TOPOLOGICAL INVARIANTS OF ANOSOVREPRESENTATIONSOLIVIER GUICHARD AND ANNA WIENHARDAbstra t. We de�ne new topologi al invariants for Anosov represen-tations and study them in detail for maximal representations of thefundamental group of a losed oriented surfa e Σ into the symple ti group Sp(2n,R). In parti ular we show that the invariants distinguish onne ted omponents of the spa e of symple ti maximal representa-tions. Sin e the invariants behave naturally with respe t to the a tionof the mapping lass group of Σ, we obtain from this the number of omponents of the quotient by the mapping lass group a tion.For spe i� symple ti maximal representations we ompute the in-variants expli itly. This allows us to onstru t ni e model representa-tions in all onne ted omponents. The onstru tion of model represen-tations is of parti ular interest for Sp(4,R), be ause in this ase thereare −1 − χ(Σ) onne ted omponents in whi h all representations areZariski dense and no model representations were known so far. Finally,we use the model representations to draw on lusion about the holonomyof symple ti maximal representations.1. Introdu tionLet Σ be a losed oriented onne ted surfa e of negative Euler hara ter-isti , G a onne ted Lie group. The obstru tion to lifting a representationρ : π1(Σ) → G to the universal over of G is a hara teristi lass of ρ whi his an element of H2(Σ;π1(G)) ∼= π1(G).When G is ompa t it is a onsequen e of the famous paper of Atiyah andBott [2℄ that the onne ted omponents of

Hom(π1(Σ), G)/Gare in one-to-one orresponden e with the elements of π1(G). When G is a omplex Lie group the analogous result has been onje tured by Goldman[18℄ and proved by Li [29℄.When G is a real non- ompa t Lie group, this orresponden e between onne ted omponents of Hom(π1(Σ), G)/G and elements of π1(G) fails.Date: O tober 22, 2009.2000 Mathemati s Subje t Classi� ation. 57M50, 20H10.Key words and phrases. Surfa e Groups, Maximal Representations, Chara teristi In-variants, Representation Varieties, Anosov Stru tures.A.W. was partially supported by the National S ien e Foundation under agreement No.DMS-0604665 and DMS-0803216. O.G. was partially supported by the Agen e Nationalede la Re her he under ANR's proje t Repsurf : ANR-06-BLAN-0311.1

2 OLIVIER GUICHARD AND ANNA WIENHARDObviously hara teristi lasses of representations still distinguish ertain onne ted omponents of Hom(π1(Σ), G)/G, but they are not su� ient todistinguish all onne ted omponents.Here are some examples of this phenomenon:(i) For n ≥ 3, the hara teristi lass of a representation of π1(Σ) intoPSL(n,R) is an element of Z/2Z. But the spa eHom(π1(Σ),PSL(n,R))/PGL(n,R)has three onne ted omponents [24℄.(ii) For representations of π1(Σ) into PSL(2,R) the Euler number doesdistinguish the 4g − 3 onne ted omponents [18℄. For representa-tions into SL(2,R) the Euler number is not su� ient to distinguish onne ted omponents, there are 22g+1 +2g− 3 omponents, and inparti ular there are 22g omponents of maximal (or minimal) Eulernumber, ea h of whi h orresponds to the hoi e of a spin stru tureon Σ.(iii) For representations of π1(Σ) into Sp(2n,R) the hara teristi lassis an integer whi h generalizes the Euler number. It belongs toH2(Σ;π1(Sp(2n,R))) ∼= Z and is bounded in absolute value by n(g−

1). The subspa e of representations where it equals n(g−1) is alledthe spa e of maximal representations. This subspa e de omposesinto several onne ted omponents, 3×22g when n ≥ 3 [14℄ and (3×22g+2g−4) when n = 2 [19℄. The spa e of maximal representationsand its onne ted omponents are in detail dis ussed in this arti le.We introdu e new topologi al invariants for representations ρ : π1(M) →

G, whenever ρ is an Anosov representation. Let us sket h the de�nition (seeSe tion 2.1 for details). Let M be a ompa t manifold equipped with anAnosov �ow. A representation ρ : π1(M) → G is said to be a (G,H)-Anosovrepresentation if the asso iated G/H-bundle over M admits a se tion whi his onstant along the �ow with ertain ontra tion properties.Theorem 1. Let ρ : π1(M) → G be a (G,H)-Anosov representation. Thenthere is a prin ipal H-bundle over M anoni ally asso iated to ρ, whosetopologi al type gives topologi al invariants of ρ. There is well de�ned mapHomH-Anosov(π1(M), G) −→ BH(M),where HomH-Anosov(π1(M), G) denotes the subspa e of Anosov representa-tions and BH(M) the set of gauge isomorphism lasses of prin ipal H-bundlesover M . This map is natural with respe t to:� taking overs of M ,� ertain morphisms of pairs (G,H) → (G′, H ′) (see Lemma 2.8).1.1. Maximal representations into Sp(2n,R). Our main fo us lies onmaximal representations into Sp(2n,R). Maximal representations into

TOPOLOGICAL INVARIANTS 3Sp(2n,R) are (Sp(2n,R),GL(n,R))-Anosov representations [9℄. More pre- isely, let M be the unit tangent bundle T 1Σ with respe t to some hyper-boli metri on Σ, then the geodesi �ow is an Anosov �ow on M . Thefundamental group π1(M) is a entral extension of π1(Σ) and omes witha natural proje tion π : π1(M) → π1(Σ). Let ρ : π1(Σ) → Sp(2n,R) be amaximal representation, then the omposition ρ ◦ π : π1(M) → Sp(2n,R)is a (Sp(2n,R),GL(n,R))-Anosov representation. The topologi al invari-ants obtained by Theorem 1 are the hara teristi lasses of a GL(n,R)-bundle overM . We only onsider the �rst and se ond Stiefel-Whitney lassessw1(ρ ◦ π) ∈ H1(T 1Σ;F2) and sw2(ρ ◦ π) ∈ H2(T 1Σ;F2).Theorem 2. Let ρ : π1(Σ) → Sp(2n,R) be a maximal representation. Thenthe topologi al invariants sw1(ρ) = sw1(ρ ◦ π) ∈ H1(T 1Σ;F2) and sw2(ρ) =sw2(ρ ◦ π) ∈ H2(T 1Σ;F2) are subje t to the following onstraints:(i) The image of

sw1 : Hommax(Γ, G) −→ H1(T 1Σ;F2)is ontained in one oset of H1(Σ;F2).� For n even, sw1(ρ) is in H1(Σ;F2) ⊂ H1(T 1Σ;F2),� for n odd, sw1(ρ) is in H1(T 1Σ;F2) −H1(Σ;F2).(ii) The image ofsw2 : Hommax(Γ, G) −→ H2(T 1Σ;F2)lies in the image of H2(Σ;F2) → H2(T 1Σ;F2).In the ase when n = 2, that is for maximal representation ρ : π1(Σ) →Sp(4,R), let Hommax,sw1=0(π1(Σ), Sp(4,R)) denote the subspa e of maximalrepresentations where the �rst Stiefel-Whitney lass vanishes. This meansthat the GL(2,R)-bundle over T 1Σ admits a redu tion of the stru ture groupto GL+(2,R), equivalently it means that the orresponding R

2-ve tor bun-dle is orientable. A redu tion of the stru ture group to GL+(2,R) gives riseto an Euler lass, but sin e an orientable bundle does not have a anoni alorientation this redu tion is not anoni al. To ir umvent this problem, weintrodu e an enhan ed representation spa e, whi h involves the hoi e of anontrivial element γ ∈ π1(Σ). For pairs (ρ, L+) onsisting of a maximalrepresentation with vanishing �rst Stiefel-Whitney lass and an oriented La-grangian L+ ⊂ R4 whi h is �xed by ρ(γ), there is a well-de�ned Euler lass(see Se tion 4.3).Theorem 3. Let ρ : π1(Σ) → Sp(4,R) be a maximal representation with

sw1(ρ) = 0. Let γ ∈ π1(Σ) − {1} and L+ an oriented Lagrangian �xedby ρ(γ). Then the Euler lass eγ(ρ, L+) ∈ H2(T 1Σ;Z) lies in the image ofH2(Σ;Z) → H2(T 1Σ;Z).For every possible topologi al invariant satisfying the above onstraintswe onstru t expli it representations ρ : π1(Σ) → Sp(2n,R) realizing thisinvariant (see Se tion 1.3 and Se tion 3). From this we dedu e a lower

4 OLIVIER GUICHARD AND ANNA WIENHARDbound on the onne ted omponents of the spa e Hommax(π1(Σ), Sp(2n,R))of maximal representations.Proposition 4. Let Hommax(π1(Σ), Sp(2n,R)) be the spa e of maximal rep-resentations.(i) If n ≥ 3 the spa e Hommax(π1(Σ), Sp(2n,R)) has at least 3 × 22g onne ted omponents.(ii) The spa e Hommax(π1(Σ), Sp(4,R)) has at least 3 × 22g + 2g − 4 onne ted omponents.Our method (so far) only gives a lower bound on the number of onne ted omponents. To obtain an exa t ount of the number of onne ted ompo-nents using the invariants de�ned here, a loser analysis for surfa es withboundary would be ne essary (see [18℄ for the ase when n = 1).Fortunately, the orresponden e between (redu tive) representations andHiggs bundles allows to use algebro-geometri methods to study the topologyof Rep(π1(Σ), G) := Hom(π1(Σ), G)/G. These methods have been developedby Hit hin [23℄ and applied to representations into Lie groups of Hermitiantype in [19, 15, 14, 7, 8℄ leading to the exa t ount mentioned above (iii).Combining Proposition 4 with this exa t ount we an on lude that theinvariants de�ned here distinguish onne ted omponents.More pre isely, let HomHit hin(π1(Σ), Sp(2n,R)) be the spa e of Hit hinrepresentations; by de�nition it is the union of the onne ted omponents ofHom(π1(Σ), Sp(2n,R)) ontaining representations of the form φirr ◦ ι whereι : π1(Σ) → SL(2,R) is a dis rete embedding and φirr : SL(2,R) →Sp(2n,R) is the irredu ible representation of SL(2,R) of dimension 2n.Hit hin representations are maximal representations.Theorem 5. Let n ≥ 3. Then the topologi al invariants of Theorem 2distinguish onne ted omponents of Hommax − HomHit hin. More pre isely,

Hommax(π1(Σ), Sp(2n,R)) − HomHit hin(π1(Σ), Sp(2n,R))sw1,sw2−−−−−→ H1(M ;F2) ×H2(M ;F2)is a bije tion onto the set of pairs satisfying the onstraints of Theorem 2.It is easy to see that, when n is even, the �rst Stiefel-Whitney lass ofa Hit hin representation vanishes, i.e. one has the in lusion HomHit hin ⊂

Hommax,sw1=0.Theorem 6. The Euler lass de�nes a mapHommax,sw1=0(π1(Σ), Sp(4,R))−HomHit hin(π1(Σ), Sp(4,R)) −→ H2(M ;Z)whi h is a bije tion onto the image of H2(Σ;Z) in H2(M ;Z). In parti u-lar, the Euler lass distinguishes onne ted omponents in Hommax,sw1=0 −HomHit hin.

TOPOLOGICAL INVARIANTS 5The omponents of Hommax−Hommax,sw1=0 are distinguished by the �rstand se ond Stiefel-Whitney lasses, i.e. the mapHommax(π1(Σ), Sp(4,R)) − Hommax,sw1=0(π1(Σ), Sp(4,R))

sw1,sw2−−−−−→(H1(Σ;F2) − {0}

)×H2(Σ;F2)is a bije tion.Remark 1. Hit hin representations are not only (Sp(2n,R),GL(n,R))-Anosov representations, but (Sp(2n,R), A)-Anosov representations, where

A is the subgroup of diagonal matri es [27℄. Applying Theorem 1 to the pair(G,H) = (Sp(2n,R), A) one an de�ne �rst Stiefel-Whitney lasses swA1 (ρ)in H1(T 1Σ;F2), similarly to the above dis ussion those invariants are shownto belong to H1(T 1Σ;F2)−H1(Σ;F2) and distinguish the 22g onne ted om-ponents of HomHit hin(π1(Σ), Sp(2n,R)).Remark 2. The existen e of 22g Hit hin omponents is due to the pres-en e of a enter in Sp(2n,R): they all proje t to the same omponent inHom(π1(Σ),PSp(2n,R)). The abundan e of non-Hit hin onne ted ompo-nents in the spa e of maximal representations pertains even when we on-sider representations into the adjoint group, and is in fa t explained by theinvariants we are de�ning here. In this spe ial ase these invariants an benon-trivial due to the non-trivial topology of GL(n,R) (or PGL(n,R)).1.2. The a tion of the mapping lass group. The �rst and se ondStiefel-Whitney lasses of a maximal representation ρ do not hange if ρ is onjugated by an element Sp(2n,R). Thus, they give well de�ned fun tions:(1) swi : Repmax(π1(Σ), Sp(2n,R)) −→ Hi(T 1Σ;F2).The mapping lass group Mod(Σ) a ts by pre omposition on Repmax; thisa tion is properly dis ontinuous [34, 28℄ and by Theorem 1 the map (1) isequivariant with respe t to this a tion and the natural a tion of Mod(Σ) onHi(T 1Σ;F2).For the Euler lass eγ (see Theorem 3) there is a orresponding statementof equivarian e for the subgroup of Mod(Σ) �xing the homotopy lass of γ.This allows us to determine the number of onne ted omponents ofRepmax(π1(Σ), Sp(2n,R))/Mod(Σ).Theorem 7. If n ≥ 3, the spa e Repmax(π1(Σ), Sp(2n,R))/Mod(Σ) has 6 onne ted omponents.The spa e Repmax(π1(Σ), Sp(4,R))/Mod(Σ) has 2g+2 onne ted ompo-nents.1.3. Model representations. Given two representations it is in generalvery di� ult to determine whether they lie in the same onne ted om-ponent or not. The invariants de�ned here an be omputed rather ex-pli itly and hen e allow us to de ide in whi h onne ted omponent ofHommax(π1(Σ), Sp(2n,R)) a spe i� representation lies in. We apply this

6 OLIVIER GUICHARD AND ANNA WIENHARDto onstru t parti ularly ni e model representations in all onne ted ompo-nents.An easy way to onstru t maximal representations ρ : π1(Σ) → Sp(2n,R)is by omposing a dis rete embedding π1(Σ) → SL(2,R) with a tight ho-momorphism of SL(2,R) into Sp(2n,R) (see [11℄ for the notion of tight ho-momorphism). For example, omposing with the 2n-dimensional irredu iblerepresentation of SL(2,R) into Sp(2n,R) we obtain an irredu ible Fu hsianrepresentation. Hit hin representations are pre isely deformations of su hrepresentations. Composing with the diagonal embedding of SL(2,R) intothe subgroup SL(2,R)n < Sp(2n,R) we obtain a diagonal Fu hsian represen-tation. These representations have a big entralizer be ause the entralizer ofthe image of SL(2,R) under the diagonal embedding is isomorphi to O(n).Any representation an be twisted by a representation into its entralizer,thus any diagonal Fu hsian representation an be twisted by a representationπ1(Σ) → O(n), de�ning a twisted diagonal representations. A representationobtained by one of these onstru tions will be alled a standard maximalrepresentation (see Se tion 3.2).Theorem 8. Let n ≥ 3. Then every maximal representation ρ : π1(Σ) →Sp(2n,R) an be deformed to a standard maximal representation.Our omputations of the topologi al invariants in Se tion 5 give morepre ise information on when a maximal representation an be deformed toan irredu ible Fu hsian or a diagonal Fu hsian representation.Corollary 9. Let ρ : π1(Σ) → Sp(2n,R) be a maximal representation. Thenρ an be deformed either to an irredu ible Fu hsian representation or to adiagonal Fu hsian representation if(i) for n = 2m, m > 2, sw1(ρ) = 0 and sw2(ρ) = −mχ(Σ)

2 mod 2,(ii) for n = 2m+ 1, sw2(ρ) = −mχ(Σ)2 mod 2.Remark 3. Another orollary of Theorem 8 is that for n ≥ 3 every maximalrepresentation an be deformed to a maximal representation whose image is ontained in a proper losed subgroup of Sp(2n,R). This on lusion analso be obtained from [14℄ be ause the Higgs bundles for standard maximalrepresentation an be des ribed quite expli itly.The ase of Sp(4,R) is di�erent. If one ombines the ount of the on-ne ted omponents of Repmax(π1(Σ), Sp(4,R)) in [19℄ with results abouttight homomorphisms in [11℄ one an on lude that there are 2g − 3 ex ep-tional omponents, in whi h every representation is Zariski dense (see [6℄ fora detailed dis ussion of this).To onstru t model representations in these omponents, we de ompose

Σ = Σl ∪ Σr into two subsurfa es and de�ne a representation of π1(Σ) byamalgamation of an irredu ible Fu hsian representation of π1(Σl) with adeformation of a diagonal Fu hsian representation of π1(Σr). We all theserepresentations hybrid representations (see Se tion 3.3.1 for details).

TOPOLOGICAL INVARIANTS 7We ompute the topologi al invariants of these representations expli itly.Allowing the Euler hara teristi of the subsurfa e Σl to vary between 3−2gand −1, we obtain 2g−3 hybrid representations whi h exhaust the 2g−3 ex- eptional omponents of the spa e of maximal representations into Sp(4,R).We on ludeTheorem 10. Every maximal representation ρ : π1(Σ) → Sp(4,R) an bedeformed to a standard maximal representation or a hybrid representation.Remark 4. To obtain Theorem 10 it is essential that we are able to omputethe topologi al invariants expli itly. Geometri ally there is no obvious reasonwhy di�erent hybrid representations lie in di�erent onne ted omponents.In parti ular, our results on the topologi al invariants imply that other on-stru tions by amalgamation (see Se tion 3.3.3) give representations whi h an be deformed to twisted diagonal representations.1.4. Holonomies of maximal representations. A dire t onsequen eof the fa t that maximal representations ρ : π1(Σ) → Sp(2n,R) are(Sp(2n,R),GL(n,R))-Anosov representations is that the holonomy ρ(γ) is onjugate to an element of GL(n,R) for every γ ∈ π1(Σ). More pre iselyρ(γ) �xes two transverse Lagrangians, one, Ls, being attra tive, the otherbeing repulsive. From this it follows that the holonomy ρ(γ) is an elementof GL(Ls) whose eigenvalues are stri tly bigger than one.For representations in the Hit hin omponents we have moreover thatρ(γ) ∈ GL(Ls) is a regular semi-simple element [27℄. This does not holdfor other onne ted omponents of Hommax(π1(Σ), Sp(2n,R)). Using thedes ription of model representations in Theorem 8 and Theorem 10 we proveTheorem 11. Let H be a onne ted omponent ofRepmax(π1(Σ), Sp(2n,R)) − RepHit hin(π1(Σ), Sp(2n,R)),and let γ ∈ π1(Σ) − {1} be an element orresponding to a simple urve. Ifγ is separating, n = 2 and the genus of Σ is 2, we require that H is not the onne ted omponent determined by sw1 = 0 and eγ = 0. Then there exist(i) a representation ρ ∈ H su h that the Jordan de omposition of ρ(γ)in GL(Ls) ∼= GL(n,R) has a nontrivial paraboli omponent.(ii) a representation ρ′ ∈ H su h that the Jordan de omposition of ρ(γ)in GL(Ls) ∼= GL(n,R) has a nontrivial ellipti omponent.1.5. Other maximal representations. Maximal representations ρ : π1(Σ) →G an be de�ned whenever G is a Lie group of Hermitian type, and they arealways (G,H)-Anosov representations [10℄, where H is a spe i� subgroupof G (see Theorem 2.15). When G is not lo ally isomorphi to Sp(2n,R)there is no analogue of Hit hin representations1 and we onje ture:1Hit hin omponents an be de�ned for any R-split semisimple Lie group (see [24℄)and the only simple R-split Lie groups of Hermitian type are the symple ti groups.

8 OLIVIER GUICHARD AND ANNA WIENHARDConje ture 12. Let G be a simple Lie group of Hermitian type. If G is notlo ally isomorphi to Sp(2n,R), then the topologi al invariants of Theorem 1distinguish onne ted omponents of Repmax(π1(Σ), G).If the real rank of G is n, then there is an embedding of SL(2,R)n intoG, it is unique up to onjugation. Thus, there is always a orrespondingdiagonal embedding of SL(2,R) intoG. The entralizer of whi h is a ompa tsubgroup of G. In parti ular, one an always onstru t twisted diagonalrepresentations.Conje ture 13. Let G be of Hermitian type. If G not lo ally isomorphi to Sp(2n,R), then every maximal representation ρ : π1(Σ) → G an bedeformed to a twisted diagonal representation.If Conje ture 13 holds the analogue of Theorem 11 will also hold.1.6. Comparison with Higgs bundle invariants. We already mentionedthat the orresponden e between (redu tive) representations and Higgs bun-dles permits to use algebro-geometri methods to study the stru ture ofRep(π1(Σ), G), and in parti ular to ount the number of onne ted om-ponents. Where these methods have been applied to study representationsinto Lie groups of Hermitian type, see [19, 15, 14, 7, 8℄, the authors asso iatespe ial ve tor bundles to the Higgs bundles, whose hara teristi lasses giveadditional invariants for maximal representations π1(Σ) → G; then theyshow that for any possible value of the invariants the orresponding modulispa e of Higgs bundles is non-empty and onne ted.For maximal representations into Sp(2n,R) the Higgs bundle invariantsare �rst and se ond Stiefel-Whitney lasses taking values in H∗(Σ;F2); forn = 2 there is also an Euler number taking values in H2(Σ;Z).We on lude the introdu tion with several remarks on erning the relationbetween the topologi al invariants de�ned here and the invariants obtainedvia Higgs bundles:(i) The Higgs bundle invariants depend on various hoi es, i.e. theydepend on the hoi e of a omplex stru ture on Σ and on the hoi eof a square-root of the anoni al bundle of Σ (i.e. a spin stru tureon Σ). The Stiefel-Whitney lasses we de�ne here are natural and donot depend on any hoi es. In parti ular, they are equivariant underthe a tion of the mapping lass group of Σ. The invariants de�nedhere are also natural with respe t to taking �nite index subgroupsof π1(Σ) and with respe t to tight homomorphisms.(ii) The invariants de�ned here an be omputed for expli it represen-tations, whi h is very di� ult for the Higgs bundle invariants. This omputability is essential to determine in whi h onne ted ompo-nents spe i� representations lie. This is of parti ular interest forthe 2g−3 ex eptional onne ted omponents when n = 2, be ause inthese onne ted omponents no expli it representations were knownbefore.

TOPOLOGICAL INVARIANTS 9(iii) The Higgs bundle approa h has the feature that the L2-norm of theHiggs �eld gives a Morse-Bott fun tion on the moduli spa e, whi hallows to perform Morse theory on the representation variety. Infa t, for the symple ti stru ture on the moduli spa e, this fun tionis the Hamiltonian of a ir le a tion, so that its riti al points areexa tly the �xed points of this ir le a tion. These �xed points areHiggs bundles of a very spe ial type, alled �variations of Hodgestru tures�. An additional information oming from this frameworkis that one is able to read the index of the riti al submanifoldsfrom the eigenvalues of the ir le a tion on the tangent spa e at a�xed point. This is used to give an exa t ount of the onne ted omponents in many ases, as well as to obtain further importantinformation about the topology of the representation variety. Formore details on this strategy we refer the reader to Hit hin's arti le[24℄.For symple ti maximal representations there is a simple relation be-tween the two invariants although they live naturally in di�erent ohomologygroups.Proposition 14. Let ρ : π1(Σ) → Sp(2n,R) be a maximal representa-tion. Denote by wi(ρ, v) ∈ Hi(Σ;F2), i = 1, 2 the �rst and se ond Stiefel-Whitney lasses asso iated to the Higgs bundle orresponding to ρ, wherev ∈ H1(T 1Σ;F2) − H1(Σ;F2) is the hosen spin stru ture on Σ. Then wehave the following equality in Hi(T 1Σ;F2):

sw1(ρ) = w1(ρ, v) + n · vsw2(ρ) = w2(ρ, v) + sw1(ρ) · v + (g − 1) mod 2.When n = 2, the Higgs bundle invariant orresponding to the Euler lassis the �rst Chern lass of a line bundle on Σ, whi h we denote by c(ρ) ∈H2(Σ;Z). The suspe ted relation is

c(ρ) = εe(ρ) + (g − 1)in Tor(H2(T 1Σ;Z)), where ε depends on the hoi es of orientation involvedin the de�nition of e(ρ) and c(ρ).The existen e of su h relations is not surprising sin e the invariants arisebasi ally from the same ompa t Lie group. Nevertheless, it would be inter-esting to provide a general proof for these relations for all maximal repre-sentations. The relations in Proposition 14 are obtained from ase by ase onsiderations for model representations, using the tensor produ t formulasof Equation (3).1.7. Stru ture of the paper. In Se tion 2 we re all the de�nition andproperties of Anosov representations and of maximal representations. Ex-amples of su h representations are dis ussed in Se tion 3. The topologi alinvariants are de�ned in Se tion 4 and omputed for symple ti maximalrepresentations in Se tion 5. Se tion 6 dis usses the a tion of the mapping

10 OLIVIER GUICHARD AND ANNA WIENHARD lass group; Se tion 7 derives onsequen es for the holonomy of symple -ti maximal representations. In Appendix A we establish several importantfa ts about positive urves and maximal representations; in Appendix B wereview some fa ts about the ohomology of the unit tangent bundle T 1Σ.A knowledgments. The dis ussions leading to this paper started duringa workshop on �Surfa e group representations� taking pla e at the Ameri- an Institute for Mathemati s in 2007. The authors thank the University ofChi ago, the Institut des Hautes Études S ienti�ques and Prin eton Univer-sity for making mutual intensive resear h visits possible.2. Preliminaries2.1. Anosov representations. Holonomy representations of lo ally homo-geneous geometri stru tures are very spe ial. The Thurston-Ehresmanntheorem [17, 5℄ states that every deformation of su h a representation anbe realized through deformations of the geometri stru ture. The on ept ofAnosov stru tures, introdu ed by Labourie in [27℄, gives a dynami al gener-alization of this, whi h is more �exible, but for whi h it is still possible toobtain enough rigidity of the asso iated representations.2.1.1. De�nition. Let� M be a ompa t manifold with an Anosov �ow φt,� G a onne ted semisimple Lie group and (P s, P u) a pair of oppositeparaboli subgroups of G,� H = P s ∩ P u their interse tion, and� Fs = G/P s (resp. Fu = G/P u) the �ag variety asso iated to P s(resp. P u).There is a unique open G-orbit X ⊂ Fs × Fu. We have X = G/H and asopen subset of Fs × Fu it inherits two foliations Es and Eu whose orre-sponding distributions are denoted by Es and Eu, i.e. (Es)(fs,fu)∼= TfsFsand (Eu)(fs,fu)

∼= TfuFu.De�nition 2.1. A �at G-bundle P over (M,φt) is said to have an H-redu tion σ that is �at along �ow lines if:� σ is a se tion of P ×G X ; i.e. σ : M → P ×G X de�nes the H-redu tion2� the restri tion of σ to every orbit of φt is lo ally onstant with respe tto the indu ed �at stru ture on P ×G X .The two distributions Es and Eu on X are G-invariant and hen e de�nedistributions, again denoted Es and Eu, on P×GX . These two distributionsare invariant by the �ow, again denoted by φt, that is the lift of the �ow onM by the onne tion.2For details on the bije tive orresponden e between H-redu tions and se tions ofP ×G G/H we refer the reader to [31, Se tion 9.4℄.

TOPOLOGICAL INVARIANTS 11To every se tion σ of P ×G X we asso iate two ve tor bundles σ∗Es andσ∗Eu on M by pulling ba k to M the ve tor bundles Es and Eu. If further-more σ is �at along �ow lines, so that it ommutes with the �ow, then thesetwo ve tor bundles σ∗Es and σ∗Eu are equipped with a natural �ow.De�nition 2.2. A �at G-bundle P → M is said to be a (G,H)-Anosovbundle if:(i) P admits an H-redu tion σ that is �at along �ow lines, and(ii) the �ow φt on σ∗Es (resp. σ∗Eu) is ontra ting (resp. dilating).We all σ an Anosov se tion or an Anosov redu tion of P.By (ii) we mean that there exists a ontinuous family of norms (‖·‖m)m∈Mon σ∗Es (resp. σ∗Eu) and onstants A, a > 0 su h that for any e in σ∗(Es)m(resp. σ∗(Eu)m) and for any t > 0 one has

‖φte‖φtm ≤ A exp(−at)‖e‖m (resp. ‖φ−te‖φ−tm ≤ A exp(−at)‖e‖m).Sin e M is ompa t this de�nition does not depend on the norm ‖ · ‖ or theparametrization of φt.De�nition 2.3. A representation π1(M) → G is said to be (G,H)-Anosov(or simply Anosov) if the orresponding �at bundle is a (G,H)-Anosov bun-dle.Remark 2.4. Note that the terminology for Anosov representations is not ompletely uniform. A (G,H)-Anosov representation is sometimes alled a(G,X )-Anosov representation or an Anosov representation with respe t tothe paraboli subgroup P s or P u.2.1.2. Properties. For the de�nition of topologi al invariants of Anosov rep-resentations in Se tion 4 the following proposition will be ru ial.Proposition 2.5. Let P →M be an Anosov bundle, then there is a uniquese tion σ : M → P ×G X su h that properties (i) and (ii) of De�nition 2.2hold. In parti ular, a (G,H)-Anosov bundle admits a anoni al H-redu tion.To prove Proposition 2.5 we will use the following lassi al fa t.Fa t 2.6. Suppose that g ∈ G has two �xed points fs ∈ Fs, fu ∈ Fu su hthat (i) fs and fu are in general position, i.e. (fs, fu) belongs to X ⊂

Fs ×Fu; and(ii) the (linear) a tion of g on the tangent spa e TfsFs is ontra ting,the a tion on TfuFu is expanding.Then fs is the only attra ting �xed point of g in Fs. Its attra ting set isthe set of all �ags that are in general position with fu, and fu is the onlyrepelling �xed point for g in Fu.Proof of Proposition 2.5. Let σ be an Anosov se tion of the �at bundle P →M . The �ow φt on M is (by hypothesis) an Anosov �ow. Be ause of the

12 OLIVIER GUICHARD AND ANNA WIENHARDdensity of losed orbits in M , it is enough to show that the restri tion of σto any losed orbit γ is uniquely determined.We identify γ with Z\R and write the restri tion of P to γ as P|γ =

Z\(R×G) where Z a ts on R×G by n · (t, g) = (n+ t, hnγg) for some hγ inG. (The element hγ is the holonomy of P along γ.)Therefore, the restri tion σ|γ is a se tion of Z\(R × X ) (the Z-a tionbeing n · (t, x) = (n + t, hnγ · x)). Sin e σ is lo ally onstant along γ thereexists x = (fs, fu) in X su h that the lift of σ|γ to R is the map R →R × X , t 7→ (t, x). Hen e fs and fu are hγ-invariant. Furthermore therestri tions of σ∗Es, resp. σ∗Eu to γ are in this ase Z\(R × TfsFs), resp.Z\(R×TfuFu), so that the ontra tion property of De�nition 2.2(ii) exa tlytranslates into the assumption of Fa t 2.6. This implies the uniqueness of(fs, fu) and hen e the uniqueness of σ|γ . �An important feature of Anosov representations is their stability underdeformation.Proposition 2.7. [27, Proposition 2.1.℄[20℄ The set of Anosov representa-tions is open in Hom(π1(M), G). Moreover, the H-redu tion given by theAnosov se tion σ depends ontinuously on the representation.2.1.3. Constru tions. Some simple onstru tions allow to obtain new Anosovbundles from old ones.Lemma 2.8. (i) Let ρ : π1(M) → G be a (G,H)-Anosov representa-tion, where H = P s ∩P u for two opposite paraboli subgroups in G.Let Qs, Qu be opposite paraboli subgroups in G su h that P s < Qsand P u < Qu. Then ρ is also a (G,H ′)-Anosov representation,where H ′ = Qs ∩Qu.(ii) Let P be an (G,H)-Anosov bundle over M with anoni al H-redu tion PH and E a �at L-bundle over M . Then the �bered prod-u t P×E is a (G×L,H×L)-Anosov bundle overM , whose anoni al

(H × L)-redu tion is the �bered produ t PH × E.(iii) Let P be an (G,H)-Anosov bundle over M with anoni al H-redu tion PH , where H = P s∩P u. Let f : G→ L be a homorphismof Lie groups, let Qs, Qu be a pair of opposite paraboli subgroupsin L su h that f−1(Qs) = P s, f−1(Qu) = P u and they are maximalwith respe t to this property, i.e. for any Q ontaining Qs (resp.Qu) satisfying f−1(Q) = P s (resp. P u) one has Q = Qs (resp.Q = Qu).Then P ×G L is an (L,M)-Anosov bundle, where M = Qs ∩ Quand its anoni al M -redu tion is the �bered produ t PH ×H M .2.1.4. De�nition in terms of the universal over of M. A (G,H)-Anosovbundle over M an be equivalently de�ned in terms of equivariant mapsfrom the universal over of M to X ∼= G/H. Let M̃ be the universal over

TOPOLOGICAL INVARIANTS 13of M . Then any �at G-bundle P on M an be written as:P = π1(M)\(M̃ ×G), γ(m̃, g) = (γ · m̃, ρ(γ)g)for some representation ρ : π1(M) → G.Let φt be the lift of the �ow on M to M̃ . This �ow lifts to φt(m̃, g) =

(φt(m̃), g), de�ning a �ow on M̃ ×G.An H-redu tion σ is the same as a ρ-equivariant mapσ̃ : M̃ −→ G/H ∼= X .The se tion σ is �at along �ow lines if, and only if, the map σ̃ is φt-invariant.The ontra tion property of the �ow is now expressed as follows:(i) There exists a ontinuous family (‖ · ‖m̃)

m̃∈fMsu h that� for all m̃, ‖ · ‖m̃ is a norm on (Es)σ̃(m̃) ⊂ Tσ̃(m̃)X ,� and (‖ · ‖m̃)

m̃∈fMis ρ-equivariant, i.e. for all m̃ in M̃ , γ in

π1(M) and e in (Es)σ̃(m̃) then ‖ρ(γ) · e‖γ·m̃ = ‖e‖m̃.(ii) The �ow φt is ontra ting. i.e. there exist A, a > 0 su h thatfor any t > 0 and m̃ in M̃ and e in (Es)σ̃(m̃) then ‖e‖φt·m̃ ≤A exp(−at)‖e‖m̃. (This inequality makes sense be ause, sin e σ̃(φt ·m̃) = σ̃(m̃), e belongs to (Es)σ̃(φt·m̃).)2.1.5. Spe ialization to T 1Σ. We restri t now to the ase when M = T 1Σ isthe unit tangent bundle of a losed oriented onne ted surfa e Σ of negativeEuler hara teristi and φt is the geodesi �ow on M with respe t to somehyperboli metri on Σ.Let ∂π1(Σ) be the boundary at in�nity of π1(Σ). Then ∂π1(Σ) is a topo-logi al ir le that omes with a natural orientation and an a tion of π1(Σ).There is an equivariant identi� ation

T 1Σ̃ ∼= ∂π1(Σ)(3+)of the unit tangent bundle of Σ̃ with the set of positively oriented triples in∂π1(Σ). The orbit of the geodesi �ow through the point (ts, t, tu) is

G(ts,t,tu) = G(ts,tu) = {(rs, r, ru) ∈ ∂π1(Σ)(3+) | rs = ts, ru = tu},and the set of geodesi leaves is parametrized by ∂π1(Σ)(2) = ∂π1(Σ)2 − ∆,the omplementary of the diagonal in ∂π1(Σ)2. (For more details we referthe reader to [21, Se tion 1.1℄.)Let π : T 1Σ → Σ be the natural proje tion.De�nition 2.9. A �at G-bundle P over Σ is said to be Anosov if its pullba k π∗P over M = T 1Σ is Anosov.A representation ρ : π1(Σ) → G is Anosov if the omposition π1(M) →π1(Σ)

ρ→ G is an Anosov representation.Remark 2.10. Note that for a (G,H)-Anosov bundle P over Σ, the pull-ba k π∗P over T 1Σ admits a anoni al H-redu tion, but this H-redu tion in

14 OLIVIER GUICHARD AND ANNA WIENHARDgeneral does not ome from a redu tion of P. Indeed, the invariants de�nedin Se tion 4 also give obstru tions for this to happen.Let ρ : π1(Σ) → G be an Anosov representation and P the orresponding�at G-bundle over Σ. Using the des ription in Se tion 2.1.4 it follows thatthere exists a ρ-equivariant map:σ̃ : T 1Σ̃ −→ X ⊂ Fs ×Fuwhi h is invariant by the geodesi �ow.In parti ular we get a ρ-equivariant map

(ξs, ξu) : ∂π1(Σ)(2) −→ Fs ×Fu.In view of the ontra tion property of σ̃ (De�nition 2.2(ii)) it is easy tosee that ξs(ts, tu) (resp. ξu(ts, tu)) depends only of ts (resp. tu). Hen e, weobtain ρ-equivariant maps ξs : ∂π1(Σ) → Fs, and ξu : ∂π1(Σ) → Fu.Corollary 2.11. Let ρ : π1(Σ) → G be a (G,H)-Anosov representation.Then for every γ ∈ π1(Σ)−{1} the image ρ(γ) is onjugate to an element inH, having a unique pair of attra ting/repelling �xed points (ξs(tsγ), ξ

u(tuγ)) ∈X , where (tsγ , t

uγ) denotes the pair of attra ting/repelling �xed points of γ in

∂π1(Σ).Remark 2.12. In the situation when P s is onjugate with P u, so that thereis a natural identi� ation between Fs and Fu, it is easy to show the equalityξs = ξu (see [27℄ or [20℄ for a similar dis ussion), thus in those ases we willdenote the equivariant map simply by ξ : ∂π1(Σ) → F .2.2. Maximal representations.2.2.1. De�nition and properties. Let G be an almost simple non ompa tLie group of Hermitian type, i.e. the symmetri spa e asso iated to Gis an irredu ible Hermitian symmetri spa e of non- ompa t type. Thenπ1(G) ∼= Z modulo torsion, and there is a hara teristi lass τ(ρ) ∈H2(Σ;π1(G)) ∼= π1(G) ∼= Z, often alled the Toledo invariant of the repre-sentation ρ : π1(Σ) → G. The Toledo invariant τ(ρ) is bounded in absolutevalue by a onstant C(G,Σ) depending only on the real rank of G and theEuler hara teristi of Σ:

|τ(ρ)| ≤ C(G,Σ)De�nition 2.13. A representation ρ : π1(Σ) → G is maximal ifτ(ρ) = C(G,Σ).The spa e of maximal representation is denoted by Hommax(π1(Σ), G).The spa e of maximal representations is a union of onne ted omponentsof Hom(π1(Σ), G).

TOPOLOGICAL INVARIANTS 15Remark 2.14. In the de�nition of maximal representations we hoose thepositive extremal value of the Toledo invariant. The spa e where the negativeextremal value is a hieved an be easily seen to be isomorphi to the spa eof maximal representations by �a hange of orientation� ( e.g. for a standardgenerating set {ai, bi}1≤i≤g of π1(Σ) the automorphism σ of π1(Σ) de�ned byai 7→ bg+1−i, bi 7→ ag+1−i pre isely hanges the sign of τ : τ(ρ ◦ σ) = −τ(ρ)for any representation ρ).Maximal representations have been extensively studied in the last years[18, 33, 22, 7, 12, 13, 9, 34, 8℄. They enjoy several interesting properties, e.g.maximal representations are dis rete embeddings, but more importantly forour onsiderations is the followingTheorem 2.15. [9, 10℄ A maximal representation ρ : π1(Σ) → G is anAnosov representation. More pre isely, ρ is a (G,H)-Anosov representation,where H < G is the stabilizer of a pair of transverse points in the Shilovboundary of the symmetri spa e asso iated to G.We will make use of the following gluing theorem, whi h follows immedi-ately from additivity properties of the Toledo invariant established in [13,Proposition 3.2.℄, to onstru t maximal representations.Theorem 2.16. [13℄ Let Σ = Σ1 ∪γ Σ2 be the de omposition of Σ along asimple losed separating geodesi and π1(Σ) = π1(Σ1) ∗〈γ〉 π1(Σ2) the or-responding de omposition as amalgamated produ t. Let ρi : π1(Σi) → G bemaximal representations whi h agree on γ, then the amalgamated represen-tation

ρ = ρ1 ∗ ρ2 : π1(Σ) −→ Gis maximal.2.2.2. Maximal representations into Sp(2n,R). Let V = R2n be a symple -ti ve tor spa e and (ei)1≤i≤2n a symple ti basis, with respe t to whi h thesymple ti form ω is given by the anti-symmetri matrix:

J =

(0 Idn

−Idn 0

).Let G = Sp(2n,R) = Sp(V ).Let Ls0 := Span(ei)1≤i≤n be a Lagrangian subspa e and P s < Sp(2n,R)be the paraboli subgroup stabilizing Ls0. The stabilizer of the Lagrangian

Lu0 = Span(ei)n<i≤2n is a paraboli subgroup P u, it is opposite to P s. Thesubgroup H = P s ∩ P u is isomorphi to GL(n,R):GL(n,R)∼−→ H ⊂ G

A 7−→(A 00 tA−1

).(Su h isomorphisms are in one to one orresponden e with symple ti bases

(ǫi)1≤i≤2n �i.e. ω(ǫi, ǫj) = ω(ei, ej) for all i, j� for whi h (ǫi)1≤i≤n is abasis of Ls0). Note that P u is onjugate to P s in G, so that the �ag variety

16 OLIVIER GUICHARD AND ANNA WIENHARDFs is anoni ally isomorphi to Fu; we will denote this homogeneous spa eby L. The spa e L is the Shilov boundary of the symmetri spa e asso iatedto Sp(2n,R); it an be realized as the spa e of Lagrangian subspa es in R

2nand the homogeneous spa e X ⊂ L × L is the spa e of pairs of transverseLagrangians.Let ρ : π1(Σ) → Sp(2n,R) be a maximal representation and P the orre-sponding �at prin ipal Sp(2n,R)-bundle over T 1Σ and E the orresponding�at symple ti R2n-bundle over T 1Σ. Then ρ is an (Sp(2n,R),GL(n,R))-Anosov representation (Theorem 2.15). The anoni al GL(n,R)-redu tionof P is equivalent to a ontinuous splitting of E into two (non-�at) �ow-invariant transverse Lagrangian subbundles

E = Ls(ρ) ⊕ Lu(ρ),Notation 2.17. We all this splitting the Lagrangian redu tion of the �atsymple ti Anosov R2n-bundle.The existen e of a Lagrangian redu tion is equivalent to the existen e ofa ontinuous ρ-equivariant urve ξ : ∂π1(Σ) → L, sending distin t pointsin ∂π1(Σ) to transverse Lagrangians. The onstru tion of a ρ-equivariantlimit urve ξ : ∂π1(Σ) → L is a tually the �rst step to prove that a maximalrepresentation ρ : π1(Σ) → Sp(2n,R) is an (Sp(2n,R),GL(n,R))-Anosovrepresentation, see [9℄. This limit urve satis�es an additional positivityproperty whi h we now des ribe.Let (Ls, L, Lu) be a triple of pairwise transverse Lagrangians in R

2n, thenL an be realized as the graph of FL ∈ Hom(Ls, Lu).De�nition 2.18. A triple (Ls, L, Lu) of pairwise transverse Lagrangians is alled positive if the quadrati form q(v) := ω(v, FL(v)) on Ls is positivede�nite.De�nition 2.19. A urve ξ : ∂π1(Σ) → L from the boundary at in�nity ofπ1(Σ) to the spa e of Lagrangians is said to be positive, denoted by ξ > 0,if for every positively oriented triple (ts, t, tu) in ∂π1(Σ)(3+) the triple ofLagrangians (ξ(ts), ξ(t), ξ(tu)) is positive.Important fa ts about the spa e of positive urves in L are established inAppendix A.1.Theorem 2.20. [13℄ Let ρ ∈ Hom(π1(Σ), Sp(2n,R)) be a maximal represen-tation and ξ : ∂π1(Σ) → L the equivariant limit urve. Then ξ is a positive urve.As a onsequen e of Proposition 2.7 we haveFa t 2.21. The positive limit urve ξ : ∂π1(Σ) → L depends ontinuouslyon the representation.Given a ρ-equivariant limit urve ξ : ∂π1(Σ) → L the onstru tion of asplitting

E = Ls(ρ) ⊕ Lu(ρ)

TOPOLOGICAL INVARIANTS 17is immediate: for any triple v = (ts, t, tu) ∈ ∂π1(Σ)(3+) ∼= T 1Σ̃, we set(Ls(ρ))v = ξ(ts) and (Lu(ρ))v = ξ(tu).In the following we will often swit h between the three di�erent viewpoints: GL(n,R)-redu tion of P, splitting E = Ls(ρ) ⊕ Lu(ρ) or (positive)equivariant urve ξ : ∂π1(Σ) → L.3. Examples of representations3.1. Anosov representations. We give examples of Anosov representa-tions. By Proposition 2.7 every small deformation of one of these represen-tations is again an Anosov representation.3.1.1. Hyperbolizations. Let Σ be a onne ted oriented losed hyperboli sur-fa e and M = T 1Σ its unit tangent bundle equipped with the geodesi �owφt. Hyperbolizations give rise to dis rete embeddings π1(Σ) → PSL(2,R)whi h are examples of Anosov representations (see [27, Proposition 3.1℄).More generally, a dis rete embedding of π1(Σ) into any �nite over L ofPSL(2,R) is an Anosov representation. Sin e PSL(2,R) has rank one thereis no hoi e for the paraboli subgroup.Later we will be interested in parti ular in Anosov bundles arising fromdis rete embeddings ι : π1(Σ) → SL(2,R). In that ase H = GL(1,R) isthe subgroup of diagonal matri es and the H-redu tion orresponds to asplitting of the �at R

2-bundle over M into two line bundles Ls(ι) ⊕ Lu(ι).3.1.2. Hit hin representations. A representation of π1(Σ) intoG = SL(n,R),Sp(2m,R) or SO(m,m + 1) is said to be a Hit hin representation if it an be deformed into a representation π1(Σ)ι−→ SL(2,R)

τn−→ G, wherethe homomorphism ι : π1(Σ) → SL(2,R) is a dis rete embedding andτn : SL(2,R) → G is the n-dimensional irredu ible representation, wheren = 2m when G = Sp(2m,R) and n = 2m + 1 when G = SO(m,m + 1).Hit hin representations are (G,H)-Anosov, where H is the subgroup of di-agonal matri es in of G [27℄. In parti ular, the H-redu tion orresponds toa splitting of the �at R

n-bundle over M into n line bundles.3.1.3. Other examples.(i) Any quasi-Fu hsian representation π1(Σ) → PSL(2,C) is Anosov.(ii) Embed SL(2,R) into PGL(3,R) as stabilizer of a point and on-sider the representation ρ : π1(Σ) → SL(2,R) → PGL(3,R). Thenspe ial deformations of ρ are (PGL(3,R), H)-Anosov, where H isthe interse tion of two opposite minimal paraboli subgroups ofPGL(3,R) [3℄.(iii) Let G be a semisimple Lie group, G′ < G a rank one subgroup, Λ <G′ a o ompa t torsionfree latti e and N = Λ\G′/K, where K < G′is a maximal ompa t subgroup. Let M = T 1N be the unit tangentbundle of N . Then the omposition π1(M) → Λ → G′ < G is a(G,H)-Anosov representation, where H is the onne ted omponent

18 OLIVIER GUICHARD AND ANNA WIENHARDof the identity of the entralizer in G of a real split Cartan subgroupin G′ ( ompare with Lemma 2.8), see [27, Proposition 3.1℄.(iv) In [30, 4℄ a notion of quasi-Fu hsian representations for a o om-pa t latti e Λ < SO◦(1, n) into SO◦(2, n) is introdu ed, and it isshown that these quasi-Fu hsian representations are Anosov repre-sentations.3.2. Standard maximal representations. In this se tion we des ribe the onstru tion of several maximal representationsρ : π1(Σ) −→ Sp(2n,R)to whi h we will refer as standard representations. All these representations ome from homomorphisms of SL(2,R) into Sp(2n,R), possibly twisted bya representation of π1(Σ) into the entralizer of the image of SL(2,R) inSp(2n,R). By onstru tion the image of any su h representation will be ontained in a proper losed Lie subgroup of Sp(2n,R).3Let us �x a dis rete embedding ι : π1(Σ) → SL(2,R).3.2.1. Irredu ible Fu hsian representation. Consider V0 = R1[X,Y ] ∼= R

2the spa e of homogeneous polynomials of degree one in the variables X andY , endowed with the symple ti form determined by

ω0(X,Y ) = 1.The indu ed a tion of Sp(V0) on V = Sym2n−1V0∼= R2n−1[X,Y ] ∼= R

2npreserves the symple ti form ωn = Sym2n−1ω0, whi h is given byωn(Pk, Pl) = 0 if k + l 6= 2n− 1 and ωn(Pk, P2n−1−k) = (−1)k

(n

k

)−1

,where Pk = X2n−1−kY k.This de�nes the 2n-dimensional irredu ible representation of Sp(V0) ∼=SL(2,R) into Sp(V ) ∼= Sp(2n,R),φirr : SL(2,R) −→ Sp(2n,R),whi h, by pre omposition with ι : π1(Σ) → SL(2,R), gives rise to an irre-du ible Fu hsian representation

ρirr : π1(Σ) −→ SL(2,R) −→ Sp(2n,R).Fa ts 3.1. (i) Let Ls(ι) be the line bundle over T 1Σ asso iated to theembedding ι : π1(Σ) → SL(2,R), and Eι, Eρirrthe �at symple ti bundles over T 1Σ. As Eρirr

= Sym2n−1Eι and Eι = Ls(ι)⊕Lu(ι) =Ls(ι) ⊕ Ls(ι)−1, the Lagrangian redu tion Ls(ρirr) over T 1Σ asso- iated to ρirr is

Ls(ρirr) = Ls(ι)2n−1 ⊕ Ls(ι)2n−3 ⊕ · · · ⊕ Ls(ι).3More pre isely, ρ(π1(Σ)) will preserve a totally geodesi tight disk in the symmetri spa e asso iated to Sp(2n,R) (the notion of tight disk is not used in this paper, theinterested reader is referred to [11℄).

TOPOLOGICAL INVARIANTS 19(ii) When n = 2, let us hoose the symple ti identi� ation Sym3R

2 ∼=R3[X,Y ] ∼= R

4 given by X3 = e1, X2Y = −e2/

√3, Y 3 = −e3, XY 2 =

e4/√

3. With respe t to this identi� ation the irredu ible represen-tation φirr : SL(2,R) → Sp(4,R) is given by the following formulaφirr

(a bc d

)=

a3 −√

3a2b −b3 −√

3ab2

−√

3a2c 2abc+ a2d√

3b2d 2abd+ b2c

−c3√

3c2d d3√

3cd2

−√

3ac2 2acd+ bc2√

3bd2 2bcd+ ad2

.In parti ular, for g =

(em 00 e−m

) one hasφirr(g) =

e3m 0 0 00 em 0 00 0 e−3m 00 0 0 e−m

.3.2.2. Diagonal Fu hsian representations. Let

R2n = W1 ⊕ · · · ⊕Wnwith Wi = Span(ei, en+i) be a symple ti splitting of R2n. Identifying Wi

∼=R

2, this splitting gives rise to an embeddingψ : SL(2,R)n −→ Sp(W1) × · · · × Sp(Wn) ⊂ Sp(2n,R).Pre omposing with the diagonal embedding of SL(2,R) → SL(2,R)n weobtain a diagonal embedding

φ∆ : SL(2,R) −→ Sp(2n,R).By pre omposition with ι : π1(Σ) → SL(2,R) we obtain a diagonal Fu hsianrepresentationρ∆ : π1(Σ) −→ SL(2,R) −→ Sp(2n,R).Fa ts 3.2. (i) Let Ls(ι) be the Lagrangian line bundle over T 1Σ asso i-ated to ι : π1(Σ) → SL(2,R), then the Lagrangian redu tion Ls(ρ∆)of the �at symple ti R

2n-bundle over T 1Σ asso iated to ρ∆ is givenbyLs(ρ∆) = Ls(ι) ⊕ · · · ⊕ Ls(ι).(ii) When n = 2 and with respe t to the symple ti basis (ei)i=1,...,4 themap ψ is given by the following formula

ψ

((a bc d

),

(α βγ δ

))=

a 0 b 00 α 0 βc 0 d 00 γ 0 δ

.

20 OLIVIER GUICHARD AND ANNA WIENHARD3.2.3. Twisted diagonal representations. We now vary the onstru tion of theprevious subse tion. For this note that the image φ∆(SL(2,R)) < Sp(2n,R)has a fairly large entralizer, whi h is a ompa t subgroup of Sp(2n,R)isomorphi to O(n).Remark 3.3. For any maximal representation ρ : π1(Σ) → Sp(2n,R) the entralizer of ρ(π1(Σ)) is a subgroup of O(n). This is be ause the entralizerof ρ(π1(Σ)) will �x the positive urve in the spa e of Lagrangians pointwise.In parti ular, it will be ontained in the stabilizer of one positive triple ofLagrangians whi h is isomorphi to O(n).That the entralizer of φ∆(SL(2,R)) is pre isely O(n) an be seen in thefollowing way. Let (W, q) be an n-dimensional ve tor spa e equipped witha de�nite quadrati form q and let again V0 = R1[X,Y ] ∼= R2 with itsstandard symple ti form ω0. The tensor produ t V0 ⊗W inherits a bilinearnondegenerate form ω0 ⊗ q whi h is easily seen to be antisymmetri , so thatwe an hoose a symple ti identi� ation R

2n ∼= V0 ⊗ W . This gives anembedding SL(2,R) ×O(n) ∼= Sp(V0) ×O(W, q)φ∆−−→ Sp(2n,R),whi h extends the morphism φ∆ de�ned above.Now given ι : π1(Σ) → SL(2,R) and a representation Θ : π1(Σ) → O(n),we set

ρΘ = ι⊗ Θ : π1(Σ) −→ Sp(V ) ∼= Sp(2n,R)

γ 7−→ φ∆(ι(γ),Θ(γ)).We will all su h a representation a twisted diagonal representation.Fa ts 3.4. (i) The �at bundle E over Σ asso iated to ρΘ : π1(Σ) →Sp(2n,R) ∼= Sp(V0 ⊗W ) is of the formE = E0 ⊗W,where (with a slight abuse of notation) W is the �at n-plane bundleasso iated to Θ and E0 the �at plane bundle over Σ asso iated to ι.(ii) Let Ls(ι) be the line bundle over T 1Σ asso iated to ι. Let W denotethe �at n-bundle over T 1Σ given by the pull-ba k of W . Then theLagrangian redu tion Ls(ρΘ) is given by the tensor produ t

Ls(ρΘ) = Ls(ι) ⊗W.3.2.4. Standard representations for other groups. Let G be an almost simpleLie group of Hermitian type of real rank n. Then there exist a (up to onju-gation by G) unique embedding SL(2,R)n → G. We all the pre ompositionof su h an embedding with the diagonal embedding SL(2,R) → SL(2,R)na diagonal embeddingφ∆ : SL(2,R) → G.The entralizer of φ∆(SL(2,R)) in G is always a ompa t subgroup K ′ of G.

TOPOLOGICAL INVARIANTS 21The omposition ρ∆ = φ∆ ◦ ι : Γ → G is a maximal representation.Given a representation Θ : Γ → K ′ we an again de�ne a twisted diagonalrepresentationρΘ : π1(Σ) → Sp(V )

γ 7→ ρ∆(γ) · Θ(γ).Remark 3.5. In the general ase the subgroup K ′ an be hara terized asbeing the interse tion of the maximal ompa t subgroup K in G with thesubgroup H < G, whi h is the stabilizer of a pair of transverse point in theShilov boundary of the symmetri spa e asso iated to G. Equivalently, it isthe stabilizer in G of a maximal triple of points in the Shilov boundary. (Forthe de�nition of maximal triples see [13, Se tion 2.1.3.℄.)3.3. Amalgamated representations. Due to Theorem 2.16 we an on-stru t maximal representations of π1(Σ) by amalgamation of maximal rep-resentations of the fundamental groups of subsurfa es.Let Σ = Σl∪γΣr be a de omposition of Σ along a simple losed separatingoriented geodesi γ into two subsurfa es Σl, lying to the left of γ, and Σr,lying to the right of γ. Then π1(Σ) is isomorphi to π1(Σl)∗〈γ〉π1(Σr), wherewe identify γ with the element it de�nes in π1(Σ).We will all a representation onstru ted by amalgamation of two repre-sentations ρl : π1(Σl) → G and ρr : π1(Σr) → G with ρl(γ) = ρr(γ) anamalgamated representation ρ = ρl ∗ ρr : π1(Σ) → G. By Theorem 2.16,the amalgamated representation ρ is maximal if and only if ρl and ρr aremaximal.43.3.1. Hybrid representations. In this se tion we des ribe the most impor-tant lass of maximal representations ρ : π1(Σ) → Sp(4,R) obtained viaamalgamation. We all these representation hybrid representations to dis-tinguish them from general amalgamated representations.Let ι : π1(Σ) → SL(2,R) be a dis rete embedding. The basi idea of the onstru tion of hybrid representations is to amalgamate the restri tion ofthe irredu ible Fu hsian representation ρirr = φirr ◦ ι : π1(Σ) → Sp(4,R) toΣl and the restri tion of the diagonal Fu hsian representation ρ∆ = φ∆ ◦ ι :π1(Σ) → Sp(4,R) to Σr. This does not dire tly work be ause the holonomiesof ρirr and ρ∆ along γ do not agree, but a slight modi� ation works.Assume that ι(γ) =

(em 00 e−m

) with m > 0. Setρl := φirr ◦ ι : π1(Σ) → Sp(4,R),4Note that it is important that the Toledo invariant for both ρl and ρr are of the samesign. Amalgamating a maximal representation with a minimal representation does notgive rise to a maximal representation.

22 OLIVIER GUICHARD AND ANNA WIENHARDwith φirr de�ned in Fa ts 3.1(ii). Then ρl(γ) =

e3m 0 0 00 em 0 00 0 e−3m 00 0 0 e−m

.Let (τ1,t)t∈[0,1] and (τ2,t)t∈[0,1] be ontinuous paths of dis rete embeddings

π1(Σ) → SL(2,R) su h that τ1,0 = τ2,0 = ι, and, for all t ∈ [0, 1],τ1,t(γ) =

(el1,t

e−l1,t

) and τ2,t(γ) =

(el2,t

e−l2,t

),where l1,t > 0 and l2,t > 0, l1,0 = l2,0 = m, l1,1 = 3m and l2,1 = m. Theexisten e of τi,t is a lassi al fa t from Tei hmüller theory, for the reader's onvenien e we in lude the statement we are using in Lemma A.3. Set

ρr := ψ ◦ (τ1,1, τ2,1) : π1(Σ) −→ Sp(4,R).Then ρr is a ontinuous deformation of φ∆ ◦ ι whi h satis�es ρr(γ) = ρl(γ).We now de�ne a hybrid representation by(2) ρ := ρl|π1(Σl) ∗ ρr|π1(Σr) : π1(Σ) = π1(Σl) ∗〈γ〉 π1(Σr) −→ Sp(4,R).Sin e ρl|π1(Σl) and ρr|π1(Σr) are maximal representations, the representationρ is maximal (see Theorem 2.16).Remark 3.6. The spe ial hoi es for the embeddings φirr and ψ (Fa ts 3.1(ii)and 3.2(ii)) will be important for the al ulation of the Euler lass of a hybridrepresentation. Obviously one an always hange one of this two embeddingsby onjugation by an element of the entralizer of ρ(γ), i.e. an element,whi h, with respe t to a suitable basis, is of the form diag(a, b, a−1, b−1).In order to keep tra k of this situation we de�neDe�nition 3.7. Let γ be a urve5 on Σ and let ρl and ρr two representationsof π1(Σ) into Sp(4,R) with ρl(γ) = ρr(γ) and su h that ρl is a Hit hinrepresentation and ρr is a maximal representation into SL(2,R)×SL(2,R) <Sp(4,R).The pair (ρl, ρr) is said to be positively adjusted with respe t to γ if thereexists a symple ti basis (ǫi)i=1,...,4 and ontinuous deformations (ρl,t)t∈[0,1]and (ρr,t)t∈[0,1] su h that:� ρl,1 = ρl and ρr,1 = ρr,� ρl,0 = φirr ◦ ι is an irredu ible Fu hsian representation, ρr,0 = φ∆ ◦ ιis a diagonal Fu hsian representation and for ea h t in [0, 1] ρr,t isa maximal representation into SL(2,R) × SL(2,R) < Sp(4,R),� for ea h t, ρl,t(γ) and ρr,t(γ) are diagonal in the base (ǫi).The pair (ρl, ρr) is said to be negatively adjusted with respe t to γ ifthe pair (gρlg

−1, ρr) is positively adjusted where g is diagonal in the base(ǫi)i=1,...,4 with eigenvalues of di�erent signs, i.e. g = diag(a, b, a−1, b−1)with ab < 0.5The urve γ does not need to be separating or simple.

TOPOLOGICAL INVARIANTS 23Remark 3.8. In fa t, it is not really ne essary here that the representationsρl, ρr were representations of π1(Σ), what matters in the omputation isthat the representation ρl|π1(Σl) is the restri tion of a Hit hin representationof some losed surfa e, and similarly that ρr|π1(Σr) is the restri tion of adeformation of a diagonal maximal representation of some losed surfa e.Sin e hybrid representations are maximal representations ρ : π1(Σ) →Sp(4,R) the asso iated �at symple ti R

4-bundle E admits a Lagrangiansplitting E = Ls(ρ) ⊕ Lu(ρ). We are unable to des ribe the Lagrangianbundles Ls(ρ) and Lu(ρ) expli itly as we did above for standard maximalrepresentations. Nevertheless, omputing the topologi al invariants (see Se -tion 5) we de�ned for Anosov representations, we will be able to determinethe topologi al type of Ls(ρ). The topologi al type will indeed only dependon the Euler hara teristi of Σl.De�nition 3.9. If χ(Σl) = k we all the representation de�ned by (2) ak-hybrid representation.3.3.2. Hybrid representations: general onstru tion. In the onstru tionabove we de ompose Σ along one simple losed separating geodesi , so theEuler hara teristi of Σl will be odd. To obtain k-hybrid representations ofπ1(Σ) for all χ(Σ) + 1 ≤ k ≤ −1 we have to onsider slightly more generalde ompositions of Σ, in parti ular Σl or Σr might not be onne ted.Let us �x some notation to des ribe this more general onstru tion. LetΣ be losed oriented surfa e of genus g and Σ1 ⊂ Σ be a subsurfa e withEuler hara teristi equal to k.The (non-empty) boundary ∂Σ1 is the union of disjoints ir les γd ford ∈ π0(∂Σ1). We orient the ir les su h that, for ea h d, the subsurfa e Σ1is on the left of γd. Write the surfa e Σ− ∂Σ1 as the union of its onne ted omponents:

Σ − ∂Σ1 =⋃

c∈π0(Σ−∂Σ1)

Σc.For any d in π0(∂Σ1) the urve γd bounds exa tly 2 onne ted omponentsof Σ − ∂Σ1: one is in luded in Σ1 and denoted by Σl(d) with l(d) in π0(Σ1);the other one is in luded in the omplementary of Σ1 and denoted by Σr(d)with r(d) in π0(Σ−Σ1). Note that l(d) and r(d) are elements of π0(Σ−∂Σ1).In parti ular, l(d) might equal l(d′) for some d 6= d′, similarly for r(d) andr(d′).We assume that� The graph with verti es set π0(Σ − ∂Σ1) and edges given by thepairs {(l(d), r(d))}d∈π0(∂Σ1) is a tree.The fundamental group π1(Σ) an be des ribed as the amalgamatedprodu t of the groups π1(Σc), c in π0(Σ − ∂Σ1), over the groups π1(γd),d ∈ π0(∂Σ1). The above assumption ensures that no HNN-extensions ap-pear in this des ription.

24 OLIVIER GUICHARD AND ANNA WIENHARDWith these notations, we an now de�ne general k-hybrid representations.For ea h c in π0(Σ1) we hoose a representationρc : π1(Σ) −→ Sp(4,R)belonging to one of the 22g Hit hin omponents. We set with a slight abuseof notation ρc = ρc|π1(Σc) for ea h c in π0(Σ1).For any d in π0(∂Σ1), ρl(d)(γd) (this makes sense sin e l(d) ∈ π0(Σ1)) is onjugate to a unique element of the form

ρl(d)(γd) ∼= ǫ(d)

el1(d)

e−l1(d)

el2(d)

e−l2(d)

∈ Sp(4,R)with ǫ(d) ∈ {±1}, l1(d) > l2(d) > 0.The onstru tion of ρc′ for c′ in π0(Σ − Σ1) now goes as follow. ByLemma A.3 one an hoose a ontinuous path

τc′,t : π1(Σ) −→ SL(2,R), t ∈ [1, 2]su h that τc′,t : π1(Σ) → SL(2,R) are dis rete embeddings for all t ∈ [1, 2]and su h that, for any d in π0(∂Σc′) ⊂ π0(∂Σ1), (hen e r(d) = c′) one hasτc′,i(γd) is onjugate to ǫ(d)( eli(d)

e−li(d)

), for i = 1, 2.Set

ρc′ =

(τc′,1|π1(Σc′ )

τc′,2|π1(Σc′ )

): π1(Σc′) −→ SL(2,R)×SL(2,R) ⊂ Sp(4,R).In order to de�ne the amalgamated representation, we need to hooseelements gc in Sp(4,R) for any c in π0(Σ − ∂Σ1) su h that, for any d,

gl(d)ρl(d)(γd)g−1l(d) = gr(d)ρr(d)(γd)g

−1r(d). As mentioned in Remark 3.6 theseelements should be hosen su h that� for any d in π0(∂Σ1) the pair of representations(

gl(d)ρl(d)g−1l(d), gr(d)ρr(d)g

−1r(d)

)is positively adjusted with respe t to γd (De�nition 3.7).Su h a family (gc) always exists by our hypothesis that the graph asso iatedto the de omposition of the surfa e Σ is a tree. One then onstru ts thek-hybrid representation

ρ : π1(Σ) −→ Sp(4,R)by amalgamation of the representations gl(d)ρl(d)g−1ld and gr(d)ρr(d)(γd)g−1

r(d)Remark 3.10. The hypothesis that the dual graph of Σ − ∂Σ1 is a tree isne essary. For example, if this graph has a double edge, one would try to on-stru t a Hit hin representation whose restri tion to the disjoint union of two losed simple urves γ1 and γ2 is ontained in some SL(2,R) × SL(2,R) <

TOPOLOGICAL INVARIANTS 25Sp(4,R). But it is not di� ult to see that the restri tion of a Hit hin rep-resentation to the subgroup generated by γ1 and γ2 is irredu ible and hen e annot be ontained in SL(2,R) × SL(2,R) < Sp(4,R).3.3.3. Other amalgamated representations. Let us des ribe a variant of the onstru tion of hybrid representation. Assume that Σ is de omposed alonga simple losed separating geodesi γ into two subsurfa es Σl and Σr asabove. On π1(Σl) we hoose again the irredu ible Fu hsian representationρirr = φirr ◦ ι into Sp(4,R), for the fundamental group of π1(Σr) we hoosea maximal representation into SL(2,R)× SL(2,R) ⊂ Sp(4,R) whi h agreeswith the irredu ible representation along γ, but sends an element α ∈ π1(Σr) orresponding to a non-separating simple losed geodesi to an element ofSL(2,R) × SL(2,R) with eigenvalues of di�erent sign. The orrespondingamalgamated representation will be maximal. But, the �rst Stiefel-Whitney lass of this representation is non-zero. Using the omputations made inSe tion 5 we an on lude that ρ an be deformed to a twisted diagonalrepresentation.The analogous onstru tions an be made to obtain maximal representa-tions into Sp(2n,R) or also into other Lie groups G of Hermitian type. Butin the ase when G is not lo ally isomorphi to Sp(4,R), we expe t that allmaximal representations an be deformed to a twisted diagonal representa-tion. 4. Topologi al invariants4.1. De�nition. Let us denote by HomH-Anosov(π1(M), G) the set of (G,H)-Anosov representations and let BH(M) the set of gauge isomorphism lassesof H-bundles over M . Summarizing Proposition 2.5 and Proposition 2.7 wehaveProposition 4.1. For any pair (G,H), there is a well-de�ned lo ally on-stant map

HomH-Anosov 7−→ BH(M),asso iating to an Anosov representation its H-Anosov redu tion.This map is natural with respe t to taking �nite overs of M and withrespe t to the onstru tions des ribed in Lemma 2.8.In general BH(M) ould be rather ompli ated, so instead of the wholespa e of gauge isomorphism lasses BH(M) we will onsider the obstru tion lasses asso iated to the H-bundle as topologi al invariants of an Anosovrepresentation, i.e. the invariants are elements of the ohomology groups ofM , possibly with lo al oe� ients.Remark 4.2. The group H is the Levi omponent of P s, therefore P s andH are homotopy equivalent. We hen e have BH = BP s, so instead of workingwith the H-redu tion we ould equally well work with the orresponding P s-redu tion (or similarly with the P u-redu tion) of the �at G-bundle

26 OLIVIER GUICHARD AND ANNA WIENHARD4.2. First and se ond Stiefel-Whitney lasses. The in lusionHommax(π1(Σ), Sp(2n,R)) ⊂ HomGL(n,R)-Anosov(π1(Σ), Sp(2n,R)),whi h is des ribed in Se tion 2.2.2, allows to apply Proposition 4.1 to asso- iate to a maximal representation into Sp(2n,R) the �rst and se ond Stiefel-Whitney lasses of a GL(n,R)-bundles over T 1Σ.Proposition 4.3. Let G = Sp(2n,R) and Hommax(π1(Σ), G) the spa eof maximal representations. Then the obstru tion lasses of the anoni alGL(n,R)-redu tion give maps:

Hommax(π1(Σ), G)sw1−−→ H1(T 1Σ;F2)and Hommax(π1(Σ), G)sw2−−→ H2(T 1Σ;F2).The following geometri interpretation makes the �rst Stiefel-Whitney lass sw1(ρ) easy to ompute. Re all that given a losed geodesi γ on

Σ there is a natural lift to a losed loop γ on T 1Σ, this gives rise to a naturalmap π1(Σ) − {1} → H1(T1Σ;Z).Lemma 4.4. Let ρ : π1(Σ) → Sp(2n,R) be a maximal representation and

ξ : ∂π1(Σ) → L the equivariant limit urve. Thensw1(ρ)([γ]) = sign(det ρ(γ)|ξ(tsγ)),where [γ] is the lass of H1(T

1Σ;F2) represented by γ, tsγ ∈ ∂π1(Σ) is theattra tive �xed point of γ and F2 is identi�ed with {±1}.Proof. The �rst Stiefel-Whitney lass of sw1(ρ)([γ]) is the obstru tion to theorientability of the bundle Ls(ρ)|γ ∼= Z\(R × ξ(tsγ)) over γ ∼= Z\R. Thus,sw1(ρ)([γ]) = 1 if ρ(γ) ∈ GL(ξ(tsγ)) lies in the onne ted omponent of theidentity and sw1(ρ)([γ]) = −1 otherwise. �4.3. An Euler lass. For n = 2, the invariants obtained in Proposition 4.3do not allow to distinguish the onne ted omponents of

Hommax(π1(Σ), Sp(4,R)) − HomHit hin(π1(Σ), Sp(4,R)).In parti ular, the se ond Stiefel-Whitney lass does not o�er enough infor-mation to distinguish the onne ted omponents ofHommax,sw1=0(π1(Σ), Sp(4,R)).The reason for this is that when sw1(ρ) = 0 the Lagrangian bundle Ls(ρ) isorientable. So there should be an Euler lass whose image in H2(T 1Σ;F2) isthe se ond Stiefel-Whitney lass sw2(ρ). Sin e an orientable ve tor bundlehas no anoni al orientation, we need to introdu e an �enhan ed� represen-tation spa e to obtain a well-de�ned Euler lass.

TOPOLOGICAL INVARIANTS 274.3.1. Enhan ed representation spa es. Let ρ : π1(Σ) → Sp(4,R) be a max-imal representation. Let γ ∈ π1(Σ)−{1} and Ls(γ) ∈ L the attra tive �xedLagrangian of ρ(γ). We denote by L+ the spa e of oriented Lagrangians andby π : L+ → L the proje tion. LetHomL,γmax,sw1=0(π1(Σ), Sp(4,R)) =

{(ρ, L) ∈ Hommax,sw1=0(π1(Σ), Sp(4,R)) × L | L = Ls(γ)

}andHom

L+,γmax,sw1=0(π1(Σ), Sp(4,R)) ={(ρ, L+) ∈ Hommax,sw1=0(π1(Σ), Sp(4,R)) × L+ | π(L+) = Ls(γ)

}.The map Hom

L+,γmax,sw1=0(π1(Σ), Sp(4,R)) → HomL,γmax,sw1=0(π1(Σ), Sp(4,R))is a 2-fold over.We introdu ed the spa e HomL,γmax,sw1=0(π1(Σ), Sp(4,R)), whi h is easilyidenti�ed with Hommax,sw1=0(π1(Σ), Sp(4,R)), to emphasize the fa t thatρ(γ) has an attra tive Lagrangian.Lemma 4.5. The natural map

HomL,γmax,sw1=0(π1(Σ), Sp(4,R)) −→ L(ρ, L) 7−→ Lis ontinuous.This lemma follows immediately from the ontinuity of the eigenspa e ofa matrix.4.3.2. The Euler lass. As a onsequen e of the following proposition forevery element (ρ, L+) in Hom

L+,γmax,sw1=0 there is a natural asso iated orientedLagrangian bundle over T 1Σ.Proposition 4.6. Let ρ ∈ Hommax(π1(Σ), Sp(2n,R)) and let ξ : ∂π1(Σ) →L be the orresponding equivariant positive urve. Suppose that n is even.Then there exists a ontinuous lift of ξ to L+.Let ξ+ : ∂π1(Σ) → L+ be one of the two ontinuous lifts of ξ. Then themap ξ+ is ρ-equivariant if, and only if, sw1(ρ) = 0. In this ase the otherlift of ξ is also equivariant.Proof. The fa t that a ontinuous lift exists depends only on the homotopy lass of the urve ξ. Sin e the spa e of ontinuous and positive urves is onne ted (see Proposition A.1), the existen e of a lift an be he k for onespe i� maximal representation. Considering a Fu hsian representation givesthe restri tion on n.Let ξ+ be a ontinuous lift of ξ and denote by ξ′+ the other lift of the urveξ. Sin e ξ is equivariant, for any γ in π1(Σ), the following alternative holds:

γ · ξ+ = ξ+ or γ · ξ+ = ξ′+.

28 OLIVIER GUICHARD AND ANNA WIENHARDFurthermore, given any t in ∂π1(Σ), by onne tedness, one observes thatγ · ξ+ = ξ+ ⇐⇒ ξ+(γ · t) = ρ(γ) · ξ+(t).Using this last equation for the attra tive �xed point tsγ of γ in ∂π1(Σ), theequivalen e be omesγ · ξ+ = ξ+ ⇐⇒ det ρ(γ)|ξ(tsγ) > 0.Using now the equality sw1(ρ)([γ]) = sign(det ρ(γ)|ξ(tsγ)) (Lemma 4.4), theproposition follows. �Proposition 4.6 gives a natural way to lift the equivariant urve given anyelement (ρ, L+) ∈ Hom

L+,γmax,sw1=0(π1(Σ), Sp(4,R)):De�nition 4.7. Let (ρ, L+) ∈ HomL+,γmax,sw1=0(π1(Σ), Sp(4,R)) and let ξ :

∂π1(Σ) → L be the ρ-equivariant urve.The lift (uniquely determined and equivariant) ξ+ of ξ su h thatξ+(tsγ) = L+is alled the anoni al oriented equivariant urve for the pair (ρ, L+). Here

tsγ ∈ ∂π1(Σ) is the attra tive �xed point of γ.This anoni al oriented equivariant urve de�nes a GL+(2,R)-redu tionof the GL(2,R)-Anosov redu tion asso iated to ρ. It also indu es a splittingof the �at bundle E asso iated to ρE = Ls+(ρ) ⊕ Lu+(ρ)into two oriented Lagrangians subbundles. Similarly to Notation 2.17 we allthis splitting the oriented Lagrangian redu tion of E.De�nition 4.8. The Euler lass eγ(ρ, L+)

eγ : HomL+,γmax,sw1=0(π1(Σ), Sp(4,R)) −→ H2(T 1Σ;Z)

(ρ, L+) 7−→ eγ(ρ, L+)is the Euler lass of the GL+(2,R)-redu tion given by De�nition 4.7.The map eγ is ontinuous.4.3.3. Conne ted omponents. We onsider now a subspa e of HomL+,γmax,sw1=0by �xing the oriented Lagrangian. Let L0+ ∈ L+ be an oriented Lagrangian;we set

HomL+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R)) =

{(ρ, L+) ∈ HomL+,γmax,sw1=0(π1(Σ), Sp(4,R)) | L+ = L0+}.

TOPOLOGICAL INVARIANTS 29Lemma 4.9. The natural mapHom

L+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R)) −→Hommax,sw1=0(π1(Σ), Sp(4,R))/Sp(4,R)is onto. Its �bers are onne ted.Proof. Let ρ ∈ Hommax,sw1=0(π1(Σ), Sp(4,R)). Sin e the �rst Stiefel-Whitney lass of ρ vanishes, there exists an attra ting oriented Lagrangian

L+ for ρ(γ). Let g ∈ Sp(4,R) su h that g · L+ = L0+, then (gρg−1, L0+)belongs to HomL+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R)) and proje ts to [ρ].Now, let (ρ, L0+) ∈ Hom

L+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R)). Then the �ber of theproje tion ontaining (ρ, L0+) is isomorphi to the quotient:{g ∈ Sp(4,R) | g · L0+ = L0+}/{z ∈ Z(ρ) | z · L0+ = L0+},where Z(ρ) denotes the entralizer of ρ(π1(Σ)) in Sp(4,R). The group {g ∈Sp(4,R) | g · L0+ = L0+} is onne ted, thus the �ber is onne ted. �Lemma 4.9 implies that Hom

L+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R)) has as the samenumber of onne ted omponents as Hommax,sw1=0(π1(Σ), Sp(4,R)).De�nition 4.10. Let γ ∈ π1(Σ) − {1} and L0+ ∈ L+. The relative Eu-ler lass eγ,L0+(ρ) ∈ H2(T 1Σ;Z) of the lass of a maximal representation

ρ ∈ Hommax,sw1=0(π1(Σ), Sp(4,R)) is de�ned by being the Euler lass of one(any) inverse image of [ρ] in the spa e HomL+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R)).4.4. Constraints on invariants. Sin e the �at Sp(2n,R)-bundle over T 1Σwhose GL(n,R)-redu tion gives rise to the invariants of the maximal repre-sentation ρ : π1(Σ) → Sp(2n,R) (Proposition 4.3) is the pull-ba k of a �atbundle over Σ, one expe ts that not every ohomology lass an arise.Proposition 4.11. Let G = Sp(2n,R). Then(i) The image of

sw1 : Hommax(π1(Σ), G) −→ H1(T 1Σ;F2)is ontained in one oset of H1(Σ;F2). More pre isely:� for n even, sw1(ρ) ∈ H1(Σ;F2) ⊂ H1(T 1Σ;F2),� for n odd, sw1(ρ) ∈ H1(T 1Σ;F2) −H1(Σ;F2).(ii) The image ofsw2 : Hommax(π1(Σ), G) −→ H2(T 1Σ;F2)lies in the image of H2(Σ;F2) → H2(T 1Σ;F2).(iii) Similarly, when n = 2, the image of

eγ,L0+: Hommax,sw1=0(π1(Σ), Sp(4,R)) −→ H2(T 1Σ;Z)lies in the image of H2(Σ;Z) → H2(T 1Σ;Z). This holds for anynontrivial γ in π1(Σ) and any oriented Lagrangian L0+ ∈ L+.

30 OLIVIER GUICHARD AND ANNA WIENHARDTo prove this proposition we use the positivity of the equivariant urveξ : ∂π1(Σ) → L of a maximal representation ρ : π1(Σ) → Sp(2n,R) (De�ni-tion 2.19).Proof of Proposition 4.11. For the �rst property, let x ∈ Σ and onsider theexa t sequen eH1(Σ;F2) −→ H1(T 1Σ;F2)

fx−→ H1(T 1xΣ;F2).We need to he k that the image of sw1(ρ) in H1(T 1

xΣ;F2) ∼= F2 does not de-pend on ρ, i.e. that the gauge isomorphism lass of Ls(ρ)|T 1xΣ is independentof ρ.Let x̃ ∈ Σ̃ be a lift of x so that T 1

xΣ ∼= T 1x̃ Σ̃ ∼= ∂π1(Σ), where the lastmap is given by the restri tion of ∂π1(Σ)(3+) → ∂π1(Σ)/(ts, t, tu) 7→ ts to

T 1x̃ Σ̃ ⊂ T 1Σ̃ ∼= ∂π1(Σ)(3+). The restri tion of the �at G-bundle to T 1

xΣ istrivial, thus the restri tion of the Anosov se tion σ to T 1xΣ an be regardedas a map T 1

xΣ → X .By Remark 4.2, we an work with the P s-redu tion, i.e. we onsideronly the �rst omponent of the map σ|T 1xΣ : T 1

xΣ → L × L. With theabove identi� ation T 1xΣ ∼= ∂π1(Σ) this map is exa tly the equivariant limit urve ξ : ∂π1(Σ) → L. Sin e the spa e of positive urves is onne ted(Proposition A.1), fx(sw1(ρ)) is independent of ρ. The al ulation for thediagonal Fu hsian representation (see Se tion 5) gives the desired statementabout sw1(ρ).For the se ond statement, in view of Lemma B.3 it is su� ient to provethat for any losed urve η ⊂ Σ the restri tion of sw2(ρ) (or eγ,L0+

(ρ)) tothe torus T 1Σ|η is zero.For this we write the torus T 1Σ|η as the quotient of ∂π1(Σ)×R by 〈η〉 ∼= Zwhere η · (θ, t) = (η · θ, t+ 1). With this identi� ation the �at bundle an bewritten as:〈γ〉\

(∂π1(Σ) × R ×G

) with γ · (θ, t, g) = (γ · θ, t+ 1, Ag)with A ∈ H ∼= GL(n,R). The se tion of the asso iated L-bundle lifts to∂π1(Σ) × R:

∂π1(Σ) × R −→ L(θ, t) 7−→ ξ(θ).Sin e the spa e of pairs

{(A, ξ) | A ∈ H, ξ > 0, C0 and A-equivariant}has exa tly two onne ted omponents given by the sign of detA (Propo-sition A.2), we on lude that sw2(L

s(ρ)|T 1Σ|η) depends only on this sign,hen e sw2(ρ) depends only on sw1(ρ) (see Lemma 4.4). A dire t al ulationshows that in fa t sw2(Ls(ρ)|T 1Σ|η) is always zero.For the Euler lass the proof goes along the same lines. �In view of Proposition 4.11 we make the following

TOPOLOGICAL INVARIANTS 31De�nition 4.12. Let n ∈ N be even. A pair(α, β) ∈ H1(T 1Σ;F2) ×H2(T 1Σ;F2)is admissible if α lies in the image of H1(Σ;F2) → H1(T 1Σ;F2) and β liesin the image of H2(Σ;F2) → H2(T 1Σ;F2).Let n ∈ N be odd . A pair (α, β) ∈ H1(T 1Σ;F2)×H2(T 1Σ;F2) is admis-sible if α lies in the oset H1(T 1Σ;F2)−H1(Σ;F2), and β lies in the imageof H2(Σ;F2) → H2(T 1Σ;F2).For n = 2, a lass β ∈ H2(T 1Σ;Z) is alled admissible if β lies in theimage of H2(Σ;Z) → H2(T 1Σ;Z).4.5. Invariants for other Anosov representations.4.5.1. Maximal representations into SL(2,R). Let ι : π1(Σ) → SL(2,R) =Sp(2,R) be a maximal representation. The Anosov stru ture asso iated to ρgives a redu tion of the stru ture group of the �at SL(2,R)-prin ipal bundle

P over T 1Σ to GL(1,R). Considering the R2-bundle E asso iated to P, thisredu tion orresponds to a Lagrangian subbundle F of E. As invariants weget the �rst Stiefel-Whitney lass of the line bundle F over T 1Σ,

sw1(ι) ∈ H1(T 1Σ;F2) −H1(Σ;F2).This �rst Stiefel-Whitney lass is pre isely the spin stru ture on Σ whi h orresponds to the hosen lift of π1(Σ) ⊂ PSL(2,R) to SL(2,R). The in-variant sw1(ι) an take 22g di�erent values, distinguishing the 22g onne ted omponents of Hommax(π1(Σ), SL(2,R)).4.5.2. The invariants for Hit hin representations into Sp(2n,R). Let ρ :π1(Σ) → Sp(2n,R) be a Hit hin representation. Then ρ is indeed a(Sp(2n,R), A)-Anosov representation, where A is the subgroup of diagonalmatri es. The redu tion of the stru ture group of the �at Sp(2n,R)-prin ipalbundle P over T 1Σ to A orresponds to a splitting of the asso iated R

2n-bundle E into the sum of 2n isomorphi line bundles F1 ⊕ · · · ⊕ F2n. The�rst Stiefel-Whitney lass of the line bundle F1 gives an invariantswA1 (ρ) ∈ H1(T 1Σ;F2),whi h lies in H1(T 1Σ;F2) − H1(Σ;F2). This invariant distinguishes the 22g omponents of HomHit hin(π1(Σ), Sp(2n,R)).5. Computations of the Invariants5.1. Irredu ible Fu hsian representations. Let ι : π1(Σ) → SL(2,R) bea dis rete embedding and ρirr = φirr ◦ ι : π1(Σ) → Sp(2n,R) an irredu ibleFu hsian representation. We observed in Fa ts 3.1(i) that the Lagrangianredu tion Ls(ρirr) is given by

Ls(ρirr) = Ls(ι)2n−1 ⊕ Ls(ι)2n−3 ⊕ · · · ⊕ Ls(ι),where Ls(ι) is the line bundle asso iated to ι.

32 OLIVIER GUICHARD AND ANNA WIENHARDTherefore, by the multipli ative properties of Stiefel-Whitney lasses, the�rst and se ond Stiefel-Whitney lasses are given bysw1(ρirr) = sw1(L

s(ρirr)) =( n∑

i=1

(2i− 1))sw1(L

s(ι)) = nsw1(Ls(ι)),and

sw2(ρirr) = sw2(Ls(ρirr)) =

n(n− 1)

2sw1(L

s(ι)) · sw1(Ls(ι))

=n(n− 1)

2(g − 1) mod 2,where the last equality follows from Proposition 5.13. Note that in parti ular

sw1(ρirr) = 0 if n is even.5.2. Diagonal Fu hsian representations. Let ι : π1(Σ) → SL(2,R) bea dis rete embedding and ρ∆ = φ∆ ◦ ι : π1(Σ) → Sp(2n,R) a diagonalFu hsian representation. We observed in Fa ts 3.2(i) that the Lagrangianredu tion Ls(ρ∆) is given byLs(ρ∆) = Ls(ι) ⊕ · · · ⊕ Ls(ι),where Ls(ι) is the line bundle asso iated to ι.Therefore the �rst and se ond Stiefel-Whitney lasses are given by

sw1(ρ∆) = sw1(Ls(ρ∆)) = nsw1(L

s(ι)),andsw2(ρ∆) = sw2(L

s(ρ∆)) =n(n− 1)

2sw1(L

s(ι)) · sw1(Ls(ι))

=n(n− 1)

2(g − 1) mod 2.Again, sw1(ρ∆) = 0 if n is even.5.3. Twisted diagonal representations. Let ι : π1(Σ) → SL(2,R) be adis rete embedding, Θ : π1(Σ) → O(n) an orthogonal representation and

ρΘ = ι ⊗ Θ : π1(Σ) → Sp(2n,R) the orresponding twisted diagonal repre-sentation. We observed in Fa ts 3.4(ii) that the Lagrangian redu tion Ls(ρΘ)is given byLs(ρΘ) = Ls(ι) ⊗W,where Ls(ι) is the line bundle asso iated to ι and W is the pull-ba k to T 1Σof the �at n-plane bundle W over Σ asso iated to Θ.Sin e W is the pull-ba k of the �at bundle W over Σ asso iated to Θ wehave swi(W ) = π∗swi(W ), where π∗ : Hi(Σ;F2) → Hi(T 1Σ;F2) is indu edby the proje tion π : T 1Σ → Σ.Thus, to ompute the invariants for twisted diagonal representations wehave to study the �rst and se ond Stiefel-Whitney lasses of orthogonal rep-resentations Θ : π1(Σ) → O(n).

TOPOLOGICAL INVARIANTS 335.3.1. The �rst Stiefel-Whitney lass of an orthogonal representation. LetΘ : π1(Σ) → O(n) be an orthogonal representation and andW the asso iated�at orthogonal n-plane bundle over Σ. Let us denote by sw1(Θ) = sw1(W ) ∈H1(Σ;F2) the �rst Stiefel-Whitney lass.Let

det : O(n) → {±1}be the determinant homomorphism, where we identify {±1} with F2. Thenthe homomorphismdet ◦Θ : π1(Σ) → {±1} orresponds to the �rst Stiefel-Whitney lass sw1(ρ) ∈ H1(Σ;F2) under theidenti� ation H1(Σ;F2) ∼= Hom(H1(Σ;Z),F2) ∼= Hom(π1(Σ),F2). In par-ti ular, the �rst Stiefel-Whitney lass is zero if the representation has imagein SO(n).Lemma 5.1. Let α ∈ H1(Σ;F2). Then there exists a representation Θα :

π1(Σ) → O(n) su h that sw1(Θα) = α.We postpone the proof of this lemma and �rst onstru t an orthogonalrepresentation. For this let us �x a standard presentationπ1(Σ) = 〈a1, b1, . . . , ag, bg |Πg

i=1[ai, bi] = 1〉.Then the homology lasses orresponding to the urves ai, bi, i = 1, . . . , ggenerate the �rst homology group H1(Σ;Z).Let us �x an embedding l : O(2) → O(n) as the left upper blo k. Wedenote an element of l(SO(2)) ⊂ O(n) by eiθ and an element of l(O(2) −SO(2)) ⊂ O(n) by Reiθ, where R is the element of O(n) − SO(n) withR2 = Id. Then det(eiθ) = 1 and det(Reiθ) = −1.For every k ∈ Z we de�ne the representation Θk : π1(Σ) → O(n) by

Θk(ai) = Θk(bi) = 1 for 2 ≤ i ≤ g

Θk(a1) = R and Θk(b1) = eiπk, where k ∈ Z.Note thatReiθR−1 = e−iθ.Thus we have [Θk(a1),Θ

k(b1)] =[R, eiπk

]= e−i2πk. Therefore

Πgi=1[Θ

k(ai),Θk(bi)] = e−i2πk,and Θk is indeed a representation.Lemma 5.2. The �rst Stiefel-Whitney lass of Θk is the homomorphism

sw1(Θk) ∈ H1(Σ;F2) determined by

sw1(Θk)(ai) = det ◦Θk(ai) = 1 if 2 ≤ i ≤ g

sw1(Θk)(bi) = det ◦Θk(bi) = 1 if 1 ≤ i ≤ g

sw1(Θk)(a1) = det ◦Θk(a1) = −1.

34 OLIVIER GUICHARD AND ANNA WIENHARDProof of Lemma 5.1. If α = 0 ∈ H1(Σ;F2) we an hoose the trivial rep-resentation Θ0 : π1(Σ) → O(n) whi h satis�es sw1(Θ0) = 0 ∈ H1(Σ;F2).Hen e we an assume that α ∈ H1(Σ;F2) − {0}.Sin e the mapping lass group a ts transitively on H1(Σ;F2) − {0}, weneed to onstru t only one representation Θ su h that sw1(Θ) ∈ H1(Σ;F2)−{0}. Then we obtain a representation with �rst Stiefel-Whitney lass α bypre omposing Θ with a suitable element of the mapping lass group.By Lemma 5.2 the representations Θk satisfy sw1(Θ

k) ∈ H1(Σ;F2) −{0}. �5.3.2. The se ond Stiefel-Whitney lass of an orthogonal representation. LetΘ : π1(Σ) → O(n) be an orthogonal representation. The se ond Stiefel-Whitney lass of Θ an be des ribed as the obstru tion to lift the represen-tation ρ to the nontrivial double over Pin(n).For this onsider the entral extension

1 → {±1} → Pin(n) → O(n) → 1,where we again identify {±1} with F2. Sin e this extension is entral wehave a ommutator map[·, ·]e : O(n) ×O(n) → Pin(n).Given a representation Θ : π1(Σ) → O(n) the se ond Stiefel-Whitney lass

sw2(Θ) ∈ H2(Σ;F2) is given byΠgi=1[Θ(ai),Θ(bi)]

e∈ F2.Lemma 5.3. Let Θk : π1(Σ) → O(n) be the representation de�ned above,then the se ond Stiefel-Whitney lass of Θk equals1 if k is even

−1 if k is odd.Proof. It is enough to do the al ulation when n = 2. In this ase Pin(2) ∼=O(2) and the restri tion to SO(2) of the over Pin(2) → O(2) is SO(2) →SO(2), x 7→ x2. We an hen e lift Θk(b1) = eiπk to Θ̃k(b1) = eiπ2k and

Θk(a1) = R to Θ̃k(a1) = R. Then we haveΠgi=1[Θ

k(ai),Θk(bi)]

e= e−iπk.

�Lemma 5.4. Let α 6= 0 ∈ H1(Σ;F2) and β ∈ H2(Σ;F2) arbitrary. Thenthere exists a representation Θα,β : π1(Σ) → O(n) with sw1(Θα,β) = α andsw2(Θα,β) = β.Proof. As above, by the equivarian e under Mod(Σ), the problem redu esto �nding representations with a non-trivial �rst Stiefel-Whitney lass andarbitrary se ond Stiefel-Whitney lass. Su h representations are given byΘk, k = 1, 2. �

TOPOLOGICAL INVARIANTS 35Lemma 5.5. Let n ≥ 3. Let β ∈ H2(Σ;F2) be arbitrary. Then there existsa representation Θ0,β : π1(Σ) → O(n) su h that sw1(Θ0,β) = 0 ∈ H1(Σ;F2)and sw2(Θ0,β) = β ∈ H2(Σ;F2).Proof. It β = 0 ∈ H2(Σ;F2) we an hoose Θ0,0 to be the trivial repre-sentation. Thus we an assume that β ∈ H2(Σ;F2) − {0}. It is su� ientto onstru t a representation into O(3) with the desired properties. Forn ≥ 3 we an then post ompose this representation with the embedding ofO(3) → O(n) as upper left blo k. Thus we are looking for a representationΘ0,β : π1(Σ) → SO(3) (i.e. sw1(Θ0,β) = 0) whi h does not lift to Spin(3).Let us realize Spin(3) ∼= S3 as the quaternions of norm one. The overingmap Spin(3) → SO(3) is realized by the a tion by onjugation on the spa eof imaginary quaternions.Let us denote by {1, i, j, k} the standard basis of the quaternions. Wede�ne a representation Θ0,β : π1(Σ) → SO(3) by sending a1 to the proje tionin SO(3) of i, b1 to the proje tion of j and all other generators of π1(Σ) tothe trivial element. Sin e [Θ0,β(a1),Θ0,β(b1)]

e = [i, j] = −1 this de�nes ahomomorphism into SO(3) whi h does not lift to Spin(3). �Corollary 5.6. Let n ≥ 3. Let α ∈ H1(Σ;F2) and β ∈ H2(Σ;F2). Thenthere exists a representation Θα,β : π1(Σ) → O(n) su h that sw1(Θα,β) = αand sw2(Θα,β) = β.Remark 5.7. Note that for Θ in Hom(π1(Σ),O(2)) the equality sw1(Θ) = 0automati ally implies sw2(Θ) = 0.5.4. Consequen es for maximal representations. Let us now omeba k and onsider maximal representations ρ : π1(Σ) → Sp(2n,R) and their�rst and se ond Stiefel-Whitney lasses sw1(ρ) ∈ H1(T 1Σ;F2), sw2(ρ) ∈H2(T 1Σ;F2).Proposition 5.8. Let n ≥ 3. Let (α, β) ∈ H1(T 1Σ;F2) × H2(T 1Σ;F2) bean admissible pair (see De�nition 4.12). Then there exists a twisted diagonalrepresentationρΘ : π1(Σ) → Sp(2n,R)su h that sw1(ρΘ) = α and sw2(ρΘ) = β.Proof. We have to show that we an realize all 22g di�erent hoi es for sw1(ρ)in the �xed oset of H1(T 1Σ;F2) and the 2 hoi es for sw2(ρ) in the imageof H2(Σ;F2) in H2(T 1Σ;F2).Let ι : π1(Σ) → SL(2,R) be the �xed dis rete embedding, Θ : π1(Σ) →O(n) an orthogonal representation and ρΘ : π1(Σ) → Sp(2n,R) the orre-sponding twisted diagonal representation.We have the following formulas for the �rst and se ond Stiefel-Whitney lasses [32, Corollary 5.4.℄:(3) sw1(ρΘ) = sw1(L

s(ι) ⊗W ) = nsw1(Ls(ι)) + sw1(W )

36 OLIVIER GUICHARD AND ANNA WIENHARDsw2(ρΘ) = sw2(L

s(ι) ⊗W ) =n(n− 1)

2sw1(L

s(ι)) · sw1(Ls(ι))

+ (n− 1)sw1(Ls(ι)) · sw1(W ) + sw2(W ),whereW is the pull-ba k to T 1Σ of the �at n-plane bundle over Σ asso iatedto Θ. Note that swi(W ) is the image of swi(Θ) under the natural map in ohomology.Corollary 5.6 implies that by hoosing di�erent O(n)-representations for

Θ we an realize 22g × 2 di�erent hoi es for sw1(Θ) and sw2(Θ). Sin ensw1(ι) is �xed, as we vary sw1(W ) over the 22g distin t lasses in theimage of H1(Σ;F2) in H1(T 1Σ;F2) we realize all possible 22g lasses in theH1(Σ;F2)- oset in H1(T 1Σ;F2) determined by nsw1(ι). Similarly for anysw1(W ) we an realize the two possibilities for sw2(W ), so we an realizethe two possibilities for sw2 by an appropriately hosen twisted diagonalmaximal representation ρΘ : π1(Σ) → Sp(2n,R). �Every representation in the Hit hin omponent is irredu ible [27℄, there-fore twisted diagonal representations are never ontained in Hit hin ompo-nents. Proposition 5.8 implies then that for n ≥ 3 the spa e

Hommax(π1(Σ), Sp(2n,R)) − HomHit hin(π1(Σ), Sp(2n,R)) onsists of at least 2 × 22g onne ted omponents, and the �rst and se ondStiefel-Whitney lasses of the Anosov bundle asso iated to the representationdistinguish them; this proves Theorem 5. The total number of onne ted omponents of Hommax(π1(Σ), Sp(2n,R)), if n ≥ 3, is 3×22g [14℄. In the 22gHit hin omponents any representation an be deformed into an irredu ibleFu hsian representation. The remaining 2 × 22g onne ted omponents all ontain a twisted diagonal representation. This gives Theorem 8 of theintrodu tion. Corollary 9 then follows dire tly from the omputations inSe tion 5.1 and Se tion 5.2.When n = 2 the above omputations implyCorollary 5.9. Let n = 2. Let (α, β) ∈ H1(T 1Σ;F2) × H2(T 1Σ;F2) beadmissible. Then there exist a twisted diagonal representation ρΘ : π1(Σ) →Sp(4,R) with sw1(ρΘ) = α and sw2(ρΘ) = β if and only if α 6= 0 or (α, β) =(0, (g − 1) mod 2).When n = 2 twisted diagonal representations ρΘ with sw1(ρΘ) = 0 willalways have Euler lass eγ,L0+

(ρΘ) = (g − 1)[Σ] ∈ H2(T 1Σ;Z).Remark 5.10. The lass [Σ] ∈ H2(T 1Σ;Z) is the image of the orientation lass in H2(Σ;Z) under the natural map H2(Σ;Z) → H2(T 1Σ;Z); it is atorsion lass of order 2g − 2.To realize all admissible pairs α = 0 ∈ H1(T 1Σ;F2) and β ∈ H2(T 1Σ;Z)we have to onsider hybrid representations.

TOPOLOGICAL INVARIANTS 375.5. Hybrid representations. The goal of this se tion is to prove the fol-lowingTheorem 5.11. Let γ ∈ π1(Σ)−{1} and L0+ ∈ L+ an oriented Lagrangiansubspa e of R4. Then the relative Euler lass distinguishes the onne ted omponents of

Hommax,sw1=0(π1(Σ), Sp(4,R)) − HomHit hin(π1(Σ), Sp(4,R)).More pre isely, given any l in Z/(2g − 2)Z ∼= Tor(H2(T 1Σ;Z)),� if l 6= (g − 1) the set e−1γ,L0+

({l}) is a onne ted omponent ofHommax,sw1=0(π1(Σ), Sp(4,R)). Every representation in e−1

γ,L0+({l}) an be deformed to a k-hybrid representation, where k = g − 1 − l

mod 2g − 2;� the set e−1γ,L0+

({1 − g}) is the union of the onne ted omponent ofHommax,sw1=0(π1(Σ), Sp(4,R)) onsisting of representations whi h an be deformed to a diagonal Fu hsian representation in Sp(4,R)and of the Hit hin omponents HomHit hin(π1(Σ), Sp(4,R)).For simpli ity we restri t the dis ussion to the ase of k-hybrid representa-tions whi h are onstru ted from the de omposition of the surfa e Σ = Σl∪Σralong a simple losed separating geodesi γ (see Se tion 3.3.1). The ompu-tation in the general ase is a generalization of this ase.The �rst step is to al ulate the �rst Stiefel-Whitney lass.Proposition 5.12. Let ρ : π(Σ) → Sp(4,R) be a k-hybrid representation asde�ned in Se tion 3.3.1, then

sw1(ρ) = 0.Proof. By Proposition 4.11 we have sw1(ρ) ∈ H1(Σ;F2) ⊂ H1(T 1Σ;F2), thusit is su� ient to show that sw1(ρ) is zero on a basis of the �rst homologygroup of Σ.Let π1(Σ) = 〈a1, b1, . . . , ag, bg |Πgi=1[ai, bi] = 1〉 be a standard presenta-tion, then a1, b1, . . . , ag, bg form a basis of H1(Σ;F2). We an hoose su h astandard presentation of π1(Σ) su h that for any element h of the generatingset, either h ∈ π1(Σl) or h ∈ π1(Σr), where the hybrid representation ρ was onstru ted with respe t to a de omposition Σ = Σl ∪ Σr. Then the sign of

det(ρ(h)|Ls(h)), where Ls(h) is the attra ting Lagrangian for ρ(h), is positivefor every element h in the above generating set. This an be he ked inde-pendently for the irredu ible Fu hsian representation ρirr and (deformationsof) the diagonal Fu hsian representation ρ∆. In view of Lemma 4.4 thisimplies sw1(ρ) = 0. �Proposition 5.13. Let Σ = Σl ∪ Σr and ρ = ρl ∗ ρr : π1(Σ) → Sp(4,R) bea hybrid representation as de�ned in Se tion 3.3.1.Let L0+ = 〈e1, e2〉 ⊂ R4 be an oriented Lagrangian subspa e. Let γ ∈

π1(∂Σl) and suppose that L0+ is an attra ting �xed point for ρ(γ).

38 OLIVIER GUICHARD AND ANNA WIENHARDTheneγ,L0+

(ρ) = (g − 1 − χ(Σl))[Σ] ∈ H2(T 1Σ;Z).Remark 5.14. If the pair (ρl, ρr) is negatively adjusted with respe t to γ(De�nition 3.7), then the Euler number iseγ,L0+

(ρ) = (g − 1 + χ(Σl))[Σ] ∈ H2(T 1Σ;Z).5.5.1. Redu tion to the group π̂1(Σ). Our strategy to prove Proposition 5.13is to hange the representation ρ slightly.We denote by π̂1(Σ) the group {±1} × π1(Σ); using the morphism ι :

π1(Σ) → SL(2,R) there is an embedding ι̂ : π̂1(Σ) → SL(2,R), with ι̂(−1) =−I2 and ι̂|π1(Σ) = ι. Observe that

ι̂(π̂1(Σ))\SL(2,R) ∼= ι(π1(Σ))\PSL(2,R) ∼= T 1Σ,so that the group π̂1(Σ) is a quotient of the group π1(T1Σ) and hen e thenotion of Anosov representations and their invariants (see Se tion 2 andSe tion 4) are well de�ned for π̂1(Σ).Let ε : π̂1(Σ) → {±1} be the proje tion onto the �rst fa tor; for anyrepresentation ρ : π1(Σ) → Sp(4,R) we de�ne a representation ε⊗ρ of π̂1(Σ)into Sp(4,R) by setting ε⊗ρ(−1) = −Id4 and ε⊗ρ|π1(Σ) = ρ. The relationsbetween the invariants of ρ and of ε ⊗ ρ are dis ussed in Appendix A.3. Inview of these relations Proposition 5.13 follows from:Proposition 5.15. Let ρ, Σl, Σr, γ and L0+ be as in Proposition 5.13.Then

eγ,L0+(ε⊗ ρ) = −χ(Σl)[Σ].The rest of this se tion is devoted to the proof of this proposition.5.5.2. Constru ting se tions. In order to al ulate the Euler lass for ε ⊗ ρwe will onstru t a lift of eγ,L0+(ε⊗ρ) under the onne ting homomorphismH1(T 1Σ|γ ;Z) → H2(T 1Σ;Z) appearing in the Mayer-Vietoris sequen e (Ap-pendix B) and then al ulate this lift in H1(T 1Σ|γ ;Z) ∼= Z

2. Su h a lift ande onstru ted from trivializations of the Lagrangian bundle.Proposition 5.16. Let ρ, Σl, Σr, γ and L0+ be as in Proposition 5.13. LetLs+(ε⊗ ρ) be the oriented Lagrangian redu tion for the �at ε⊗ ρ-�at bundleover T 1Σ.Then the restri tions of Ls+(ε⊗ ρ) to both T 1Σ|Σl

and T 1Σ|Σr are trivial.This proposition is a onsequen e of the following three lemmas.Lemma 5.17. Let ι : π1(Σ) → SL(2,R) be a maximal representation and letφ : SL(2,R) → Sp(4,R) be a homomorphism su h that ρ = φ ◦ ι is maximal.Then the oriented Lagrangian bundle Ls+(ε⊗ ρ) is trivial.

TOPOLOGICAL INVARIANTS 39Proof. First we observe that ε⊗ρ = φ◦ ι̂ with ι̂ = ε⊗ ι. In this situation themap de�ning the oriented Lagrangian bundle is the equivariant map (re allthat T 1Σ = π̂1(Σ)\SL(2,R))SL(2,R) −→ L+

g 7−→ φ(g) · L0+,where L0+ is the attra ting Lagrangian �xed by ρ(γ).An equivariant map SL(2,R) → Sp(4,R) that trivialize the orrespondingbundle is given simply byg 7−→ φ(g).

�Lemma 5.18. Let ρ0 and ρ1 be two homotopi maximal representations.Then the Lagrangian bundles Ls(ρ0) and Ls(ρ1) are homotopi , hen e theyare isomorphi .If the �rst Stiefel-Whitney lass sw1(ρ0) = sw1(ρ1) is zero, then the or-responding oriented Lagrangian bundles are also isomorphi .Proof. This is a onsequen e of the fa t that the equivariant positive urvedepends ontinuously of the representation (Fa t 2.21). �Lemma 5.19. Let ρ and ρ′ be two maximal representations with zero �rstStiefel-Whitney lass and su h that ρ|π1Σ′ = ρ′|π1Σ′ for a subsurfa e Σ′ ⊂ Σ.Then the two bundles Ls+(ε ⊗ ρ) and Ls+(ε ⊗ ρ′) are homotopi when re-stri ted to T 1Σ|Σ′.Proof. Note that if Ls(ε ⊗ ρ) is homotopi to Ls(ε ⊗ ρ′) then Ls+(ε ⊗ ρ) ishomotopi to Ls+(ε⊗ρ′) There exists a line bundle Dε su h that Ls(ε⊗ρ) =Dε ⊗ Ls(ρ) and Ls(ε⊗ ρ′) = Dε ⊗ Ls(ρ′). Thus it is su� ient to show thatLs(ρ)|M ′ ≃ Ls(ρ′)|M ′ with M ′ = T 1Σ|Σ′ ⊂M = T 1Σ.Let Σ̃′ be the universal over of Σ′; it an be realized as a π1(Σ

′)-invariantsubset of Σ̃. More pre isely, under an identi� ation Σ̃ ∼= H2 we an setΣ̃′ = Conv(Λπ1(Σ′)) the onvex hull of the limit set Λπ1(Σ′) of π1(Σ

′) in theboundary ∂H2 of the hyperboli plane.The manifoldM := T 1Σ̃ is a π1(Σ)- over ofM and we setM ′:= T 1Σ̃|eΣ′

⊂T 1Σ̃ = M . When we identify the unit tangent bundle M with ∂π1(Σ)(3+),the set of positively oriented triples of ∂π1(Σ), we an identify M ′ with thesubset of ∂π1(Σ)(3+) whose proje tion to Σ̃ belongs to Σ̃′.The bundle Ls(ρ) is onstru ted via the ρ-equivariant map

∂π1(Σ)(3+) p−→ ∂π1(Σ)ξ−→ L

(ts, t, tu) 7−→ ts 7−→ ξ(ts)where ξ is the positive ρ-equivariant urve. Similarly Ls(ρ′) is onstru tedfrom the positive ρ′-equivariant urve ξ′.

40 OLIVIER GUICHARD AND ANNA WIENHARDThis means that the restri tion Ls(ρ)|M ′ is onstru ted from the ρ|π1(Σ′)-equivariant mapM

′ p|M

′

−−−→ ∂π1(Σ)ξ−→ L.And onversely, any ρ|π1(Σ′)-equivariant ontinuous map ∂π1(Σ) → L de�nesa Lagrangian redu tion of the symple ti R

4-bundle overM ′. Hen e we get ahomotopy of the two bundles Ls(ρ)|M ′ and Ls(ρ′)|M ′ on e we have a ρ|π1(Σ′)-equivariant homotopy between the two mapsM

′ p|M

′

−−−→ ∂π1(Σ)ξ−→ L

M′ p|

M′

−−−→ ∂π1(Σ)ξ′−→ L.For this it is su� ient to onstru t a ρ|π1(Σ′)-equivariant homotopy betweenthe two positive maps ξ and ξ′. This is the ontent of the next lemma. �Lemma 5.20. Let ρ : π1(Σ) → Sp(4,R) be a maximal representation andlet Σ′ be a subsurfa e. Then the set

C = {ξ : ∂π1(Σ) → L | ξ > 0, C0, ρ|π1(Σ′)-equivariant}is onne ted.Proof. To simplify notation we assume that the boundary ∂Σ′ = γ onsistsof one omponent. Note �rst that the restri tion to the limit set Λπ1(Σ′)of any urve ξ in C is ompletely determined by the a tion of π1(Σ′). The omplement ∂π1(Σ) − Λπ1(Σ′) is a ountable union of open intervals, whi hare transitively ex hanged by the a tion of π1(Σ

′). The interval (tuγ , tsγ),where tuγ and tsγ are the �xed points of γ in ∂π1(Σ), is one su h interval. Themap ξ is ompletely determined by its restri tion to this interval.Conversely given a positive ρ|〈γ〉-equivariant ontinuous urve β : (tuγ , t

sγ) →

L+, one obtains a ontinuous ρ|π1(Σ′)-equivariant positive urve ∂π1(Σ) −Λπ1(Σ′) → L, whi h one shows to have a ontinuous extension ∂π1(Σ) → L.Thus the map:

C −→ {β : (tuγ , tsγ) → L, C0, > 0, (ρ|γ)-equivariant}

ξ 7−→ ξ|(tuγ ,tsγ)is a homeomorphism. By Proposition A.2, this spa e is onne ted. �5.5.3. Cal ulating the Euler lass. Proposition 5.16 gives pre isely what isneeded onstru t a lift of the Euler lass eγ,L0+(ε⊗ ρ) under the onne tinghomomorphism H1(T 1Σ|γ ;Z) → H2(T 1Σ;Z). We have (see Appendix B)(4) H1(T 1Σ|γ ;Z) ∼= HomZ(π1(T

1Σ|γ),Z) ∼= Z × Z.The last identi� ation sends φ ∈ HomZ(π1(T1Σ|γ),Z) to the pair (φ(T 1

xΣ), φ(γ)),where the two ir les T 1xΣ and γ are naturally onsidered as loops in T 1Σ|γ .

TOPOLOGICAL INVARIANTS 41Proposition 5.21. Let ρ, Σl, Σr, γ and L0+ be as in Proposition 5.13.Let Ls+(ε⊗ ρ) be the oriented Lagrangian redu tion for the �at ε⊗ ρ-bundleover T 1Σ. Suppose that gl and gr are trivializations of the restri tions ofLs+(ε⊗ ρ) to T 1Σ|Σl

and T 1Σ|Σr .Then gl ◦ g−1r : T 1Σ|γ × R

2 → T 1Σ|γ × R2 is a gauge transformation ofthe trivial oriented R

2-bundle over T 1Σ|γ and de�nes a map h : T 1Σ|γ →GL+(2,R). Let h∗ ∈ Hom(π1(T1Σ|γ);π1(GL+(2,R)) denote the map in-du ed by h on the level of fundamental groups.(i) The image of h∗ ∈ Hom(π1(T

1Σ|γ);π1(GL+(2,R)) ∼= H1(T 1Σ|γ ;Z)under the onne tion homomorphism H1(T 1Σ|γ ;Z) → H2(T 1Σ;Z)is the Euler lass: δ(h∗) = eγ,L0+(ε⊗ ρ).(ii) Under the identi� ation H1(T 1Σ;Z) ∼= Z × Z in (4), h∗ is equal to

(1, 0), up to the image of the map H1(T 1Σ|Σl;Z)⊕H1(T 1Σ|Σr ;Z) →H1(T 1Σ|γ ;Z).Remark 5.22. The ambiguity given by the image of the map H1(T 1Σ|Σl

;Z)⊕H1(T 1Σ|Σr ;Z) → H1(T 1Σ|γ ;Z) a ounts for hanging the trivializations gland gr of the restri tions of Ls+(ε⊗ ρ) to T 1Σ|Σland T 1Σ|Σr .Proof of Proposition 5.15. With the notation of Proposition 5.21 δ(h∗) =

eγ,L0+(ε ⊗ ρ). Sin e h∗ is equal to (1, 0) in the identi� ation (4), Proposi-tion B.2 pre isely says that δ(h∗) = (2g(Σl) − 1)[Σ] = −χ(Σl)[Σ]. �Proof of Proposition 5.21. The �rst statement δ(h∗) = eγ,L0+

(ε⊗ρ) is lassi- al, it follows from the fa t that the Euler lass is the obstru tion to trivializethe oriented Lagrangian bundle Ls+(ε⊗ ρ) over T 1Σ.For the se ond statement, let T 1xΣ and γ be the two generators of thefundamental group of the 2-torus T 1Σ|γ . We have to show, identifying

π1(GL+(2,R)) with Z, thath∗(T

1xΣ) = 1 and h∗(γ) = 0.In both ases our strategy will be the same: we will write the homotopy lass we want to al ulate as a produ t of two homotopy lasses, the �rstdepending only on the restri tion to Σl and the se ond depending on therestri tion to Σr. With this we will be able to deform the representationsand the �ber bundles independently on Σl and on Σr without hanging thehomotopy lasses we are onsidering. By onstru tion of the hybrid repre-sentations this means that we an redu e the al ulation of the homotopy lass to the ase when the representations are restri tions of the irredu ibleFu hsian representation for Σl and the diagonal Fu hsian representation for

Σr.We start by proving se ond equality: h∗(γ) = 0.We identify γ with Z\R so that the restri tion of the symple ti bundleto γ is identi�ed with Z\(R × R4) where Z a ts diagonally on R × R

4,n · (t, v) = (n + t, (Aγ)

nv), where Aγ = ρ(γ) is the diagonal element

42 OLIVIER GUICHARD AND ANNA WIENHARDdiag(elγ , ekγ , e−lγ , e−kγ ). Furthermore, the restri tion of the oriented La-grangian redu tion Ls+(ε ⊗ ρ) to the geodesi γ is �at and hen e has theform Z\(R × L0+).A trivialization of the symple ti bundle over γ is given by an equivariantmap H : R → Sp(4,R), i.e. H(t + 1) = H(t)(Aγ)−1. Su h a trivializationindu es a trivialization of the Lagrangian bundle Ls+(ε⊗ ρ)|γ if furthermorefor all t the element H(t) stabilizes L0+. We now provide a � anoni al�trivialization in our situation. For this let A = diag(el, ek, e−l, e−k) be adiagonal element and let us denote by LA the Lagrangian bundle Z\(R×L0+)where n · (t, v) = (n+ t, Anv). Then the ontinuous map

H(A, ·) : R −→ Sp(4,R), t 7−→ H(A, t) = diag(e−tl, e−tk, etl, etk)provides su h trivialization of LA.The homotopy lass h∗(γ) we are al ulating omes from two trivializa-tions gl|γ , gr|γ : Ls+(ε⊗ ρ)|γ → γ × L0+, andgl|γ ◦ (gr)

−1|γ(t, v) = (t, h(γ(t))v),for (t, v) ∈ γ × L0+.With the trivialization H(Aγ , ·) of Ls+(ε⊗ ρ)|γ given above, we haveg⋆|γ ◦H(Aγ , ·)−1(t, v) = (t,M⋆(t)v), for ⋆ = l, r.This means that in π1(GL

+(2,R)) = Z we have the equalityh∗(γ) = [Ml] − [Mr].We now prove that both [Ml] and [Mr] are trivial.By onstru tion of the hybrid representation ρ = ρl ∗ ρr in Se tion 3.3.1we know that Aγ = ρl(γ) = ρirr(γ) = φirr ◦ ι(γ), where ι : π1(Σ) → SL(2,R)is a dis rete embedding and φirr : SL(2,R) → Sp(4,R) is the irredu iblerepresentation. The trivialization given by Lemma 5.17 and the formula for

H(·, ·) imply that Ml is the onstant map, thus [Ml] = 0.To ompute [Mr] = 0 we have to onsider Aγ = ρr(γ) = ρr,1(γ), whereρr,s, s ∈ [0, 1], is a ontinuous path of maximal representations with ρr,0 =ρ∆ = φ∆ ◦ ι and, for all s, ρr,s(γ) is diagonal and where ι is as above andφ∆ : SL(2,R) → Sp(4,R) is the diagonal embedding. Thus the family of hanges of trivializations gr,s ◦H(ρl,s(γ), ·)−1, s ∈ [0, 1], provide a homotopyfrom the loop Mr = Mr,1 to the loop Mr,0, whi h is the onstant map.Therefore [Mr] = 0.We now turn to the proof of the �rst equality: h∗(T 1

xΣ) = 1.In ontrast to the previous al ulation, here we do not have to satisfyany equivarian e properties, but the �standard� trivializations will not benatural.We identify T 1xΣ with the group PSO(2), via an orbital map in T 1Σ̃ ∼=PSL(2,R), and identify it also with the boundary ∂π1(Σ), via the proje tion

T 1Σ̃ ∼= ∂π1(Σ)(3+) → ∂π1(Σ), (ts, t, tu) 7→ ts. We an suppose that under

TOPOLOGICAL INVARIANTS 43these identi� ations the attra tive �xed point tsγ of γ is sent to [Id2] in PSO(2)whereas the repulsive �xed point tuγ is sent to [( 0 1−1 0

)].Sin e we are working with the representation ε⊗ρ, the �at Sp(4,R)-bundleover T 1xΣ ∼= PSO(2) is the quotientSO(2) ×{±1} Sp(4,R) := {±1}\(SO(2) × Sp(4,R))of the trivial bundle over SO(2) by the group {±1}, where the a tion is givenby

(−1) · (s, g) = (−s,−g).The oriented Lagrangian redu tion Ls+(ε⊗ ρ)|PSO(2) is given by the positive ontinuous urve asso iated to ρ:SO(2) → PSO(2) ∼= ∂π1(Σ)ξ+−→ L+into the spa e of oriented Lagrangians.A trivialization of the bundle SO(2) ×{±1} Sp(4,R) is then a {±1}-equivariant map

g : SO(2) −→ Sp(4,R).This trivialization indu es furthermore a trivialization of the Lagrangianredu tion Ls+(ε⊗ ρ)|PSO(2) if for all α in SO(2)

ξ+(α) = g(α) · Ls0+.We observe thatξ+(Id2) = Ls0+ and ξ+( 0 1

−1 0

)= Lu0+,where Ls0+ = 〈e1, e2〉 and Lu0+ = 〈e3, e4〉 with e1, . . . , e4 being the standardsymple ti basis of R

4.Lemma-De�nition 5.23. Let η : SO(2) → L+ be a ontinuous, {±1}-invariant, positive urve de�ning a Lagrangian redu tion Lη of the bundleSO(2) ×{±1} Sp(4,R).Suppose thatη(Id2) = Ls0+ and η

(0 1−1 0

)= Ls0+.Then for all α, η(α) and φ∆(α) · Ls0+ are transversal Lagrangians. Thismeans that there exists a unique symmetri 2 by 2 matrix M(α) su h that:

( Id2 0M(α) Id2

)φ∆(α)−1 · η(α) = Ls0+.Then the map

βη : SO(2) → Sp(4,R), α 7→ φ∆(α)

( Id2 0−M(α) Id2

)is a trivialization of Lη.

44 OLIVIER GUICHARD AND ANNA WIENHARDThe only point to prove is that η(α) and φ∆(α) · Lu0+ are transversal.This is immediate for α = ±Id2 and for α = ±(

0 1−1 0

); for other αit follows from the positivity of the 4-tuple (Ls0+, η(α), Lu0+, φ∆(α) · Lu0+)(or the 4-tuple (Lu0+, η(α), Ls0+, φ∆(α) · Lu0+) depending on whi h intervalα lies) whi h results from the positivity of the triples (Ls0+, η(α), Lu0+) and(Lu0+, φ∆(α) · Lu0+, Ls0+).Going ba k to the proof of Proposition 5.21, the trivializations βη enableus to write h∗(T 1

xΣ) as the di�eren e:h∗(T

1xΣ) = [Nl] − [Nr],where, for ⋆ = l, r, N⋆ is de�ned as the hange of trivializations g⋆|T 1

xΣ ◦β−1ξ+

.Again Nl and Nr are in fa t homotopi to the orresponding hanges oftrivializations we obtain from the representations φirr ◦ ι and φ∆ ◦ ι respe -tively. It is then immediate that Nr is homotopi to the onstant map andthat Nl is homotopi to the mapPSO(2) −→ Stab(Ls0+)

[α] 7−→ φirr(α)φ∆(α)−1

( Id2 0−M(α) Id2

),whereM(α) is the only 2×2 symmetri matrix su h that this produ t belongsto Stab(Ls0+). A dire t al ulation gives for α =

(cos(θ) sin(θ)− sin(θ) cos(θ)

)

φirr(α)φ∆(α)−1

( Id2 0−M(α) Id2

)=

(A(θ) ∗

0 tA(θ)−1

),where

A(θ) =4

cos2(2θ) + 3

(cos(2θ) −

√3

2 sin(2θ)√3

2 sin(2θ) cos(2θ)

).Hen e a path in GL+(2,R) representing Nl is [0, π] → GL+(2,R), θ 7→ A(θ).It follows that h∗(T 1

xΣ) = 1. �Remark 5.24. The above omputation of the Euler lass for hybrid represen-tations hints towards a more general gluing formula for topologi al invariantsof representations for surfa es with boundary. We intend to give a rigorousde�nition of invariants for surfa es with boundary in a subsequent paper.6. A tion of the mapping lass groupIt is known that the a tion of the mapping lass group Mod(Σ) onRepmax(π1(Σ), Sp(2n,R)) = Hommax(π1(Σ), Sp(2n,R))/Sp(2n,R)is properly dis ontinuous [28, 34℄. Furthermore this group a ts naturally onHi(T 1Σ;F2) and the mapsswi : Repmax(π1(Σ), Sp(2n,R)) −→ Hi(T 1Σ;F2)

TOPOLOGICAL INVARIANTS 45are equivariant.Therefore, understanding the a tion of Mod(Σ) on the subspa es ofHi(T 1Σ;F2) where the Stiefel-Whitney lasses take their values in, allowsus to determine the number of onne ted omponents of the quotient ofRepmax(π1(Σ), Sp(2n,R)) by Mod(Σ).Lemma 6.1. Let A = Z or F2. The mapping lass group of Σ a ts triviallyon the image of H2(Σ;A) in H2(T 1Σ;A).Proof. This is immediate from the fa t that Mod(Σ) preserves the image ofH2(Σ;A) in H2(T 1Σ;A) and a ts trivially on H2(Σ;A). �For H1(T 1Σ;F2) the pi ture is a bit more ompli ated.Proposition 6.2. The a tion of the mapping lass group on H1(T 1Σ;F2)preserves the two osets H1(Σ;F2) and H1(T 1Σ;F2) −H1(Σ;F2).(i) On H1(Σ;F2) the a tion of the mapping lass group has two orbits,0 and H1(Σ;F2) − 0.(ii) On H1(T 1Σ;F2) −H1(Σ;F2) the a tion of the mapping lass grouphas two orbits.Proof. It is obvious that the mapping lass group of Σ preserves the two osets H1(Σ;F2) and H1(T 1Σ;F2) −H1(Σ;F2).On H1(Σ;F2) ∼= F

2g2 the mapping lass group a ts by symple tomorphismsand it is a lassi al fa t that it generates the whole group Sp(2n,F2). Thisa tion has two orbits, 0 and F

2g2 − 0.The a tion of the mapping lass on the spa e of spin stru tures H1(T 1Σ;F2)−H1(Σ;F2) has two orbits [26℄. The orbits are distinguished by the Arf-invariant of the quadrati form de�ned by v ∈ H1(T 1Σ;F2)−H1(Σ;F2).6 �Corollary 6.3. Let n ≥ 3. The spa e Repmax(π1(Σ), Sp(2n,R))/Mod(Σ)has 6 onne ted omponents.Proof. Let us �rst onsider the Hit hin omponents RepHit hin(π1(Σ), Sp(2n,R)).This subset is invariant by the a tion of Mod(Σ). The 22g di�erent Hit hin omponents are indexed by elements swA1 ∈ H1(T 1Σ;F2) − H1(Σ;F2) (seeSe tion 4.5.2). The a tion of the mapping lass group has 2 orbits onH1(T 1Σ;F2)−H1(Σ;F2). Therefore the quotient RepHit hin(π1(Σ), Sp(2n,R))/Mod(Σ)has 2 onne ted omponents.The 2 × 22g onne ted omponents ofRepmax(π1(Σ), Sp(2n,R)) − RepHit hin(π1(Σ), Sp(2n,R))are indexed by the �rst and se ond Stiefel-Whitney lasses. The mapping lass group a ts trivially on the image of H2(Σ;F2) in H2(T 1Σ;F2) and has6This Arf-invariant is the same as the invariant of the orresponding spin stru turede�ned by Atiyah in [1℄.

46 OLIVIER GUICHARD AND ANNA WIENHARDtwo orbits in the oset of H1(Σ;F2) in H1(T 1Σ;F2). This implies that it has4 orbits in the set of admissible pairs (De�nition 4.12), hen e the quotient(Repmax(π1(Σ), Sp(2n,R)) − RepHit hin(π1(Σ), Sp(2n,R))

)/Mod(Σ)has 4 onne ted omponents. �6.1. The ase of Sp(4,R). To de�ne the Euler lass of a representa-tion ρ : π1(Σ) → Sp(4,R) with sw1(ρ) = 0 we had to make several hoi es (see Se tion 4.3). In parti ular, we �xed a nontrivial element

γ ∈ π1(Σ) to de�ne the spa e HomL+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R)) and the Eu-ler lass eγ,L0+

. Denoting Repmax,sw1=0(π1(Σ), Sp(4,R)) the quotient ofHommax,sw1=0(π1(Σ), Sp(4,R)) by the a tion of Sp(4,R), the equivarian eof

eγ,L0+: Repmax,sw1=0(π1(Σ), Sp(4,R)) −→ H2(T 1Σ;Z)only holds for the subgroup Stab(γ) ⊂ Mod(Σ) of the mapping lass groupstabilizing the homotopy lass of γ:Lemma 6.4. Let γ ∈ π1(Σ)−{1} and L0+ ∈ L+ an oriented Lagrangian. Let

ψ be an element of Stab(γ) ⊂ Mod(Σ) ∼= Out(π1(Σ)) �xing the homotopy lass of γ and let ψ be an automorphism of π1(Σ) representing ψ and �xingγ. Then ψ a ts on Hom

L+,γmax,sw1=0(π1(Σ), Sp(4,R)) and preserves the subspa eHom

L+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R)). Furthermore, the following diagram is om-mutativeHom

L+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R))ψ∗−→ Hom

L+,γ,L0+max,sw1=0(π1(Σ), Sp(4,R))

↓ ↓Repmax,sw1=0(π1(Σ), Sp(4,R))ψ∗−→ Repmax,sw1=0(π1(Σ), Sp(4,R)).Consequently, we have the the equalities eγ,L0+

◦ ψ∗ = ψ∗ ◦ eγ,L0+= eγ,L0+

.This impliesProposition 6.5. Let C be a onne ted omponent of Repmax,sw1=0(π1(Σ), Sp(4,R))−RepHit hin(π1(Σ), Sp(4,R)). Then C is sent to itself by the a tion of the map-ping lass group of Σ.Proof. It is a lassi al fa t that the mapping lass groupMod(Σ) is generatedby Dehn twists along simple losed urves (see for example [25℄). As a onsequen e Mod(Σ) is generated by {Stab(γ)}γ∈π1(Σ)−{1}. With this saidit is enough to show the invarian e of C under every ψ belonging to someStab(γ). However we know that Hommax,sw1=0−RepHit hin is invariant underMod(Σ) and thateγ,L0+

: Repmax,sw1=0(π1(Σ), Sp(4,R)) − RepHit hin(π1(Σ), Sp(4,R))

−→ Tor(H2(T 1Σ;Z))

TOPOLOGICAL INVARIANTS 47is a bije tion. The invarian e property given in Lemma 6.4 implies the laim. �Corollary 6.6. The spa e Repmax(π1(Σ), Sp(4,R))/Mod(Σ) has 2g+2 on-ne ted omponents.Proof. Let us write Repmax(π1(Σ), Sp(4,R)) as a union of subspa esRepmax(π1(Σ), Sp(4,R)) =Repmax,sw1 6=0(π1(Σ), Sp(4,R))⋃RepHit hin(π1(Σ), Sp(4,R))

⋃

((Repmax,sw1=0(π1(Σ), Sp(4,R)) − RepHit hin(π1(Σ), Sp(4,R))

).The mapping lass group a tion preserves this omposition. Be ause themapping lass group has two orbits on H1(T 1Σ;F2) − H1(Σ;F2), the spa eRepHit hin(π1(Σ), Sp(4,R))/Mod(Σ) has 2 onne ted omponents. Themapping lass group a ts transitively on H1(Σ;F2) − {0} and trivially onthe image of H2(Σ;F2) ∈ H2(T 1Σ;F2), thus the spa eRepmax,sw1 6=0(π1(Σ), Sp(4,R))has 2 onne ted omponents. By Proposition 6.5 the mapping lass groupstabilizes the others omponents. This gives a total of 2g + 2 onne ted omponents. �7. Holonomy of maximal representationsLet ρ : π1(Σ) → Sp(2n,R) be a maximal representation and ξ :

∂π1(Σ) → Sp(2n,R) the ρ-equivariant limit urve. Then, for every non-trivial γ ∈ π1(Σ), the image ρ(γ) �xes the two transverse Lagrangiansξ(tsγ), ξ(tuγ), where tsγ , tuγ ∈ ∂π1(Σ) are the �xed points of γ. Moreover,ξ(tsγ) is the unique attra ting Lagrangian for ρ(γ). This readily impliesthat ρ(γ) is an element of GL(ξ(tγ+)) whose eigenvalues are bigger than1 (see Corollary 2.11). For representations in the Hit hin omponentsHomHit hin(π1(Σ, Sp(2n,R)) ⊂ Hommax(π1(Σ), Sp(2n,R)), Corollary 2.11implies moreover that ρ(γ) ∈ GL(ξ(tsγ)) is a semi-simple element. The fol-lowing statement shows that we annot expe t anything similar for maximalrepresentations in general.Theorem 7.1. Let H be a onne ted omponent ofRepmax(π1(Σ), Sp(2n,R)) − RepHit hin(π1(Σ), Sp(2n,R)),and let γ be an element in π1(Σ) − {1} orresponding to a simple urve.If γ is separating, n = 2 and the genus of Σ is 2, we require that H is notthe onne ted omponent determined by sw1 = 0 and eγ = 0.Then there exist(i) a representation ρ ∈ H su h that the Jordan de omposition of ρ(γ)in GL(n,R) has a nontrivial paraboli omponent.(ii) a representation ρ′ ∈ H su h that the Jordan de omposition of ρ′(γ)in GL(n,R) has a nontrivial ellipti omponent.

48 OLIVIER GUICHARD AND ANNA WIENHARDWe �rst establish some preliminary results towards the proof of the theo-rem.Lemma 7.2. Let Σ be a surfa e with one boundary omponent γ = ∂Σ. LetG be a semisimple Lie group and ρ0 : π1(Σ) → G a representation whose entralizer in G is �nite.Then the di�erential of the map

Hom(π1(Σ), G)τγ−→ G

ρ 7−→ ρ(γ)at the point ρ0 is surje tive.Proof. One an always �nd a set {a1, . . . , ak, b1, . . . , bk} freely generatingπ1(Σ) and su h that γ = [a1, b1] · · · [ak, bk]. The result is then simply areformulation of [16, Proposition 3.7.℄. �Lemma 7.3. Let γ be a simple losed separating urve on a losed surfa eΣ; denote by Σ1 and Σ2 the omponents of Σ − γ. Let H be a omponentof Hommax(π1(Σ), Sp(2n,R)) su h that the onditions of Theorem 7.1 aresatis�ed.Then there exists ρ in H su h that� ρ(γ) ( onsidered as an element of GL(n,R) < Sp(2n,R)) is a mul-tiple of the identity,� the restri tion of ρ to π1(Σ1) (resp. π1(Σ2)) has �nite entralizer.Proof. By Theorem 8 and Theorem 10 we only need to prove that there arerepresentations satisfying the on lusions of the lemma in a neighborhood ofa model representation (i.e. a standard maximal representation or a hybridrepresentation).First onsider a diagonal Fu hsian representation ρ0 = φ∆ ◦ ι. Using stan-dard Fri ke-Klein theory of the Tei hmüller spa e, there exists deformationsι1,t, . . . , ιn,t (t ∈ [0, 1]) su h that ιi,t(γ) = ι(γ) and ιi,0 = ι and, for all t > 0,the representation ρt = (ι1,t, . . . , ιn,t) sends π1(Σ1) and π1(Σ2) into a Zariskidense subgroup of SL(2,R)n < Sp(2n,R) (this onstru tion was alreadyused in [13, Se tion 9℄). As the entralizer of SL(2,R)n inside Sp(2n,R) is�nite, the statement of the lemma follows.Now onsider a twisted diagonal representation ρ0 = ι⊗ Θ whi h annotbe deformed to a diagonal Fu hsian representation. The analysis in Se -tion 5.3 implies that it is su� ient to onsider the ase when Θ takes valuesin O(2) < O(n) and has �nite image or the ase when Θ takes values inSO(3) and has �nite image. In the �rst ase, the representation ρ0 takes val-ues in Sp(4,R) × SL(2,R)n−2 < Sp(2n,R); we write ρ0 = (ι ⊗ Θ, ι, . . . , ι).As above, we an �nd a deformation ρt = (ι1,t ⊗ Θ, ι2,t, . . . , ιn−2,t) of ρ0su h that, fot all t, ρt(γ) = ρ(γ) and, for all t > 0, the Zariski losureof ρt(π1(Σ1)) (resp. ρt(π1(Σ2))) ontains φ∆(SL(2,R)) × SL(2,R)n−2 <

TOPOLOGICAL INVARIANTS 49Sp(4,R) × SL(2,R)n−2 < Sp(2n,R). This already means that the entral-izer of ρt(π1(Σ1)) is ontained in O(2) < O(n) < Sp(2n,R); it also impliesthat the image Θ(π1(Σ1)) is ontained in the Zariski losure of ρt(π1(Σ1)).Suppose now that the image of π1(Σ1) by Θ is not ontained in SO(2).Then there exists a re�e tion R ∈ O(2) − SO(2) that belongs to the Zariski losure ρt(π1(Σ1))Z . Therefore the entralizer of ρt(π1(Σ1))

Z (whi h equalsthe entralizer of ρt(π1(Σ1))) will be ontained in the entralizer of R in O(2)whi h is �nite.If the restri tion of Θ to π1(Σ1) is ontained in SO(2), we an supposethat this restri tion is the trivial representation. In this situation we writeρ0 as amalgamated representation ρ0 = ρ1 ∗ ρ2, where ρi = ρ0|π1(Σi). Thenthe restri tion of Θ to π1(Σ2) is not ontained in SO(2) (otherwise ρ0 wouldbe in the same onne ted omponent as a diagonal Fu hsian representation).We then deform ρ0 as an amalgamated representation: ρ(1)

t ∗ ρ(2)t , where ρ(1)

tis a deformation of ρ1 onsidered as a diagonal Fu hsian representation (the�rst ase we investigated) and ρ(2)t is a deformation of the twisted diagonalrepresentation ρ2. The entralizer of π1(Σ2) is �nite by the same argumentas above.The ase when Θ is in SO(3) with �nite image an be treated in a verysimilar fashion and is left to the reader.Let us now assume that n = 2 and and ρ0 is a hybrid representation in

Hommax,sw1=0(π1(Σ), Sp(4,R)) − HomHit hin(π1(Σ), Sp(4,R)). The de�ni-tion of a hybrid representation ρ involves the hoi e of a subsurfa e Σ′ in Σ,for whose fundamental group we hoose an irredu ible Fu hsian representa-tion. Sin e we want the holonomy around γ to be a multiple of the identityin GL(2,R) the urve γ has to be ontained in Σ−Σ′ and not homotopi toa boundary omponent of Σ′. This requires the Euler hara teristi χ(Σ′) tobe di�erent from 3 − 2g.The way hybrid representations are onstru ted in Se tion 3.3, and withthe hypothesis on γ and Σ′, one an ensure that ρ(γ) is a multiple of theidentity in GL(2,R) < Sp(4,R) and that the Zariski losure ρ(π1(Σ1))Z(resp. ρ(π1(Σ2))

Z) ontains SL(2,R) × SL(2,R) or an irredu ible SL(2,R)in Sp(4,R). Therefore, the hybrid representation ρ satis�es all the desired on lusions. Considering not only hybrid representation onstru ted frompositively adjusted pairs as in Se tion 3.3.1, but also su h onstru ted fromnegatively adjusted pairs (De�nition 3.7) we get representations with Euler lass g − 1 ± χ(Σ′) ∈ Z/(2g − 2)Z (see Proposition 5.13 and Remark 5.14).Varying the subsurfa e Σ′, every Euler hara teristi χ(Σ′) di�erent form3 − 2g an be attained, hen e we obtain representations in any onne ted omponent of

Hommax,sw1=0(π1(Σ), Sp(4,R)) − HomHit hin(π1(Σ), Sp(4,R)),

50 OLIVIER GUICHARD AND ANNA WIENHARDex ept if g = 2, when we annot obtain representations with Euler lasseγ = 0 by the above onstru tion. �Proof of Theorem 7.1. Suppose that γ is a separating urve and let ρ0 in Hbe a representation satisfying the on lusions of Lemma 7.3. Let us denoteby Σ1 and Σ2 the omponents of Σ−γ. We all τ1 and τ2 the two evaluationmaps:

τi : Hom(π1(Σi), Sp(2n,R)) −→ Sp(2n,R)ρ 7−→ ρ(γ).The representations spa e for π1(Σ) is the �ber produ t of the representationsspa es for π1(Σ1) and π1(Σ2) over Sp(2n,R):

Hom(π1(Σ), Sp(2n,R)) ={(ρ1, ρ2) ∈ Hom(π1(Σ2), Sp(2n,R)) × Hom(π1(Σ2), Sp(2n,R)) |

τ1(ρ1) = τ2(ρ2)}.By Lemma 7.2 the map τ1 (resp. τ2) is lo ally surje tive in a neighborhoodof ρ0|π1(Σ1) (resp. ρ0|π1(Σ2)). This implies that the map

τ : Hom(π1(Σ), Sp(2n,R)) −→ Sp(2n,R)ρ 7−→ ρ(γ)is lo ally surje tive in a neighborhood of ρ0. As τ(ρ0) = ρ0(γ) is a multiple ofthe identity in GL(n,R) we obtain representations ρ and ρ′ in a neighborhoodof ρ0 having the desired properties.When γ is not separating, the proof follows a similar strategy, with thearguments being a little easier. We leave this proof to the reader. �Appendix A. Maximal representationsA.1. The spa e of positive urves. In this se tion we establish ertain onne tedness properties of the spa e of positive urves into the LagrangianGrassmannian.We will use the notation from Se tion 2.2: R

2n is a symple ti ve -tor spa e, with symple ti basis (ei)i=1,...,e2n ; X ⊂ L × L is the spa eof pairwise transverse Lagrangian subspa es of V , Ls0 = Span(ei)1≤i≤n,Lu0 = Span(ei)n+1≤i≤2n are two transverse Lagrangian subspa es of R

2n,P u, P s ⊂ Sp(2n,R) their stabilizer. The unipotent radi al of P s is

U s =

{us(M) =

( Idn M0 Idn ) |M ∈ M(n,R), tM = M

}.A Lagrangian L an be written as us(M) · Lu0 for some M if and only if Land Ls0 are transverse, in whi h ase M is uniquely determined by L. Thetriple of Lagrangians (Ls0, L, L

u0) is positive (De�nition 2.18) if and only ifthe symmetri matrix M su h that L = us(M) · Lu0 is positive de�nite.Re all that a urve ξ : S1 → L is said positive if it sends every positivetriple of S1 to a positive triple of Lagrangians.

TOPOLOGICAL INVARIANTS 51Proposition A.1. The spa e P of ontinuous and positive urves from S1to L is onne ted. In fa t, �xing two points xs 6= xu in S1, the �bers of themapP −→ X ⊂ L× Lξ 7−→ (ξ(xs), ξ(xu))are ontra tible.Proof. Consider the set

P0 = {ξ ∈ P | ξ(xs) = Ls0, ξ(xu) = Lu0}.Sin e Sp(2n,R) a ts transitively on X , it is enough to show that P0 is on-tra tible. For every ξ in P0 and every x 6= xs, ξ(x) and ξ(xs) are transverse,therefore we an regard P0 as a subset of the spa e of maps from S1 − {xs}to Sym(n,R). It is pre isely the set of maps ξ̃ : S1 − {xs} → Sym(n,R)su h that� limx→xs us(ξ̃(x)) · Lu0 = Ls0� ξ̃ is ontinuous� x→ us(ξ̃(x)) ·Lu0 is positive (i.e. it sends positive triples to positivetriples).� ξ̃(xu) = 0.This set of maps is a onvex subset of the spa e of all maps from S1 − {xs}into Sym(n,R) (this follows from the fa t that the set of positive de�nitematri es is onvex). The ontra tibility of P0 follows. �Proposition A.2. Let γ be a nontrivial element of π1(Σ). Then the spa eof pairs

Pγ ={(ρ, ξ) ∈ Hom(〈γ〉, Sp(2n,R)) × P | ξ is ρ-equivariant}has two onne ted omponents. If tsγ and tuγ are the �xed points of γ in

∂π1(Σ), then the onne ted omponents are dete ted by whi h element ofπ0

(Stab(ξ(tsγ)) ∩ Stab(ξ(tuγ))) ∼= F2 ontains ρ(γ).Proof. There are at least two onne ted omponents sin e the sign of

det(ρ(γ)|ξ(tsγ)) varies ontinuously.It is su� ient to understand the onne ted omponents of the �bers ofthe mapφ : Pγ −→ X

(ρ, ξ) 7−→ (ξ(tsγ), ξ(xu)).Again it is enough to al ulate the omponents of Pγ

0 := φ−1(Ls0, Lu0). Thepoints tsγ and tuγ divide the ir le ∂π1(Σ) in two intervals Isu and Ius: theyare hosen so that x belongs to Isu (respe tively Ius) if, and only if, the triple

(tsγ , x, tuγ) (respe tively (tuγ , x, t

sγ)) is positively oriented. These two intervalsare homeomorphi to R and isomorphisms are hosen so that the a tion of

γ is onjugate to t 7→ t+ 1 on R.

52 OLIVIER GUICHARD AND ANNA WIENHARDIt is not di� ult to show that a urve ξ : ∂π1(Σ) → L su h that ξ(ts/uγ ) =

Ls/u0 is positive if, and only if, the following onditions are satis�ed:� for all x in Isu the triple (Ls0, ξ(x), L

u0) is positive.� for all x in Ius the triple (Lu0 , ξ(x), Ls0) is positive.� the restri tion of ξ to Isu is positive (i.e. it sends positive triples topositive triples).� the restri tion of ξ to Ius is positive.Therefore we an onsider the two intervals Isu and Ius separately. Usingthe parametrization by symmetri matri es as above, it is su� ient to showthat the set

S ={(A, ξ̃) ∈ GL(n,R) × C0(R, Sym>0(n,R)) |

ξ̃(t+ 1) = Aξ̃(t) tA,∀s < t ξ̃(t) − ξ̃(s) > 0}has two onne ted omponents that are distinguished by the sign of detA.(Note that the ρ-equivarian e of ξ guarantees that limt→∞ us(ξ̃(t)) ·Lu0 = Ls0and limt→−∞ us(ξ̃(t)) · Lu0 = Lu0). Taking into a ount the natural a tion ofGL(n,R) on S redu es the question to determining the onne ted ompo-nents of the subset S0 := {(A, ξ̃) ∈ S | ξ̃(0) = Idn}. The map

S0 −→ GL(n,R)

(A, ξ̃) 7−→ A.has onvex, hen e ontra tible �bers. Its image is{A ∈ GL(n,R) | A tA− Idn ∈ Sym>0(n,R)

}.Using the Cartan de omposition of GL(n,R), it is easy to show that this sethas pre isely two onne ted omponents given by the sign of detA. �A.2. Deforming maximal representations in SL(2,R). The followingfa t follows from lassi al Fri ke-Klein theory for Tei hmüller spa e.Lemma A.3. Let (γi)i=1,...,k be a family of pairwise disjoint simple losedgeodesi s on Σ and denote by γi ∈ π1(Σ) the orresponding elements of thefundamental group. Let ι0 : π1(Σ) → SL(2,R) a dis rete embedding with

ι0(γi) =

(eλi0 00 e−λi0

). Let (λit)t∈[0,1] be ontinuous paths in R.Then there exists a ontinuous path of dis rete embeddings ιt : π1(Σ) →SL(2,R), t ∈ [0, 1] of ι su h that, for any t ∈ [0, 1], ιt(γi) =

(eλit 00 e−λit

).A.3. Twisting representations. In this se tion we explain the strategywhi h we used to al ulate the topologi al invariants for maximal represen-tations.

TOPOLOGICAL INVARIANTS 53A.3.1. The group π̂1(Σ). We �x a dis rete embedding of π1(Σ) into PSL(2,R):π1(Σ) < PSL(2,R). Let π : SL(2,R) → PSL(2,R) be the proje tion.We set π̂1(Σ) = π−1(π1(Σ)) ⊂ SL(2,R). The group π̂1(Σ) is a two-to-one over of π1(Σ), whi h is isomorphi to {±1}×π1(Σ). The isomorphism an be hosen so that it intertwines π with the se ond proje tion {±1} × π1(Σ) →π1(Σ). Any hoi e of su h an isomorphism amounts to hoosing a lift ofπ1(Σ) < PSL(2,R) to SL(2,R); su h lifts are in one-to-one orresponden ewith spin-stru tures on Σ.For the rest of this se tion we �x su h an isomorphism π̂1(Σ) = {±1} ×π1(Σ).A.3.2. Maximal representation of π̂1(Σ).De�nition A.4. A representation ρ̂ : π̂1(Σ) = {±1} × π1(Σ) → Sp(4,R) issaid to be maximal if the restri tion ρ̂|π1(Σ) is maximal (see De�nition 2.13).The set of maximal representation is denoted by Hommax(π̂1(Σ), Sp(4,R)).Let ρ̂ : π̂1(Σ) → Sp(4,R) be a maximal representation and ε : π̂1(Σ) →{±1} be any representation, then the representation ε · ρ̂, de�ned by γ 7→ε(γ)ρ̂(γ), is also maximal.If ρ : π1(Σ) → Sp(4,R) is a maximal representation, then ρ̂ = ρ ◦ pr2 :

π̂1(Σ) → Sp(4,R) is a maximal representation, where pr2 : π̂1(Σ) = {±1} ×π1(Σ) → π1(Σ) denotes the proje tion onto the se ond fa tor.Sin e T 1Σ ∼= π̂1(Σ)\SL(2,R) the notion of Anosov representations andAnosov redu tions (see Se tion 2.1) an be easily extended to representa-tion of π̂1(Σ). The following lemma is an immediate onsequen e of Theo-rem 2.15.Lemma A.5. Every maximal representation ρ̂ : π̂1(Σ) → Sp(4,R) isAnosov.Similarly to the dis ussion in Se tion 4, the Anosov redu tion leads to amap

Hommax(π̂1(Σ), Sp(4,R))sw1−→ H1(T 1Σ;F2)Fixing a nontorsion element γ̂ of π̂1(Σ), we introdu e the spa e

HomL+,γ̂max,sw1=0(π̂1(Σ), Sp(4,R))of pairs (ρ, L+) onsisting of a maximal representation ρ of π̂1(Σ) whose �rstStiefel-Whitney lass is zero and an attra ting oriented Lagrangian L+ for

ρ(γ̂). For those pairs, following the dis ussion in Se tion 4.3.2, we de�ne anEuler lassHom

L+,γ̂max,sw1=0(π̂1(Σ), Sp(4,R))ebγ−→ H1(T 1Σ;Z)

54 OLIVIER GUICHARD AND ANNA WIENHARDA.3.3. Relations between the invariants. In this se tion we des ribe the re-lations between topologi al invariants of maximal representations of π1(Σ)and π̂1(Σ). More pre isely:� If ρ : π1(Σ) → Sp(4,R) is a maximal representation, we omparethe invariants of ρ and ρ̂ = ρ ◦ pr2.� If ρ̂ : π̂1(Σ) → Sp(4,R) is a maximal representation and ε : π̂1(Σ) →{±1} a homomorphism, we ompare the invariants of ρ̂ and ε · ρ̂.Lemma A.6. Let ρ be a maximal representation of π1(Σ) and let ρ̂ = ρ◦pr2.Then

sw1(ρ) = sw1(ρ̂).When sw1(ρ) = 0, let γ be a nontrivial element of π1(Σ), γ̂ one of thetwo elements of π̂1(Σ) proje ting to γ and let L+ be an attra ting orientedLagrangian for ρ(γ) = ρ̂(γ̂). Theneγ(ρ, L+) = ebγ(ρ̂, L+).Proof. The (oriented) Lagrangian redu tions asso iated to ρ and for ρ̂ areexa tly the same, hen e their hara teristi lasses oin ide. �Lemma A.7. Let ρ̂ be a maximal representation of π̂1(Σ) and ε : π̂1(Σ) →

{±1} a representation. Then the �rst Stiefel-Whitney lass of ρ̂ and ε · ρ̂ oin ide:sw1(ρ̂) = sw1(ε · ρ̂).Proof. Let L1 and L2 be the two Lagrangians redu tions over T 1Σ asso iatedto ρ̂ and ε · ρ̂ respe tively, then

sw1(ρ̂) = sw1(L1) and sw1(ε · ρ̂) = sw1(L2).LetDε be the �at real line bundle (over T 1Σ) asso iated to the representationε. Then the �at bundle asso iated with ε · ρ̂ is the tensor produ t of Dε andthe �at bundle asso iated with ρ̂, therefore we have

L2 = Dε ⊗ L1.In parti ular, using Equation (3), we getsw1(ε · ρ̂) = sw1(L2) = 2sw1(Dε) + sw1(L1) = sw1(ρ̂).

�Proposition A.8. Let ρ̂ : π̂1(Σ) → Sp(4,R) be a maximal representationwith sw1(ρ̂) = 0. Let γ̂ ∈ π̂1(Σ) be a nontorsion element and L+ an attra t-ing oriented Lagrangian for ρ̂(γ̂). Let ε : π̂1(Σ) → {±1} be a homomorphism.Then L+ is an attra ting oriented Lagrangian for (ε · ρ̂)(γ̂) and the Euler lass (relative to γ̂) for the pair (ρ̂, L+) isebγ(ε · ρ̂) = ebγ(ρ̂) ∈ H2(T 1Σ;Z) if ε(−1) = 1,and

TOPOLOGICAL INVARIANTS 55ebγ(ε · ρ̂) = ebγ(ρ̂) + (g − 1)[Σ] ∈ H2(T 1Σ;Z) if ε(−1) = −1.Proof. Let L1 and L2 be the Lagrangian redu tions asso iated to ρ̂ and

ε · ρ̂. Denote by L1+ and L2+ the orresponding oriented Lagrangian bundlesdetermined by the hoi e of γ̂ and L+. If Dε is the �at real line bundle overT 1Σ asso iated to the representation ε, we have

L2+ = Dε ⊗ L1+.(Be ause L1 has even dimension there is a anoni al orientation on Dε⊗L1+,even if Dε is not oriented nor ne essarily orientable)Let S1 and S2 be the asso iated S1-bundles orresponding to L1+ and L2+and let Sε be the �at S1-bundle asso iated to the representation ε : π̂1(Σ) →{±1} ⊂ S1, then the above equality an be restated as

S2 = Sε ×S1 S1.This implies for the Euler lasses:e(S2) = e(Sε) + e(S1).And sin e e(S2) = ebγ(ε · ρ̂) and e(S1) = ebγ(ρ̂), the proposition will followfrom the following lemma. �Lemma A.9. Let ε : π̂1(Σ) → S1 be a representation and let Sε be theasso iated �at S1-bundle. Thene(Sε) = 0 if ε(−1) = 1,and

e(Sε) = (g − 1)[Σ] if ε(−1) = −1.Proof. First we note that e(Sε) varies ontinuously with ε. Hen e e onlydepends on whi h onne ted omponent of Hom(π̂1(Σ), S1) ε belongs to.Sin e Hom(π̂1(Σ), S1) = Hom({±1}× π1(Σ), S1) = Hom({±1}×Z2g, S1) ∼=

{±1} × (S1)2g, this spa e has two onne ted omponents distinguished pre- isely by the value of ε(−1).We only need to al ulate the Euler lass for two spe i� representa-tions. The �rst is the trivial representation for whi h the result is obvious.The se ond one is the proje tion pr1 : π̂1(Σ) ∼= {±1} × π1(Σ) → {±1}onto the �rst fa tor. Consider the (non-�at) S1-bundle over the surfa e:π1(Σ)\SL(2,R) → π1(Σ)\H, its Euler lass is given by the Toledo numberof the inje tion π1(Σ) → SL(2,R), whi h is (g − 1). The S1-bundle Sε isthe pullba k of this S1-bundle by the natural proje tion π̂1(Σ)\SL(2,R) ∼=T 1Σ → π1(Σ)\H ∼= Σ. This implies the laim. �

56 OLIVIER GUICHARD AND ANNA WIENHARDAppendix B. The ohomology of T 1ΣIn this se tion we ompute the ohomology of the unit tangent bundle T 1Σwith Z and F2 oe� ients and study the onne ting homomorphism in theMayer-Vietoris sequen e. The results are used in Se tion 4.4 and Se tion 5.5.Proposition B.1. Let Σ be a losed, onne ted, oriented surfa e of genusg > 1. The ohomology groups of T 1Σ with Z oe� ients are:� H0(T 1Σ;Z) = Z,� H1(T 1Σ;Z) = Z

2g,� H2(T 1Σ;Z) = Z2g × Z/(2g − 2)Z,� H3(T 1Σ;Z) = Z.The ohomology groups of T 1Σ with F2 oe� ients are:� H0(T 1Σ;F2) = F2,� H1(T 1Σ;F2) = F2g+12 ,� H2(T 1Σ;F2) = F2g+12 ,� H3(T 1Σ;F2) = F2.Proof. We prove only the statement about the ohomology with Z oe�- ients, the proof for F2 oe� ients is similar.The unit tangent bundle T 1Σ → Σ is a prin ipal S1-bundle whose Euler lass e is (2−2g) in Z ∼= H2(Σ;Z). The Gysin exa t sequen e for this bundleis

0 −→ H0(Σ;Z) −→ H0(T 1Σ;Z) −→ 0 −→ H1(Σ;Z)

−→ H1(T 1Σ;Z) −→ H0(Σ;Z)∪e−→ H2(Σ;Z) −→ H2(T 1Σ;Z)

−→ H1(Σ;Z) −→ 0 −→ H3(T 1Σ;Z) −→ H2(Σ;Z) −→ 0.The on lusion for H0 and H3 follows immediately from this. Sin e H0(Σ;Z)∪e−→H2(Σ;Z) is inje tive, we get the exa t sequen es:

0 −→ Z2g −→ H1(T 1Σ;Z) −→ 0,and

0 −→ Z/(2g − 2)Z −→ H2(T 1Σ;Z) −→ Z2g −→ 0.From this the result for H1 and H2 follows easily. �The above proof gives a anoni al isomorphism

Tor(H2(T 1Σ;Z)) ∼= Z/(2g − 2)Zbetween the torsion of H2(T 1Σ;Z) and Z/(2g−2)Z. In parti ular, [Σ] is the anoni al generator of Tor(H2(T 1Σ;Z)).Let γ be a simple losed oriented separating urve on the surfa e Σ, i.e.Σ− γ has two onne ted omponents, Σl denotes the omponent on the leftof γ and Σr the omponent on the right (this uses the orientations of γ andΣ). This indu es a de omposition of the unit tangent bundle: T 1Σ is the

TOPOLOGICAL INVARIANTS 57union of T 1Σ|Σland T 1Σ|Σr identi�ed along T 1Σ|γ . The Mayer-Vietorissequen e for this de omposition reads as

0 −→ H0(T 1Σ;Z) −→ H0(T 1Σ|Σl;Z) ⊕H0(T 1Σ|Σr ;Z)

−→ H0(T 1Σ|γ ;Z) −→ H1(T 1Σ;Z) −→ H1(T 1Σ|Σl;Z) ⊕H1(T 1Σ|Σr ;Z)

−→ H1(T 1Σ|γ ;Z) −→ H2(T 1Σ;Z) −→ H2(T 1Σ|Σl;Z) ⊕H2(T 1Σ|Σr ;Z)

−→ H2(T 1Σ|γ ;Z) −→ H3(T 1Σ;Z) −→ 0.This sequen e an also be used to ompute the ohomology of the unit tan-gent bundle. We on entrate on the onne ting morphismδ : H1(T 1Σ|γ ;Z) −→ H2(T 1Σ;Z)and its kernel.We realize γ as a simple losed geodesi on Σ, it then has a natural lift tothe unit tangent bundle T 1Σ whi h we denote again by γ. This lift indu esa trivialization T 1Σ|γ ∼= S1 × γ and hen e isomorphismsH1(T 1Σ|γ ;Z) ∼= H1(S1;Z) ⊕H1(γ;Z) ∼= Z ⊕ Z.The �rst identi� ation is the map in ohomology orresponding to the proje -tions T 1Σ|γ → S1 and T 1Σ|γ → γ whereas the se ond identi� ation involvesthe orientations on S1 and γ (the orientation on S1 ∼= T 1

xΣ is indu ed by theorientation on Σ).Proposition B.2. Let γ be an oriented losed simple separating geodesi onthe surfa e Σ.Then the orientation lass oγ ∈ H1(γ;Z) ∼= Z is sent to [Σ] by the on-ne ting homomorphism of the Mayer-Vietoris sequen e:H1(γ;Z) ⊂ H1(T 1Σ|γ ;Z)δ−→ H2(T 1Σ;Z).The kernel of δ is generated by the elements:

(1, 1 − 2g(Σl)) and (−1, 1 − 2g(Σr)) ∈ Z × Z ∼= H1(T 1Σ|γ ;Z).Proof. The onne ting homomorphisms for the de ompositions of the surfa eand the unit tangent bundle �t in a ommutative diagram:H1(γ;Z)δ−→ H2(Σ;Z)

↓ ↓H1(T 1Σ|γ ;Z)δ−→ H2(T 1Σ;Z)So the �rst result follows from the equality δ(oγ) = oΣ, where oΣ is theorientation lass in H2(Σ;Z). This equality is easy to establish. In fa t theMayer-Vietoris sequen e for the surfa e:H1(γ;Z) −→ H2(Σ;Z) −→ H2(Σl;Z) ⊕H2(Σr;Z)already shows that δ : H1(γ;Z) → H2(Σ;Z) is surje tive so that δ(oγ) =

±oΣ. The signs onventions are pre isely arranged so that δ(oγ) = oΣ.

58 OLIVIER GUICHARD AND ANNA WIENHARDDue to the exa tness of the Mayer-Vietoris sequen e the kernel of δ is theimage of H1(T 1Σ|Σl;Z) ⊕H1(T 1Σ|Σr ;Z) −→ H1(T 1Σ|γ ;Z).It is therefore enough to show that the image of(5) H1(T 1Σ|Σl

;Z) −→ H1(T 1Σ|γ ;Z) ∼= Z ⊕ Zis generated by (1, 1 − 2g(Σl)). (The al ulation for Σr is similar.) The ommutative square H1(Σl;Z) −→ H1(γ;Z)↓ ↓H1(T 1Σ|Σl

;Z) −→ H1(T 1Σ|γ ;Z)implies that the omposition H1(Σl;Z) ⊂ H1(T 1Σ|Σl;Z) → H1(T 1Σ|γ ;Z) iszero, be ause the map H1(Σl;Z) → H1(γ;Z) is zero as γ is a boundary in

Σl. The restri tion T 1Σ|Σlis the trivial bundle S1 × Σl so thatH1(T 1Σ|Σl;Z) ∼= H1(S1;Z) ×H1(Σl;Z).This means that the image of the above map (5) has rank 1 and is the imageof

Z ∼= H1(S1;Z) −→ H1(T 1Σ|γ ;Z) ∼= Z × Z.The �rst omponent of this last map is the identity Z ∼= H1(S1;Z) →H1(S1;Z) ∼= Z so that the image of (5) is generated by (1, n) for some integern. To al ulate this integer n let us onsider the losed surfa e Σ1 = Σl∪γD2obtained by gluing a disk along γ. The genus of Σ1 is g(Σ1) = g(Σl) andthe two S1-bundles T 1Σ|Σl

and T 1Σ1|Σlare isomorphi . From the Mayer-Vietoris sequen e for the de omposition of T 1Σ1 we getH1(T 1Σ|Σl

;Z) ⊕H1(T 1Σ1|D2 ;Z) −→ H1(T 1Σ|γ ;Z) −→ H2(T 1Σ1;Z)

−→ H2(T 1Σ|Σl;Z) ⊕H2(T 1Σ1|D2 ;Z) −→ H2(T 1Σ|γ ;Z).A small al ulation shows that

Z ∼= H1(T 1Σ1|D2 ;Z) → H1(T 1Σ|γ ;Z) ∼= Z × Zsends 1 to (−1, 1). The exa t sequen e redu es to0 −→ Z

2/〈(1, n), (−1, 1)〉 −→ H2(T 1Σ1;Z) −→ Z2g(Σl) −→ 0.This exa t sequen e implies that the torsion of H2(T 1Σ1;Z) is isomorphi to

Z/(n+ 1)Z. This torsion being isomorphi to Z/(2g(Σl) − 2)Z by Proposi-tion B.1 the value of n is 1 − 2g(Σl) or 2g(Σl) − 3.Doubling Σl along γ to obtain a losed surfa e, a similar argument showsthat the torsion of the double of Σl is isomorphi to Z/(2n)Z. Thus the onlypossible value for n is 1 − 2g(Σl) be ause the genus of the double of Σl is2g(Σl)). �

TOPOLOGICAL INVARIANTS 59Lemma B.3. Let {ηi} be urves in Σ whose images generate the homologygroup H1(Σ;Z); we denote by fi : ηi → Σ the orresponding map. Witha slight abuse of notation we write T 1Σ|ηifor the pulled ba k ir le bundle

f∗i T1Σ and denote again by fi : T 1Σ|ηi

→ T 1Σ the orresponding map.Let c be a lass in H2(T 1Σ;A) su h that, for all i,c|T 1Σ|ηi

:= f∗i (c) = 0.Then c belongs to Im(H2(Σ;A) → H2(T 1Σ;A)).Proof. Using the Gysin exa t sequen e with A oe� ients we only need toshow that the image a of c in H1(Σ;A) is zero. As {ηi} generates the ho-mology it will be the ase if f∗i (a) = 0 for all i. Observe that the Gysinsequen es for T 1Σ and for T 1Σ|ηi�t in a ommutative diagramH2(T 1Σ;A) −→ H1(Σ;A)

↓ ↓H2(T 1Σ|ηi;A) −→ H1(ηi;A).So the property follows from the hypothesis f∗i (c) = 0. �Referen es[1℄ M. F. Atiyah. Riemann surfa es and spin stru tures. Ann. S i. É ole Norm. Sup. (4),4:47�62, 1971.[2℄ M. F. Atiyah and R. Bott. The Yang-Mills equations over Riemann surfa es. Philos.Trans. Roy. So . London Ser. A, 308(1505):523�615, 1983.[3℄ T. Barbot. Three-dimensional Anosov �ag manifolds. arXiv/0505500, 2005.[4℄ T. Barbot. Quasi-Fu hsian AdS representations are Anosov. arXiv/0710.0969, 2007.[5℄ N. Bergeron and T. Gelander. A note on lo al rigidity. Geom. Dedi ata, 107:111�131,2004.[6℄ S. B. Bradlow, O. Gar ía-Prada, and P. B. Gothen. Deformations of maximal repre-sentations in Sp(4, R). preprint.[7℄ S. B. Bradlow, O. Gar ía-Prada, and P. B. Gothen. Surfa e group representationsand U(p, q)-Higgs bundles. J. Di�erential Geom., 64(1):111�170, 2003.[8℄ S. B. Bradlow, O. Gar ía-Prada, and P. B. Gothen. Maximal surfa e group represen-tations in isometry groups of lassi al Hermitian symmetri spa es. Geom. Dedi ata,122:185�213, 2006.[9℄ M. Burger, A. Iozzi, F. Labourie, and A. Wienhard. Maximal representations of sur-fa e groups: Symple ti Anosov stru tures. Pure and Applied Mathemati s Quaterly.Spe ial Issue: In Memory of Armand Borel., 1(2):555�601, 2005.[10℄ M. Burger, A. Iozzi, and A. Wienhard. Maximal representations and anosov stru -tures. in preparation.[11℄ M. Burger, A. Iozzi, and A. Wienhard. Tight homomorphisms and embeddings. toappear in GAFA.[12℄ M. Burger, A. Iozzi, and A. Wienhard. Surfa e group representations with maximalToledo invariant. C. R. A ad. S i. Paris, Sér. I, 336:387�390, 2003.[13℄ M. Burger, A. Iozzi, and A. Wienhard. Surfa e group representations with maximalToledo invariant. math/0605656, to appear in Annals of Mathemati s, 2006.[14℄ O. Gar ía-Prada, P. B. Gothen, and I. Mundet i Riera. Representations of surfa egroups in real symple ti groups. arXiv/0809.0576, 2008.[15℄ O. Gar ía-Prada and I. Mundet i Riera. Representations of the fundamental groupof a losed oriented surfa e in Sp(4, R). Topology, 43(4):831�855, 2004.

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