On the divisibility of characteristic classes of non-oriented surface bundles

ON THE DIVISIBILIT Y OF CHARA CTERISTIC CLASSES OF NON-ORIEN TED SU RF A CE BUNDLES JOHANNES EBER T, OSCAR RANDAL-WILLIAMS Abstract. In this note w e introduce a construction which assigns to an arbi- trary manifold bundle its fib erwise orientation co v ering. This i s used to show that the zeta classes of unorien ted surface bundles are not divisible in t he stable r ange. 1. Introduction The mapping class group N g of a non-or ie n table surface S g of genus g (that is, the co nnected sum of g copies of RP 2 ) is defined to b e N g := π 0 (Diff ( S g )) the group of comp onents of the diffeomor phis m group of tha t surface. If g ≥ 3, the co mpo nen ts o f Diff ( S g ) ar e contractible [3], hence B N g ≃ B Diff ( S g ), and so the coho mology of B N g (or the group co homology of N g ) can be interpreted a s the ring o f c haracteris tic classes for S g -bundles. W ahl [9] has prov ed a homologic a l stability theorem for these groups, which says that in degr e es ∗ ≤ ( g − 3) / 4 the cohomolo gy groups H ∗ ( N g ) are indepe ndent of the genus g . W e c a ll this r ange of degr ees the stable r ange . With rational co efficients these stable groups can be identified: there ar e certain in tegral c haracteristic classes ζ i in degree 4 i and the map Q [ ζ 1 , ζ 2 , ζ 3 , ... ] → H ∗ ( N g ; Q ) is an isomor phis m in the stable rang e. In [8] the second author ca lculates these stable g roups with co efficient s in a finite field, a nd tabulates so me low-dimensional int egral gr oups. The classes ζ i are a nalogues of the even Miller– Mo rita–Mumford classes , for unoriented sur face bundles. Galatius, Madsen and Tillmann [6] hav e s tudied the divisibility of the Miller –Morita– Mumford clas s es κ i ∈ H ∗ (Γ ∞ ; Z ) in the free par t of the integral cohomolog y of the stable mapping cla ss group Γ ∞ . They find that the even classes are divisible by 2 and the o dd clas ses a re divisible by a denominator of a B ernoulli num ber. In [5] the first author studied the div isibilit y of the Miller– Morita–Mumfor d classes for surface bundles with spin structur es, a nd it was shown that the divisibilit y increas es by a certain p ow er of 2 re lative to the non-spin case. Contin uing the study of divisibility o f c haracteris tic cla sses of surface bundles, we prov e Theorem A. The universal zeta classes, ζ n ∈ H 4 n ( N g ; Z ) , ar e not divisible in the stable r ange. Inde e d, they ar e n ot divisible in the fr e e quotient of c ohomol o gy H 4 n f r ee ( N g ; Z ) in this r ange. 2000 Mathematics Subje ct Classific ation. 57R20. J. Eb ert is s upp orted by a fellowship within the Postdoc-Programme of the German Academic Exc hange Servi ce (D AAD); O. Randal-Williams is supp orted by an EPSRC Student ship, DT A grant n um ber EP/P502667/1. 1 2 JOHANNES EBER T, OS CAR RANDAL-WILLIAMS This g ives the trend that extra structure on the vertical ta ngent bundle, such as an orientation o r a s pin structure, gives extra divisibility of characteris tic clas ses of surface bundles. 2. Lifting diffeomorphisms to orient a tion coverings In this se c tion, we will constr uc t a natura l ho momorphism from the diffeomor - phism gr oup Diff ( M ) o f a smo oth d -manifold to the group Diff + ( ˜ M ) of orientation- preserving diffeomorphisms of the orientation cov ering of M . This implies that any smo oth fib e r bundle p : E → B admits a t wo-fold covering π : ˜ E → E , suc h that p ◦ π : ˜ E → B is an or iented smo oth fibe r bundle and that the r e s triction of π : ˜ E → E to a fib er of p is the orientation cov ering. Let M be a smo oth d -manifold, d > 0, a nd let Λ d T M b e the highest e xterior power o f the tangent bundle, which is a rea l line bundle. The total space of the orientation c overing of M can b e defined as (2.1) ˜ M := (Λ d T M \ 0) / R > 0 . The canonical map π : ˜ M → M is a tw o-sheeted cov ering. The space ˜ M is a smo oth manifold, which is orientable. In fact, there is a preferre d orientation o f ˜ M . T o see this, recall tha t an orientation of a d -dimensional rea l vector space V is a co mpo nent of Λ d V \ 0, o r, in other words, one of the tw o p o in ts of (Λ d V \ ) / R > 0 . Thus a p oint in x ∈ ˜ M is by definition an or ient ation of the ta ng ent space T π ( x ) M . On the other hand, the ma p π is a covering and hence a lo c a l diffeomor phis m. The differ ent ial T x π at x ∈ ˜ M is a linear isomorphism T x ˜ M → T π ( x ) M . It follows that T x ˜ M has a preferred or ientation: the p oint x defines an orie ntation o f T π ( x ) M and therefore one of T x ˜ M via the linear is omorphism T x π . Using lo ca l co or dina tes on M , it is easy to see that the orientations of the tang ent spaces T x ˜ M constr ucted ab ov e fit contin uo usly together and define an o rientation of ˜ M , the pr efe rr e d orientation of M . Moreov er, this constr uction is na tural: a diffeomorphism f : M → N of smo o th manifolds induces a diffeomorphis m ˜ f : ˜ M → ˜ N which cov ers f . It is easy to see that ˜ f is orientation-preserving pr ovided ˜ M and ˜ N are endow ed with the pr eferred orientations. If g : N → P is ano ther diffeomor phism, then ] g ◦ f = ˜ g ◦ ˜ f . Also, ˜ id M = id ˜ M . Finally , we did not use that f is a diffeo mo rphism, but o nly that the differ e n tial of f was nonsingular . It follows that the assignments M 7→ ˜ M and f 7→ ˜ f define a functor L from the ca tegory X d of s mo oth d -manifolds and lo cal diffeomor phisms to the category X + d of oriented d -manifolds and orientation- preserving lo c a l diffeomorphisms. In particular , we defined a g roup homo mo rphism L M : Diff ( M ) → Diff + ( ˜ M ). F or a ma nifo ld M , we deno te by π M the cov ering map ˜ M → M and by ι M : ˜ M → ˜ M the unique nontrivial deck tr a nsformation. If f : M → N is a (lo cal) diffeomorphism, the following relations hold (2.2) π N ◦ ˜ f = f ◦ π M ; ˜ f ◦ ι M = ι N ◦ ˜ f . The morphism spac e s of the ca tegories X d and X + d hav e a natural topolo gy , the weak C ∞ -top ology , with resp ect to which the comp ositio n maps are co ntin uous . Thus X d and X + d are top olog ic a l categ ories. Using lo cal co o rdinates, it is easy to see that the functor L is contin uous. In particular , the homo rphism L M : Diff ( M ) → Diff + ( ˜ M ) is contin uous. Let us now discuss smo oth fib er bundles. Let p : E → B b e a smo oth fib er bundle with fibe r a d -dimensiona l smo o th manifold M and structura l g r oup Diff ( M ) (with the weak C ∞ -top ology). Cons ider the asso cia ted Diff ( M )-principal bundle Q → B , which has the prop er t y that Q × Diff ( M ) M ∼ = E . Via the homomo r phism L M , the DIVISIBILITY OF CHARACT ERISTIC CLASSES 3 manifold ˜ M has a Diff ( M )- a ction by orient ation preserving diffeomor phisms. Hence the fib er bundle q : ˜ E := Q × Diff ( M ) ˜ M → M is an oriented s mo oth fib er bundle with fiber ˜ M . Be cause o f (2.2), there is a tw o fold cov ering π E : ˜ E → E , s uch that q = p ◦ π E . F urthermore , there is a fiber -preser v ing and orientation-reversing inv o lution ι E on ˜ E . W e call ˜ E the fib erwise orientation c over o f E . W e summarize the r esults of this section. Theorem 2.1 . The fib erwise orientation c over π : ˜ E → E of a smo oth fib er bun- d le p : E → B is a two-she ete d c overing whose re striction t o any fib er E b of p is t he orientation c over of E b . The c omp osition q = p ◦ π E is an oriente d fib er bund le. F urthermor e, ˜ E and π E ar e uniquely determine d by these pr op erties (up to orientation-pr eserving isomorphism). W e co nclude with a simple remar k . All the co nstructions in this section ma ke sense when the ma nifold M (or the fib er bundle E ) is o rientable. If this is the case, then ˜ M is the disjoin t sum of tw o copies o f M . T he choice of an o rientation of M singles o ut a comp onent of ˜ M . 3. Characteristic classes of surf ace bundl es In this sectio n, we g ive a brief review of the theor y of characteristic classes of surface bundles, both or ie n ted and non-o riented. First we discuss the o riented ca s e. Let π : E → B be an o riented surface bundle a nd let T v E b e the vertical tang ent bundle. It is an o riented 2-dimensio nal real vector bundle o n E a nd thus it has an Euler class e ( T v ( E )) ∈ H 2 ( E ; Z ). W e c an conside r T v E als o as a complex line bundle (there is a complex structure on it, which is unique up to isomor phism) and the Euler class agree s with the first Chern class. The Miller –Morita– Mumford classes ar e defined to b e κ n ( E ) := π ! ( e ( T v E ) n +1 ) ∈ H 2 n ( B ; Z ) , where π ! : H ∗ ( E ; Z ) → H ∗− 2 ( B ; Z ) is the umkehr, or cohomolo gical fiber-integration, map. This definition cannot b e generalized to the non-orie n ted ca se without fur- ther effort, b eca use b oth the E uler class and the umkehr map only exis t for orie nted surface bundles. The concept ne e ded for a genera liz a tion is the Becker–Gottlieb tr ansfer [1]. Let p : E → B b e a smo oth fib er bundle with compa ct fiber s diffeomorphic to F (not necessarily of dimension 2 ). The transfer is a stable map in the conv erse direc tio n, more pr ecisely , it is a map of the suspens io n sp ectra trf p : Σ ∞ B + → Σ ∞ E + . Recall that the sp ectrum cohomolo gy of the susp ensio n sp ectrum of a s pa ce Σ ∞ X + agrees with the ususal cohomolog y of the spa ce X . Th us we can for m the map trf ∗ p ◦ p ∗ : H ∗ ( B ; Z ) → H ∗ ( B ; Z ), and for all x ∈ H ∗ ( B ; Z ) we hav e (3.1) trf ∗ p ◦ p ∗ ( x ) = χ ( F ) · x , where χ ( F ) denotes the E uler num ber of the fib er ([1, Theorem 5 .5]). F urthermor e , if q : ˜ E → E is ano ther smo oth fib er bundle with compact fib ers, then p ◦ q is also such a fib er bundle. In this situation the comp os itio n of the transfer s is homotopic to the transfer o f the comp osition (se e [2, Equa tion 2.2, page 1 3 7]): (3.2) trf p ◦ q ≃ tr f q ◦ trf p . A diffeomor phism f : M → N o f manifolds can b e considered as a fiber bundle whose fib er is a po int . By (3.1), (3.3) trf ∗ f ◦ f ∗ = id H ∗ ( N ; Z ) , f ∗ ◦ trf ∗ f = id H ∗ ( M ; Z ) . 4 JOHANNES EBER T, OS CAR RANDAL-WILLIAMS In fact, trf f and Σ ∞ ( f − 1 ) are homotopic, but we do not need this fact. The transfer of an oriented fib er bundle p : E → B is closely related to the umkehr ma p. F or all x ∈ H ∗ ( E ; Z ), one ha s (see [1, Theorem 4.3]) (3.4) trf ∗ p ( x ) = p ! ( x ∪ e ( T v E )) . The identit y (3.4) implies (3.5) κ n ( E ) = tr f ∗ p ( e ( T v E ) n ) for the Miller– Morita–Mumfor d classe s of a n orie nted sur face bundle p : E → B . Because of the identit y p 1 ( L ) = e ( L ) 2 for the Pont rjagin class o f a 2 -dimensional oriented real vector bundle L , we see that (3.6) κ 2 n ( E ) = tr f ∗ p ( p 1 ( T v E ) n ) . This gener alises to the non- oriented cas e. W ahl defines ([9, page 3]) (3.7) ζ i ( E ) := trf ∗ p ( p 1 ( T v E ) i ) ∈ H 4 i ( B ; Z ) , for a non-oriented surfa c e bundle p : E → B , where p 1 ( T v E ) ∈ H 4 ( E ; Z ) is the firs t Pon trjagin cla ss of the vertical tangent bundle. Now we c an state and prov e the main result of this section. Theorem 3.1. L et p : E → B b e a non-oriente d sur fac e bund le with c omp act fib ers and let c : ˜ E → E b e its fib erwi se orientation c overi ng. Denote q := p ◦ c : ˜ E → B . Then the fol lowing r elations hold: (1) F or al l n ≥ 0 , we have κ 2 n ( ˜ E ) = 2 · ζ n ( E ) . (2) F or al l n ≥ 0 , we have 2 · κ 2 n +1 ( ˜ E ) = 0 . Pr o of. F or the identit y (1), we c ompute κ 2 n ( ˜ E ) = tr f ∗ q (( p 1 ( T v ˜ E ) n ) = tr f ∗ p (trf ∗ c ( c ∗ ( p 1 ( T v E ) n ))) = tr f ∗ p (2 · p 1 ( T v E ) n ) = 2 · ζ n ( E ) . The fir st eqality is (3 .6). Because c : ˜ E → E is a smo oth covering in every fib er, c ∗ ( T v E ) ∼ = T v ( ˜ E ), whence p 1 ( T v ˜ E ) = c ∗ ( p 1 ( T v E )). T o gether with (3.2), this fact implies the s e cond eq uality . Beca us e c is a double covering, the E uler num ber o f its fiber is 2. Thus trf ∗ c ◦ c ∗ = 2 , which gives the third equality . The fourth equa lit y is the definition. F or the pr o of of iden tit y (2), we use the orientation-reversing involution ι on ˜ E . By (3.3), trf ∗ ι = ( ι ∗ ) − 1 = ι ∗ . Because c ◦ ι = c , it follows that tr f ∗ c = trf ∗ c ◦ trf ∗ ι = trf ∗ c ◦ ι ∗ . Becaus e ι is an orientation-reversing fiber wise diffeomo rphism, it induces a n o rientation-reversing vector bundle isomorphism dι : T v ˜ E → ι ∗ T v ˜ E . Thu s e ( T v ˜ E ) = − ι ∗ e ( T v ˜ E ). Thus κ 2 n +1 ( ˜ E ) = tr f ∗ p trf ∗ c ( e ( T v ( ˜ E ) 2 n +1 ) = trf ∗ p trf ∗ c ι ∗ ( e ( T v ( ˜ E ) 2 n +1 ) = ( − 1) 2 n +1 trf ∗ p trf ∗ c ( e ( T v ( ˜ E ) 2 n +1 ) = − κ 2 n +1 ( ˜ E ) .  R emark 3.2 . An implication o f this theor em is that for an or ient ed surface bundle E ′ → B , the characteristic clas s es 2 · κ 2 n +1 ( E ′ ) are o bstructions to E ′ admitting an orientation-reversing fixed- po int free fiber wise inv olution. F ur thermore, for bundles DIVISIBILITY OF CHARACT ERISTIC CLASSES 5 which do admit such an inv olution, it gives an interpretation of 1 2 κ 2 n ( E ′ ) as the zeta clas ses of the asso ciated quotient bundle of non-or ientable surfac es. 4. An example In this section, w e consider the rather easy ex a mple of a g enus zero surface bundle. Let γ 3 → B S O (3) b e the universal 3 -dimensional oriented Riemannian real vector bundle and let S ( γ 3 ) → B S O (3) b e its unit sphere bundle. It is k nown that this is the univ ersal smo o th oriented bundle with fib er S 2 , but we do not ne e d this fact. It is not har d to see that κ 2 n ( S ( γ 3 )) = 2 p 1 ( γ 3 ) n , co mpare [4 , pa ge 49]. The bundle S ( γ 3 ) a dmits an orientation-reversing, fixed-p oint free inv olution on its fib ers, namely the antipo dal ma p − id. The quotient is P ( γ 3 ), the RP 2 -bundle asso ciated to γ 3 . By Theorem 3.1, we have (4.1) 2 ζ n ( P ( γ 3 )) = 2 p 1 ( γ 3 ) n . The integral coho mology ring of B S O (3) is not to o hard to compute. Denote by χ ∈ H 3 ( B S O (3); Z ) the universal Euler class. Then w e hav e (4.2) H ∗ ( B S O (3); Z ) ∼ = Z [ p 1 , χ ] / (2 χ ) . T o see this, one uses the Leray–Serre s pec tr al sequence of the fibration S 2 → B S O (2) → B S O (3). In particular , the p owers p n 1 ( γ 3 ) are not div is ible in the free quotient H ∗ f r ee ( B S O (3); Z ) of H ∗ ( B S O (3); Z ). W e have shown: Prop ositio n 4.1. The class ζ n ( P ( γ 3 )) is n ot divisible in H ∗ f r ee ( B S O (3); Z ) . R emark 4.2 . It may lo ok a bit co nfusing that the universal RP 2 -bundle has bas e space B S O (3), but this is indeed the case. The r eason is the isomo r phism of gr oups P O (3 ) ∼ = O (3) / {± 1 } ∼ = S O (3). 5. A review of the st able homotopy theor y of surf aces and proof of Theorem A In this section, we give a brief survey of the mo de r n homotopy theor y o f surface bundles developped by Tillma nn, Madsen, W eis s and Galatius. Le t us first discuss the oriented case. A go o d survey can be found in [6], which also contains refer ences to all re lev ant pa pe r s. 1 A new pro of of the main re sults which can b e gener alized to the non-oriented case can b e found in [7]. Consider the universal complex line bundle L → B S O (2). There do es not exis t a vector bundle V suc h that V ⊕ L is trivial. But we can define a n additive inv erse L ⊥ of L as a stable ve ctor bund le . The Madsen–Til lmann sp e ctrum MTSO (2 ) is by definition the Thom sp ectrum of L ⊥ . F or any o riented surface bundle E → B , there exists a na tural map α E : B → Ω ∞ 0 MTSO (2) int o the unit comp onent of the infinite lo op space of the Ma dsen–Tillmann sp ec- trum. In pa r ticular, it can b e defined for the universal oriented surfa c e bundle with fiber s a surface F g of genus g . W e obtain a univ ersal map α g : B Diff + ( F g ) → Ω ∞ 0 MTSO (2) . F or all n > 0, there exists a cohomolog y class y n ∈ H 2 n (Ω ∞ 0 MTSO (2); Z ) such that for any surface bundle (5.1) α ∗ E ( y n ) = κ n ( E ) . 1 The pap er [ 6] uses a different notation: they denote MTSO (2) by CP ∞ − 1 . 6 JOHANNES EBER T, OS CAR RANDAL-WILLIAMS The ra tional cohomo lo gy o f Ω ∞ 0 MTSO (2) is isomor phic to the po lynomial ring Q [ y 1 , y 2 , . . . ]. The map α g induces an isomorphism on ho mology groups in the stable ra nge, that is , H k ( α g ) : H k ( B Diff + ( F g ); Z ) → H k (Ω ∞ 0 MTSO (2); Z ) is an isomorphism a s long a s g ≥ 2 k + 2 . Similar r e sults are true in the non-o riented case. The Madsen–T illma nn sp ec- trum is r eplaced by MTO (2 ), which is the Thom sp ectrum of the stable inv erse of the universal 2-dimensiona l real vector bundle ov er B O (2). There exis ts an analogue o f the map α for any non-or ient ed surface bundle. There are classes x n ∈ H 4 n (Ω ∞ 0 MTO (2); Z ), n > 0, s uch that α ∗ E ( x n ) = ζ n ( E ). These things are completely analogo us to the oriented case. The rational cohomolo gy ring of Ω ∞ MTO (2) is isomorphic to the poly nomial ring Q [ x 1 , x 2 , . . . ]. The analog ue of the Madsen–W eiss theo r em is a ls o true in the non-or ient ed case. More pr ecisely (5.2) H k ( α ; Z ) : H k ( B Diff ( S g ); Z ) → H k (Ω ∞ 0 MTO (2); Z ) is a n is o morphism as long as 4 k + 3 ≤ g . The pro o f of this theorem consis ts of tw o parts: one part is the pr o of of the analo gue of the Ha rer–Iv anov s tabilit y theorem in the non-or ient ed ca se and was done by W ahl [9 ]. The other par t is the determination of the homotopy type of an appropr iate cob or dism category by Galatius, Mads en, Tillmann and W eiss [7 ]. Pr o of of The or em A. This is now stra ightforw ard. W e a s sume that the universal class ζ n is div is ible in the sta ble r ange. It follows, by (5.2) that the class x n ∈ H 4 n (Ω ∞ M T O (2); Z ) is also divis ible. W e hav e s een in P rop osition 4.1 that the image of x n ∈ H 4 n ( B S O (3); Z ) under the map α : B S O (3) → Ω ∞ MTO (2) is not divisible. This is a contradiction.  References [1] J. C. Beck er, D. H. Gottlieb: The tr ansfer map and fib er bund les , T op ology 14 (1975), 1-12. [2] G. Brumfiel, I. M adsen: Evaluation of the tr ansfer and the universal sur gery classes , Inv ent. Math. 32 (1976), 133-169. [3] C.J. Ear l e, J. Eells: A fibr e bund le description of Te ichm¨ ul ler the ory , J. Differential Geometry 3 (1969), 19-43. [4] J. Eb ert: Char acte ri stic classes of spin surfac e bund les: Applic ations of the Madsen-Weiss the ory , PhD thesis, Bonner Mathematische Sc hriften 381 (2006). [5] J. Eb ert: D ivisibility of Mil ler–Morita–Mumfor d classes of spin surfac e bund les . T o appear in Quart. J. M ath. [6] S. Galatius, I. M adsen, U. Tillmann: Div isibility of the stable Mil ler-Morita-Mumfor d classes , Journal A.M.S. 19 (2006) 759-779. [7] S. Galatius, I. Madsen, U. Ti llmann, M . W eiss: The homotopy typ e of the co b or dism c ate gory . T o appear in Acta Math., electronic prepri n t arXiv: m ath/0605 249. [8] O. Randal-Williams: The homolo gy of the stable non-orientable mapping class gr oup , elec- tronic preprint arXiv:0803.3825 . [9] N. W ahl: Homolo gic al stability f or the mapping c lass gr oup s of non-orient able surfac e s , Inv en t. Math. 171 (2008) , 389-424. Ma thema tical Institute, 24-29 St Giles’, Oxford, OX1 3LB, England E-mail addr ess : ebert@maths.ox. ac.uk, randal-w@mat hs.ox.ac .uk