, R)-Higgs bundles. The group EGL(n, R) can be seen as a closed subgroup of GL(n, C) × U(1) through The moduli space of GL(n, C) × U(1)-Higgs bundles is M GL(n,C) (d 1 ) × Jac d 2 (X), where Jac d 2 (X) is the subspace of the Picard group of the compact Riemann surface X which parametrizes holomorphic line bundles of degree d 2 . It is isomorphic to the Jacobian of X.
We shall realize M d as a subspace of M GL(n,C) (nd/2) × Jac d (X) and it will be in this subspace that we will work.
is injective.
Proof. First of all we see that we only have to take care with the form Q. Suppose that (V, Φ) is stable. The automorphism (q ′ ) −1 q of V is Φ-equivariant hence, from stability, (q ′ ) −1 q = λ ∈ C * , so ( √ λ, 1 L ) is an isomorphism between (V, L, Q, Φ) and (V, L, Q ′ , Φ).
Suppose now that (V, Φ) is strictly polystable, with Here Φ i : V i → V i ⊗ K and all the Higgs bundles (V i , Φ i ) are stable and with the same slope µ = deg(L)/2.
Consider the decomposition of q : V → V * ⊗ L as a matrix (q ij ) compatible with that of V and of . These are stable Higgs bundles and µ(V j ) = µ(V * i ⊗ L), therefore q ij must be an isomorphism. Hence for each pair i, j, if q ij is non-zero, then it is an isomorphism.
We will consider now three cases.
In the first case we suppose that (V, Φ) is a direct sum of isomorphic copies of the same Higgs bundle (W, Φ W ), with Φ W : W → W ⊗ K. Let then We have (W, Φ W ) ∼ = (W * ⊗ L, Φ t W ⊗ 1 K ⊗ 1 L ) but we can have more than one isomorphism on each column of (q ij ).
Choose i 0 and j 0 such that q i 0 j 0 : is any homomorphism, then (q −1 i 0 j 0 )q ij is an endomorphism of (W, Φ W ) and, since (W, Φ W ) is stable, q ij = α ij q i 0 j 0 where α ij ∈ C and, moreover, α ij = 0 if and only if q ij = 0. Hence the choice of q i 0 j 0 gives a way to represent (q ij ) by a symmetric k × k matrix where each entry is α ij and which can be diagonalized through a k × k matrix (λ ij ). Define the automorphism g of V given, with respect to the decomposition of V , by a k × k matrix (g ij ) where g ij = λ ij : W → W . Thus g is such that (g t ⊗ 1 L )qg =q whereq : V → V * ⊗ L is an isomorphism which is diagonal, by rk(W ) × rk(W ) blocks, with respect to the given decomposition of V .
Note also that g is Φ-equivariant. Hence, ifQ is the quadratic form associated toq, (g, 1 L ) : (V, L,Q, Φ) → (V, L, Q, Φ) is an isomorphism. So we can suppose that (q ij ) and (q ′ ij ) are diagonal and, an argument analogous to the case where (V, Φ) was stable shows then that (V, L, Q, Φ) and (V, L, Q ′ , Φ) are isomorphic, the isomorphism being (f, 1 L ) where f is given, according to the decomposition of V , by Here, each λ i ∈ C * is such that q ii = λ i q ′ ii . In the second case we consider In this case q ij = 0 if −l i − j l. So (q ij ) splits into four l × l blocks in the following way (q ij ) = 0 q 2 q 1 0 where q 1 represents q| W ⊕···⊕W : W ⊕ · · · ⊕ W −→ W ⊕ · · · ⊕ W and q 2 represents q| W * ⊗L⊕···⊕W * ⊗L : W * ⊗ L ⊕ · · · ⊕ W * ⊗ L −→ W * ⊗ L ⊕ · · · ⊕ W * ⊗ L.
Again, using the stability of (W, Φ W ) and of (W * ⊗ L, Φ t W ⊗ 1 K ⊗ 1 L ) and the fact that Φ is symmetric with respect to q, we see that each entry of q 1 is given by a scalar. The same happens with q 2 . Hence we can write where A is a non-singular l × l matrix. Now, if we write in the same way, then consider the isomorphism of V given by where we mean by this that each entry of B −1 A represents a scalar automorphism of (W, Φ W ) and f is the identity over W * ⊗ L ⊕ · · · ⊕ W * ⊗ L. So (f, 1 L ) is an isomorphism between (V, L, Q, Φ) and (V, L, Q ′ , Φ).
The last case is the generic one, where we consider a combination of the previous cases. We can always write If V a and V b are not inside the same parenthesis in (6.3) then q ab = 0 = q ′ ab . Hence we have an isomorphism f between (V, L, Q, Φ) and (V, L, Q ′ , Φ) where f is diagonal by blocks (not all of the same size), each corresponding to an isomorphism of one of the previous cases.
We identify M d with its image by the map i and therefore consider M d as a subspace of M GL(n,C) (nd/2) × Jac d (X).
The following lemma will be useful in Section 7. 6.2. Deformation theory of EGL(n, R)-Higgs bundles. In this section, we briefly recall the description of Biswas and Ramanan [7] (see also [19]) of the deformation theory of G-Higgs bundles and, in particular, the identification of the tangent space of M G at the smooth points, and then apply it to the case of EGL(n, R)-Higgs bundles.
The spaces h C and m C in the Cartan decomposition of g C verify the relation [h C , m C ] ⊂ m C hence, given v ∈ m C , there is an induced map ad(v)| h C : h C → m C . Applying this to a G-Higgs bundle (E H C , Φ), we obtain the following complex of sheaves on X: Proposition 6.8 (Biswas, Ramanan [7]). Let (E H C , Φ) represent a G-Higgs bundle over the compact Riemann surface X. (ii) there is an exact sequence 6.3. Topological classification of EGL(n, R)-Higgs bundles. Our calculations will be performed on M d so we will also need the topological invariants of EGL(n, R)-Higgs bundles.
We will define these discrete invariants using the relation between EGL(n, R)-Higgs bundles and PGL(n, R)-Higgs bundles. In fact, we already know from Theorem 3.13 that the invariants µ 1 and µ 2 completely classify PGL(n, R)-bundles over X, and also know from Proposition 5.6 that if two EGL(n, R)-Higgs bundles project to the same PGL(n, R)-Higgs bundle, then the degree of the corresponding line bundle L is equal modulo 2.
As we are dealing with topological classification of bundles, we can forget the Higgs field and consider only twisted orthogonal bundles (V, L, Q) which correspond to elements of the set H 1 (X, C(EO(n))), and PO(n)-bundles E which are in bijection with H 1 (X, C(PO(n))). There is then a relation between (V, L, Q) and E which is similar to the one between EGL(n, R)-Higgs bundles and PGL(n, R)-Higgs bundles, but now forgetting the Higgs field.
Consider then the following commutative diagram: The map p ′ 2 : H 1 (X, C(O(n) × U(1))) −→ H 1 (X, C(EO(n))) is induced from the projection O(n) × U(1) → EO(n) and hence defined, in terms of vector bundles, by . Once again we see that (V, L, Q) is in the image of p ′ 2 if and only if deg(L) is even. Moreover, if this is the case, the pre-image of (V, L, Q) under p ′ 2 is the following set of 2 2g pairs: where L 1/2 is a fixed square root of L (notice that saying that F is a 2-torsion point of the Jacobian is equivalent to say that F has a nowhere degenerate quadratic form The map p 2 is the one induced from the projection O(n) → PO(n).
Definition 6.10. Given a twisted orthogonal bundle (V, L, Q), let E be the corresponding PO(n)-bundle under the map H 1 (X, C(EO(n))) → H 1 (X, C(PO(n))). Define the first invariant µ 1 of (V, L, Q) as is then the obstruction to reducing the structure group of (V, L, Q) to ESO(n) = SO(n) × Z 2 U(1). Hence, this happens if and only if E reduces to a PSO(n)-bundle.
If deg(L) is even, then the first Stiefel-Whitney class of the real orthogonal bundle V ⊗ L −1/2 (the value of w 1 is independent of the choice of the square root of L because n is even -notice that w 1 (W ⊗ F ) = w 1 (W ) + rk(W ) 2 w 1 (F ) where rk(W ) 2 = rk(W ) mod 2).
Definition 6.11. Let (V, L, Q) be a twisted orthogonal bundle with rk(V ) = n 4. Define the second invariant µ 2 of (V, L, Q) as follows:  From Theorem 3.13, we know that µ 1 and µ 2 completely classify PO(n)-bundles. Moreover, since we also know that the difference between two (V, L, Q) and (V ′ , L ′ , Q ′ ) mapping to the same PO(n)-bundle lies in the degree of L, we have then the following: Theorem 6.13. Let X be a closed oriented surface of genus g 2 and let n 4 be even. Then twisted orthogonal bundles over X are topologically classified by the invariants Now, returning to our principal objects -EGL(n, R)-Higgs bundles -we see that Theorem 6.13 also gives the topological classification of EGL(n, R)-Higgs bundles. If µ 1 = 0 and d 1 = d 2 mod 2, then In all cases the bijection is given by where F is a holomorphic line bundle of suitable degree.

The Hitchin proper functional
Here we use the method introduced by Hitchin in [14] to study the topology of moduli space M G of G-Higgs bundles.
This function f is usually called the Hitchin functional.
Here we are using the harmonic metric (cf. [8,9]) on E H C to define Φ L 2 . So we are using the identification between M G (c) with the space of gauge-equivalent solutions of Hitchin's equations. We opt to work with M G (c), because in this case we have more algebraic tools at our disposal. We shall make use of the tangent space of M G (c), and we know from [14] that the above identification induces a diffeomorphism between the corresponding tangent spaces.

then f is a non-degenerate perfect Bott-Morse function.
Since f is proper, it attains a minimum on each connected component of M G (c). Moreover, we have the following result from general topology: In our case, the Hitchin functional f : M GL(n,C) (nd/2) −→ R is given by

Smooth minima.
Away from the singular locus of M GL(n,C) (nd/2), the Hitchin functional f is a moment map for the Hamiltonian S 1 -action on M GL(n,C) (nd/2) given by From this it follows immediately that a stable point of M GL(n,C) (nd/2) is a critical point of f if and only if is a fixed point of the S 1 -action. Let us then study the fixed point set of the given action (this is analogous to [15] and [3]).
Let (V, Φ) represent a stable fixed point. Then either Φ = 0 or (since the action is on M GL(n,C) (nd/2)) there is a one-parameter family of gauge transformations g(θ) such that g(θ) · (V, Φ) = (V, e √ −1θ Φ). In the latter case, let be the infinitesimal gauge transformation generating this family. Simpson shows in [29] that this (V, Φ) is what is called a complex variation of Hodge structure. This means that where the F j 's are the eigenbundles of the infinitesimal gauge transformation ψ: over F j , which is non-zero for all j except the maximal one.
Set From now on we will consider the restriction of f to M L . This fact will be important in the counting of components of each M d , as we shall see in Section 8.
The circle action (7.3) restricts to M L . So, if (V, L, Q, Φ) is an EGL(n, R)-Higgs bundle such that (V, Φ) is stable and is a fixed point of the S 1 -action (i.e., is a critical point of f ), then it is a variation of Hodge structure. In this case, g(θ) ∈ H 0 (X, EO(n, O)) and, since the Lie algebra of EO(n, C) is o(n, C) ⊕ C, we have ψ ∈ H 0 (X, o(n, O) ⊕ O), therefore being skew-symmetric with respect to Q. Thus, using (7.4) we have that, if v j ∈ F j and v l ∈ F l , F j and F l are therefore orthogonal under Q if l = −j, and q : V → V * ⊗ L yields an isomorphism for some m integer or half-integer.
Using these isomorphisms and the fact that Φ is symmetric under Q, we see that The Cartan decomposition of g C induces a decomposition of vector bundles where E H C (g C ) (resp. E H C (h C )) is the adjoint bundle, associated to the adjoint representation of H C on g C (resp. h C ). For the group EGL(n, R), we have E H C (g C ) = End(V ) ⊕ O where O = End(L) is the trivial line bundle on X and we already know that is the bundle of skew-symmetric endomorphisms of V with respect to the form Q. The involution in End(V ) ⊕ O defining the above decomposition is θ ⊕ 1 O where θ is the involution on End(V ) defined by Its +1-eigenbundle is Λ 2 Q V ⊕ O and its −1-eigenbundle is S 2 Q V . We also have a decomposition of this vector bundle as From (7.4), this is the √ −1k-eigenbundle for the adjoint action ad(ψ) :

Write
U i,j = Hom(F j , F i ). The restriction of the involution θ, defined in (7.6), to U i,j gives an isomorphism Observe that Φ ∈ H 0 (X, U − 1 ⊗ K). The map ad(Φ) = [Φ, −] interchanges U + with U − and therefore maps U ± k to U ∓ k+1 ⊗K. So, for each k, we have a weight k subcomplex of the complex C • (V, L, Q, Φ) defined in Proposition 6.9: is given by and, for k = 0, by From Proposition 6.9, if an EGL(n, R)-Higgs bundle (V, L, Q, Φ) is such that (V, Φ) is stable, its infinitesimal deformation space is We say that H 1 (X, C • k (V, L, Q, Φ)) is the subspace of H 1 (X, C • (V, L, Q, Φ)) with weight k. By Hitchin's computations in [15], we have the following result which gives us a way to compute the eigenvalues of the Hessian of the Hitchin functional f at a smooth (here we mean smooth in M GL(n,C) (nd/2)) critical point.
is an isomorphism for all k 1.
The following theorem is quite similar to the corresponding one in [15] and in [3] as one would naturally expect. Indeed, the proof of this theorem is inspired in the one of Theorem 4.3 of [3].
Proof. The proof that a local minimum of f must be of one of the above types is very similar to the one presented in the proof of Theorem 4.3 of [3], so we skip it.
We can write g as (7.10) and, using the decompositions (7.8), (7.9) and (7.10), this yields The summands lie in different U − i,j ⊗ K, hence [Φ, g] = 0 is equivalent to the following system of equations Take any fibre of V and choose suitable basis of V and V * ⊗ L such that, with respect to these basis, over the corresponding fibre of U + k . Then (7.11) implies that, over this fibre, g i = g j for all i, j. In particular, (7.13) g j = g −j−k for all j. From (7.12) and (7.13), we must then have g = 0. Since we considered any fibre, the result follows.
Remark 7.7. Although we are always assuming rk(V ) 4 even, we will need during the proof of Proposition 7.10 below, to consider EGL(n, R)-Higgs bundles of rank 1 and 2 and also of rank bigger or equal than 3 odd. In the first two cases it is straightforward to see that the minima of the Hitchin functional f (V, Φ) = Φ 2 L 2 , with (V, Φ) stable, in the corresponding moduli spaces are the following: For rk(V ) 3 odd, (V, L, Q, Φ) is a minimum of f if and only if either Φ = 0 or V = ⊕ m i=−m F i with rk(F i ) = 1 and Φ i is an isomorphism, for i = m. This case is completely analogous to the even case considered here. The details can be found in [20].
Let (V, L, Q, Φ) represent a local minimum of f of type (2) of Theorem 7.6. Then, where m is an half-integer. where each W α is the √ −1α-eigenbundle for an infinitesimal EO(n, C)-gauge transformation ψ and where Φ α : W α → W α+1 ⊗ K, with the possibility that Φ α = 0. We can then also write Q V , and define the following complex of sheaves associated to (V, L, Q, Φ): Hitchin's computations in [15] for showing that a given fixed point of the circle action is not a local minimum yield the following proposition.
In other words, if (V, Φ) is strictly polystable, Hitchin's arguments in [15] are also valid: if there is a non-empty subspace of H 1 (X, C • (V, L, Q, Φ)) which gives directions in which f decreases and if these directions are integrable into a one-parameter family in M L , then (V, L, Q, Φ) is not a local minimum of f .
The following result, adapted from [15], shows that there are no more non-zero minima of f besides the ones of Theorem 7.6.
Consider first the case where V 1 and V 2 are not isomorphic and V 1 ∼ = V * 1 ⊗L and V 2 ∼ = V * 2 ⊗L. Then the quadratic form Q also splits as We have therefore the EGL(n, R)-Higgs bundles (V 1 , L, Q 1 , Φ 1 ) and (V 2 , L, Q 2 , Φ 2 ) which are local minima of f on the corresponding lower rank moduli space. Let n 1 = rk(V 1 ) and n 2 = rk(V 2 ) so that n = n 1 + n 2 (here, the cases n 1 = 2 or n 2 = 2 or n i 3 odd are included). So we have Consider the complex Since Φ = 0, we have m + k > 0 and C • m+k (V, L, Q, Φ) is a subcomplex of the complex C • >0 (V, L, Q, Φ) defined in (7.15).
and, since F m (resp. G k ) is a Φ 1 (resp. Φ 2 )-invariant subbundle of V 1 (resp. V 2 ), we have, from the stability of (V 1 , Φ 1 ) and of (V 2 , Φ 2 ), It follows, by Riemann-Roch, that H 1 (X, Hom(G −k , F m )) is non-zero. Choose then and let ) is the map induced by the involution θ on End(V ) previously defined. σ is obviously non-zero and, moreover, it is annihilated by , which we also denote by σ. Now, σ defines extensions ).
The summands m−1 i=−m+1 F i and k−1 j=−k+1 G j in V σ have a quadratic form coming from Q, and we also have the canonical L-valued quadratic form on U σ ⊕ (U * σ ⊗ L). These give a L-valued quadratic form Q σ on V σ .
Notice that, if σ = 0, then (V 0 , L, Q 0 , Φ 0 ) = (V, L, Q, Φ). Now, consider the family (V tσ , L, Q tσ , Φ tσ ) of EGL(n, R)-Higgs bundles. The induced infinitesimal deformation is given by σ which, from (7.16), lies in a positive weight subspace of H 1 (X, C • (V, L, Q, Φ)). Taking Proposition 7.9 in account, this proves that (V, L, Q, Φ) is not a local minimum of f . Suppose now that V 1 ∼ = V 2 , but the form Q does not decompose. From the stability of (V 1 , Φ 1 ) and of (V 2 , Φ 2 ) we must have q = 0 q 12 is not a local minimum of f in M GL(n,C) (nd/2) (this is because the group GL(n, C) is complex). Therefore one can find a family (V 1,s , Φ 1,s ) of stable Higgs bundles near (V, Φ) such that f (V 1,s , Φ 1,s ) < f (V, Φ) for all s, i.e., Consider now the family of EGL(n, R)-Higgs bundles in M L given by where we are using the harmonic metric on V * 1,s and on V 1,s ⊕ V * 1,s induced by the one on V 1,s . We have tr( and from (7.19) we conclude that is not a local minimum of f .
If V 1 ∼ = V 2 , then we saw in the proof of Lemma 6.4 that we can decompose Q = Q 1 ⊕ Q 2 so that we can decompose the EGL(n, R)-Higgs bundles (V, L, Q, Φ) as (V 1 , L, . Hence we use the same argument as the first case to prove that (V, L, Q, Φ) is not a minimum of f .
If, for example, Φ 1 = 0 and Φ 2 = 0 then, due to the symmetry of Φ relatively to Q, the quadratic form must split into Q 1 ⊕ Q 2 , so that we have (V 1 , L, Q 1 , Φ 1 ) and (V 2 , L, Q 2 , 0) and in a similar manner to the first case considered, we prove that (V, L, Q, Φ) is not a local minimum of f . For EGL(n, R)-Higgs bundles such that (V, Φ) has more than two summands, just consider the first two and use one of the above arguments.

Connected components of the space of EGL(n, R)-Higgs bundles
In this section we compute the number of components of the subspaces of the moduli space of EGL(n, R)-Higgs bundles such that the degree of L is 0 and 1. Denote these subspaces by M 0 and M 1 , respectively. In other words, using Notation 6.14, we write M 0 as a disjoint union On the other hand, Of course, the space M of isomorphism classes of polystable EGL(n, R)-Higgs bundles has an infinite number of components because the invariant given by the degree of L can be any integer. But our computation will also give the number of components of any subspace of M with the degree of L fixed, due to the identifications given in Remark 6.15.
Before proceeding with the computation, we need some results which will be used. Let N EO(n,C) be the moduli space of holomorphic semistable principal EO(n, C)-bundles on X and N EO(n,C) (µ 1 , µ 2 ) be the subspace with invariants (µ 1 , µ 2 ).
The following is an adaptation of Proposition 4.2 of [21].
Proof. Let E ′ and E ′′ represent two classes in N EO(n,C) (µ 1 , µ 2 ). Let P be the underlying C ∞ principal bundle, and let ∂ A ′ and ∂ A ′′ be the operators on P defining, respectively, E ′ and E ′′ and given by unitary connections A ′ and A ′′ .
Let D be an open disc in C containing 0 and 1. Consider the C ∞ principal-EO(n, C) bundle E → D × X, where E = D × P . Define the connection form on E by where v is tangent to D at z and w is tangent to P at some point p. If we consider the holomorphic bundle E z given by E| {z}×X with the holomorphic structure given by A z , then we have that E 0 ∼ = E ′ and E 1 ∼ = E ′′ .
Since semistability is an open condition with respect to the Zariski topology, D \ D ′ is connected where D ′ = {z ∈ D : E z is not semistable}. Hence {E z } z∈D\D ′ is a connected family of semistable EO(n, C)-principal bundles joining E 0 and E 1 . Since E 0 ∼ = E ′ and E 1 ∼ = E ′′ , using the universal property of the coarse moduli space N GL(n,C) of GL(n, C)principal bundles, there is a connected family in N GL(n,C) joining E ′ and E ′′ . But, of course this connected family lies in N EO(n,C) (µ 1 , µ 2 ). Let us now concentrate attentions on M 0 .
For each L ∈ Jac 0 (X) = Jac(X), the Jacobian of X, M L is a subspace of M 0 and it is the fibre over L of the map In particular, any fibre is isomorphic to M O .
More precisely, after lifting to a finite cover, (8.2) becomes a product. This is a similar situation to the one which occurs on the moduli of vector bundles with fixed determinant (cf. [1]). Indeed, we have the following commutative diagram: The following result is proved in Proposition 4.6 of [3] and gives a more detailed information about the structure of M GL(n,R) .
For the space M L,0 (0, z 0 ) we have the decomposition (8.7). The space M ′ L,0 (0, z 0 ) is connected from Corollary 8.2. Let us then analyse the space M ′′ L,0 (0, z 0 ). Consider the non-zero local minima of the Hitchin functional f . From Corollary 7.8, these are such that (8.9) where r = m − 1/2 and F −1/2 is a square root of LK. There are 2 2g different choices for F −1/2 thus the space of local minima of this kind consists of 2 2g isolated points. Therefore M ′′ L,0 (0, z 0 ) has 2 2g connected components. All these are homeomorphic to a vector space and constitute the so-called Hitchin or Teichmüller components of M L,0 [15]. So M L,0 (0, z 0 ) has 2 2g + 1 components.
If follows from (8.8) and from the count above that M L,0 has 2 2g+1 + 1 connected components.
We have computed the components of each fibre of ν 0 . Let us see that the space M 0 has less components than M L,0 .
such that each EGL(n, R)-Higgs bundle is a local minimum of type (2) on the corresponding fibre of ν 0 (see (8.2)). Hence where F −1/2 (resp. F ′ −1/2 ) is a square root of LK (resp. L ′ K). Since Jac(X) is connected, there is a path L t in Jac(X) joining L to L ′ . Set is a path in M 0 joining (V, L, Q, Φ) and (V ′ , L ′ , Q ′ , Φ ′ ) and such that, for every t, (V t , L t , Q t , Φ t ) is a minimum of f in M Lt,0 of type (2). Hence we conclude that all the 2 2g Hitchin components of all fibres of ν 0 join together to form a unique component of M 0 : M ′′ 0 (0, z 0 ) = L∈Jac(X) M ′′ L,0 (0, z 0 ). Note that this is not a Hitchin component. Indeed, the group EGL(n, R) is not a split real form (due to U(1)), so the moduli space of EGL(n, R)-Higgs bundles on X was not expected to have a Hitchin component (cf. [15]).
Let us now deal with the space M 1 .

Topology of M SL(3,R)
In this subsection we shall consider the lower rank case of SL(3, R)-Higgs bundles. In holomorphic terms these are triples (V, Q, Φ) where V is holomorphic vector bundle equipped with a nowhere degenerate quadratic form Q and with trivial determinant, and Φ is a traceless K-twisted endomorphism of V , symmetric with respect to Q.
Let M SL(3,R) be the moduli space of SL(3, R)-Higgs bundles. These objects are classified by the second Stiefel-Whitney class w 2 ∈ {0, 1}, and let M SL(3,R) (w 2 ) be the subspace of M SL(3,R) whose elements have the given w 2 .
The moduli space M SL(3,R) was considered in [15] where the minimum subvarieties of the Hitchin functional were studied. There it was shown that if (V, Q, Φ) represents a fixed point of the circle action (7.3), with Φ = 0, then V is of the form for all j. From this and since rk(V ) = 3, we conclude that fixed points with non-zero Higgs field are precisely those such that with rk(F j ) = 1 and, if j = 1, Φ j : F j → F j+1 ⊗ K is an isomorphism. These are local minima of the Hitchin function f . The corresponding connected component, the Hitchin component, being isomorphic to a vector space, is contractible.
For each w 2 ∈ {0, 1}, let M ′ SL(3,R) (w 2 ) be the subspace of M SL(3,R) (w 2 ) such that the minima on each of its connected components have Φ = 0. Given (V, Q, Φ) ∈ M SL(3,R) (w 2 ), we know from [31], that This map, together with the previous discussion, provides the following result (recall that, from [15], we know that M SL(3,R) has 3 components). Proof. The first part has already been discussed. For the second part, we have to see that the map F defined in (9.1) is continuous, providing then a retraction from M ′ SL(3,R) (w 2 ) into N SO(3,C) (w 2 ), for each value of w 2 . When t = 0, the continuity of F is obvious. We will take care of the case t = 0.
The space M GL(3,C) is a quasi-projective, algebraic variety and C * acts algebraically on it as (V, Φ) → (V, tΦ). Linearise this action with respect to an ample line bundle N (such that N s is very ample) over M GL (3,C) . This C * -action induces one on N s and, therefore we obtain a C * -action on H 0 (M GL (3,C) , N s ) given by (t · s)(V, Φ) = t · (s(V, t −1 Φ)).
One can choose a rank r + 1, C * -invariant subspace W ⊆ H 0 (M GL (3,C) , N s ) and hence C * acts on W . From this action we obtain a C * -action on P r ∼ = P(W ), and there is a C * -equivariant locally closed embedding (9.2) ι : M GL(3,C) ֒→ P r .
If we linearise the given C * -action on P r with respect to the very ample O P r (1), then this is compatible with the morphism (9.2) and with the isomorphism N s ∼ = O P r (1). Now, we can decompose W as where r i = rk(W r i ) and C * acts over each W r i as v → t α i v, t ∈ C * , α i ∈ Z and α i < α j whenever i < j. So, for each r i , we have a subspace of P r given by P r i −1 = P(W r i ). With respect to the above decomposition of W , C * acts as Then, we also have the induced C * -action on the closed subspace M ′ SL(3,R) (w 2 ) and a C *equivariant topological embedding (9.4) ι| M ′ SL(3,R) (w 2 ) : M ′ SL(3,R) (w 2 ) ֒→ P r and we denote the image in P r of M ′ SL(3,R) (w 2 ) through ι| M ′ SL(3,R) (w 2 ) also by M ′ SL(3,R) (w 2 ). So we view M ′ SL(3,R) (w 2 ) not as a subvariety of P r , but simply a closed subspace (for the complex topology). But we already know that Fix C * (M ′ SL(3,R) (w 2 )) = N SO(3,C) (w 2 ) which is an irreducible variety, by Theorem 5.9 of [23]. So we conclude that (9.5) Fix C * (M ′ SL(3,R) (w 2 )) = M ′ SL(3,R) (w 2 ) ∩ P r i 0 −1 for some i 0 ∈ {1, . . . , k}.
If we take the mapF : P r × [0, 1] → P r given bỹ then it is well-defined by the definition of i 0 in (9.6) and it is clearly continuous because i 0 is constant. By the compatibility of the actions, we have that F corresponds, under (9.4), tõ F | M ′ SL(3,R) (w 2 )×[0,1] , so F is also continuous.

Connected components of spaces of representations
Recall that our main goal is to compute the number of components of R PGL(n,R) for n 4 even, but we had to work with the group EGL(n, R). The work done also gives a way to count the components of the subspace of R = R Γ,EGL(n,R) given by the disjoint union R 0 ⊔ R 1 . Denote this subspace by R 0,1 .
Proof. The result follows immediately from the existence of the surjective continuous map p : R → R PGL(n,R) satisfying the identities of Proposition 6.16 and from the previous proposition. Note that the two components of R 0 (0, z 0 ) are not mapped into only one in R PGL(n,R) (0, z 0 ) because if that were the case, every representation in PGL(n, R) with (0, z 0 ) as invariants could deform to a representation into PO(n), the maximal compact, and then the same would occur for the group EGL(n, R). We know however that this is not possible because of the analysis of the minima with invariants (0, z 0 ): the component with minima with Φ = 0 corresponds precisely to those representations which do not deform to a representation in EO(n). On the other hand, PGL(n, R) is a split real form so by [15] the space R PGL(n,R) should have a Hitchin component which in this case corresponds to the representations which do not deform to PO(n). Remark 10.3. For the proof of Theorem 10.2 is not essential to have Proposition 10.1. We could have used Propositions 8.5 and 8.7 and noticed that the vector bundles corresponding to minima of f of type (2) are projectively equivalent. This would give us the number of components of M PGL(n,R) (µ 1 , µ 2 ), therefore of R PGL(n,R) (µ 1 , µ 2 ) from Theorem 2.8.
Remark 10.4. If µ 1 = 0, then we might expect to get the same components as Hitchin did in [15] but that does not happen. We computed 4 components while Hitchin's result was 6. The difference is that we are considering PGL(n, R)-equivalence (cf. Remark 3.9), while Hitchin considered PSL(n, R)-equivalence.