The low-dimensional homotopy of the stable mapping class group

The lo w-dimensional homotop y of the stable mapping class group Johannes F. Eb ert ∗ e-mail : eb ert@m a ths.ox.ac.uk Octob er 29, 2018 Abstract Due to the deep w ork of Tillmann, Madsen, W eiss and Galatius, the cohomology of the stable map p ing class group Γ ∞ is kno wn with rational or fi nite field co efficien ts. Little is kno wn ab out the inte gral cohomology . In this pap er , we study the first four cohomology groups. Also, w e compu te the first few s teps of the P ostniko v tow er of B Γ + ∞ , the Quillen plus construction a pp lied to B Γ ∞ . Our metho d relies on the Madsen-W eiss theorem, a few kno wn computations of stable homotop y groups of spheres and p r o j ectiv e spaces and on a certain action of the binary icosahedral group on a surface. Using the latter, we can also describ e an explicit geo metric generator of the third homotop y group π 3 ( B Γ ∞ ). Keyw ords: stable mapping class group, sym m etries of Riemann surfaces, Postnik o v to w ers, cob ordism theory 1 In tro duction and o v erview Let Γ g ,b b e the gr o up of isotopy classes of or ientation-preserving diffeomorphisms of a connected compact surface F g ,n of gen us g with b b oundary comp onen ts (b o th, diffeo- morphisms a nd isotopies are assumed to fix the b oundary p oin twis e), in other w ords, the mapping class gr oup . W e w rite Γ g := Γ g , 0 . There are stabilization maps defined for b > 0 : • h : Γ g ,b → Γ g ,b +1 (gluing in a pair of pan ts along one boundar y c omp o nent), • i : Γ g ,b +1 → Γ g +1 ,b , (glueing in a p air o f pan ts along t w o b oundary comp onents ), • j : Γ g ,b → Γ g ,b − 1 , (glueing in a disc). ∗ Suppo rted by a fellowship within the Postdo c-Prgr amme of the German Academic Exchange Ser vice (D AAD) 1 The homological stability theorem of Harer [8], a s impro v ed b y Iv ano v [12 ], [1 3 ]) asserts that H k ( h ; Z ) and H k ( i ; Z ) are isomorphisms whe nev er g ≥ 2 k + 1, w hile H k ( j ; Z ) is an isomor- phism w hen g ≥ 2 k + 2. It follo ws that H k ( B Γ g ,n ) do es not dep end on g or n in the stable r a n ge , i.e. as long as g ≥ 2 k + 2. L et Γ ∞ ,b := colim( . . . → Γ g ,b +1 → Gamma g +1 ,b +1 → . . . , where the comp osition i ◦ h is used to fo r m the colimit. The homology groups of this infinite mapping class group are the stable homology groups of the mapping class group. Due to the work of Tillmann [24 ], Madsen, W eiss [15] and Galatius [6], the stable ho- mology of Γ g is completely understo o d, at least in theory . Let us desc rib e this stor y briefly . Ev en b efor e Harers w ork, P o w ell [18] sho w ed that Γ g is p erfect if g ≥ 3, i.e. H 1 ( B Γ g ; Z ) = 0. The Q uillen plus construction from a lg ebraic K-theory pro vides (see [19]) a simply-connected space B Γ + ∞ ,b and a homology equiv alence B Γ ∞ ,b → B Γ + ∞ ,b . The glueing m aps j induce a homotop y equiv alence B Γ + ∞ ,b ≃ B Γ + ∞ . While B Γ ∞ is b y defini- tion aspherical, B Γ + ∞ has man y non trivial homotop y g roups. Tillmann prov ed in [24] tha t Z × B Γ + ∞ has the homotop y t yp e of an infinite lo op space. Later, she and Madsen [14] constructed a sp ectrum MTSO (2) and maps α g ,b : B Γ g ,b → Ω ∞ 0 MTSO (2). Madsen and W eiss sho w ed that H k ( α g ,b ) is an isomorphism if g ≥ 2 k + 2, in other w ords, the limit map B Γ + ∞ → Ω ∞ 0 MTSO (2) is a homotopy equiv alence. One consequence of these results is the existence of a map α g : B Γ g → Ω ∞ 0 MTSO (2) ≃ B Γ + ∞ (without b oundaries!) whic h induces isomorphisms in homolo g y in the stable range. The rational homolog y of Ω ∞ 0 MTSO (2) is easy to determine (it is isomorphic to the ra- tional homology of B U , see [1 4]). The homolo gy with F p -co efficien ts for all primes p w as computed b y Gala t ius [6 ]. His result is v ery complicated and it is v ery hard to destill explicit inf o rmation form it, eve n in small dimensions. The aim of this note is the study of the homotop y and homology groups of B Γ + ∞ up to dimension 4. The first few homotopy gr oups of Ω ∞ MTSO (2) a nd hence a lso of B Γ + ∞ w ere already computed in [14]. The result is Theorem 1.0.1. π k ( B Γ + ∞ ) ∼ =          0 if k = 1 Z if k = 2 Z / 24 if k = 3 Z if k = 4 . This fo llo ws fro m Serre’s classical computation of the first v alues of π k (Ω ∞ Σ ∞ S 0 ) and fr om a computation by Muk ai [17]. Theorem 1.0.2. Th e first few homolo gy gr oups of the stable mapping class g r oup ar e giv en by 2 H k ( B Γ + ∞ ; Z ) ∼ =          0 if k = 1 Z if k = 2 Z / 12 if k = 3 Z 2 if k = 4 . The first homology was computed by Po w ell, as men tioned ab ov e, the second b y Harer [9]. Harer also pro v ed that H 3 ( B Γ ∞ ) is a torsion group, s ee [10] and determined the rank of H 4 ( B Γ ∞ ) (unpublished). Our metho d for the pro of of theorem 1.0.2 is a quite indirect in the sense that w e use Theorem 1.0.1 fo r it. The relation b et w een the homology and homotop y gr o ups is express ed by the first stages o f the Postnik ov to we r of B Γ + ∞ . The Postnikov tower of a simple space X tells us ho w X is built from t he Eilen b erg- MacLane spaces K ( π n ( X ) , n ). W e will compute the P ostnik o v tow er of B Γ + ∞ up to de gree 4. W e in tro duce the follo wing notation. A map f : X → Y is a π n -isomorphism , if it induces an isomorphism π n ( X ) → π n ( Y ). A space Y is N -c o c o nne cte d if π k ( Y ) = 0 for all k > n . If X is a simple space, t hen there is a n n -co connected space τ ≤ n X and a map a n : X → τ ≤ n X , whic h is a π k -isomorphism for k ≤ n . These data are unique up to homotopy . The maps a n assem ble to a homot o p y-comm utativ e diagram . . .   τ ≤ n +1 X p n +1   X a n − 1 # # G G G G G G G G G a n / / a n +1 ; ; w w w w w w w w w D D                    6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 τ ≤ n X p n   τ ≤ n − 1 X   . . . W e can ass ume that the maps p n are fibrations. The homotop y type of the space X can b e reco ve red as the inv erse limit of this tow er of fibrations. The homotopy fib er o f p n is an Eilen b erg-MacLane space K ( π n ( X ) , n ). Th us the fibration p n is classified b y a map k n : τ ≤ n − 1 X → K ( π n ( X ); n + 1) in the sense that p n is the homotopy-pullbac k of the path- lo op fibration K ( π n ( X ) , n ) → P K ( π n ( X ) , n + 1) → K ( π n ( X ) , n ) via k n . Or, equiv a len tly , τ ≤ n X is the homotop y fib er of the ma p k n . No w w e state the main result of this pap er. Theorem 1.0.3. The Po s tnikov invariants k 2 and k 4 of B Γ + ∞ ar e trivia l . The Postnikov- invariant k 3 : τ ≤ 2 B Γ + ∞ ≃ K ( Z ; 2) ≃ CP ∞ → K ( Z / 24 ; 4 ) has or der 2 when c onsider e d as an eleme nt in H 4 ( CP ∞ ; Z / 24) ∼ = Z / 24 . Giv en Theorem 1.0 .1, the t heorems 1.0.2 and 1.0.3 are equiv alen t - this follow s easily from the Leray-Serre sp ectral sequence. The first sen tence of Theorem 1.0 .3 follows f rom the 3 existence of a π 2 -isomorphism φ 2 : B Γ + ∞ → CP ∞ and a π 4 -isomorphism φ 4 : B Γ + ∞ → K ( Z ; 4). The existence of φ 2 is easy b ecause π 1 ( B Γ + ∞ ) = 0, but we sho w the follo wing. The mapping class group Γ g acts on the cohomology of the surface F g preserving the in tersection fo rm, prov iding a g r oup homo mo r phism Γ g → Sp 2 g ( Z ) → Sp 2 g ( R ). Because U ( g ) ⊂ Sp 2 g ( R ) is a maximal compact subgroup, B Sp 2 g ( R ) ≃ B U ( g ). W e o btain a map Φ g : B Γ g → B U ( g ). This map classifies a complex vec tor bundle V 1 , which is also easy to describ e. Consider the univ ersal surface bundle 1 π : B Γ 1 g → B Γ g . One can choose a complex structure on the ve rtical tangent bundle of this bundle, turning π : B Γ 1 g → B Γ g in to a family of Riemann surfaces. Let V 1 b e the ve ctor bundle whose fib er ov er x ∈ B Γ g is the dual of the g -dimensional v ector space o f holo morphic 1 -forms o n π − 1 ( x ). It is a standard result that this is indeed a v ector bundle once the to p ology is appropriately c hosen [1]. It is easy to see that the isomorphism class o f this bundle do es not dep end on the c hoice of the complex structure and that it is classified by Φ g . It w as sho wn in [1 4] that there is a map Φ : B Γ + ∞ → B U suc h that Φ g = Φ ◦ α g . The map Φ can b e used to obtain cohomolo g y classes of B Γ + ∞ . W e set ζ i := Φ ∗ c i and γ i := Φ ∗ s i ; s i ∈ H 2 i ( B U ; Z ) the univ ersal integral Chern c haracter class. It was sho wn b y Morita [16 ] that γ 2 i is a torsion class. Prop osition 1.0.4. The c omp osition B Γ + ∞ Φ / / B U c 1 / / K ( Z ; 2) is a π 2 -isomorphism. C o nse quently, H 2 (Γ + ∞ ; Z ) ∼ = Z is gener ate d by γ 1 . This was already pro v en b y Harer [9 ] using differen t metho ds. T o carr y out the neces- sary computation, w e use a b ordism-theoretic in terpretation of π i (Ω ∞ MTSO (2)) and the Hirzebruc h-Riemann-Ro ch theorem. F or the existence o f the anno unced π 4 -isomorphism w e use t he Segal splitting of Ω ∞ Σ ∞ CP ∞ [21]. T o get the informatio n ab o ut the third Postnik ov in v arian t, we first construction an a ction of the binary icosahedral g roup ˆ G , the fundamental g r oup of the famous P oincar´ e sphere, on the surface F 14 . In other words , w e construct a homomorphism ρ : ˆ G → Γ 14 , whic h yields a map B ρ : B ˆ G → B Γ + ∞ . The next step is easily deriv ed f rom the Lefsc hetz fixed p oin t formula. Prop osition 1.0.5. The char acteristic class B ρ ∗ ζ 2 ∈ H 4 ( B ˆ G ; Z ) ∼ = Z / 120 has or der 24 . This result, tog ether with π 3 ( B Γ + ∞ ) ∼ = Z / 24, suffices to finish the pro o f of theorem 1.0.3 and th us also of 1.0 .2. Finally , w e use t he action ˆ G y F 14 to construct a g enerator o f π 3 ( B Γ + ∞ ). This go es as follo ws. The group ˆ G is p erfect and acts freely on S 3 . The quotient M := S 3 / ˆ G is the P oincar ´ e sphere. This giv es a map M → B Γ 14 . W e can apply t he plus construction to it and obtain a map f : M + → B Γ + ∞ . Because M + ≃ S 3 , w e hav e an elemen t θ ∈ π 3 ( B Γ + ∞ ). The second main result of this note is 1 By definition, Γ 1 g := π 0 (Diff ( F g , p ), where p ∈ F g . It is quite obvious that the forg e tful map B Γ 1 g → B Γ g is homotopy equiv alent to the universal sur face bundle on B Diff ( F g ), a s long as g ≥ 2. 4 Theorem 1.0.6. The el e m ent θ ∈ π 3 ( B Γ + ∞ ) ∼ = Z / 24 is a gener ator The structure of the pa p er is as follow s. In section 2, w e recall the pro of of Theorem 1.0.1 and pr ov e the first sen tence of Theorem 1.0.3. The section 3 contains the construction of the ˆ G -action on whic h the determination of the third Postnik ov inv arian t, whic h is carried out in section 4, dep ends. The pro of of Theorem 1.0.6 is also giv en in section 4. Ac kno wledgemen ts: This pap er is an improv ed v ersion of a c hapter of the author’s PhD thesis [3]. The author w an ts to express his thanks to his PhD advisor, Carl- F riedric h B¨ odigheimer, for his constant supp ort and patience. Also, I wan t to thank the Max-Planck - Institute for Mathematics in Bonn fo r financial supp ort during m y time as a PhD studen t. The final writing of this pap er was done while the author sta y ed at the Mathematical Institute of the Oxford Univ ersit y , whic h w as made p ossible b y a gran t from the Deutsc her Ak ademisc her Austausc hdienst. Finally , I w ant to thank Jeffrey G iansiracusa for teac hing me Theorem 2.1.1. 2 The first few h o motop y groups of Ω ∞ MTSO (2) Giv en some classical computations o f homotop y groups, it is not hard to compute the first fiv e homotop y groups of Ω ∞ MTSO (2). Recall that there exists a fib er sequenc e of sp ectra [7] MTSO (2) ω / / Σ ∞ CP ∞ + trf S 1 / / Σ ∞ S − 1 . (2.0.1) The sec ond map is the so-called cir cle tr a nsfer . W e are going to use the calculation of the circle transfer on lo w-dimensional homotop y groups b y Muk ai [17] and the classical lo w-dimensional computations of Serre. The first ho motop y gro ups of the sp ectrum Σ ∞ S 0 are (these v alues a re tabularized in [25]) k 1 2 3 4 5 6 π k (Ω ∞ Σ ∞ S 0 ) Z / 2 Z / 2 Z / 24 0 0 Z / 2. The no n trivial elemen t in π 1 is denoted η (the Ho pf elemen t). W e also need to know the effect of the m ultiplication by η on homotopy . η 2 is nonzero and η 3 has o r der 2. The first v alues of π k (Ω ∞ Σ ∞ CP ∞ ) (without additional basep oin t) are given by ([17], p.199) k 1 2 3 4 5 6 7 π k (Ω ∞ Σ ∞ CP ∞ ) 0 Z 0 Z Z / 2 Z Z / 2. There is a pro duct decomp osition Ω ∞ Σ ∞ CP ∞ + ≃ Ω ∞ Σ ∞ CP ∞ × Ω ∞ Σ ∞ S 0 . The effect of the circle transfer on second summand of π n (Ω ∞ Σ ∞ CP ∞ + ) = π n (Ω ∞ Σ ∞ CP ∞ ) ⊕ π n (Ω ∞ Σ ∞ S 0 ) is giv en by the m ultiplication with η . Using this and Theorem 1 .1 of [17] w e o btain tha t π k (Σ ∞ CP ∞ + ) → π k +1 (Ω ∞ Σ ∞ S 0 ) is surjectiv e if k ≤ 5. Th us the long exact homot o p y sequence asso ciated to 2.0.1 breaks in to short pieces 0 → π k (Ω ∞ MTSO (2)) → π k (Ω ∞ Σ ∞ CP ∞ + ) → π k +1 ( S 0 ) → 0 for k ≤ 4. An easy diagram c hase finishes the pro of of Theorem 1 .0.1. 5 2.1 Geometric int erpretation The sp ectra in the fib er sequence 2.0.1 ar e all Thom sp ectra and therefore the long exact homotop y sequence has a manifold-theoretic interpretation 2 . The in terpretation of the ho- motop y groups of a general Thom sp ectrum is a standard result. Let X ∗ := ( X 0 ⊂ X 1 ⊂ X 2 ⊂ . . . ) b e a sequence of top ological spaces a nd let V ∗ → X ∗ b e a stable v ector bundle of dimension d ∈ Z , i.e. a sequence V n → X n of real v ector bundles of dimension n + d ( d can b e negative; then X n is fo r ced to b e empt y if d + n < 0), together with isomorphisms ǫ n : V n ⊕ R ∼ = V n +1 | X n − 1 . An imp or tan t example of a stable vec tor bundle is the stable normal bundle ν M of a closed smo oth m -manifold M ; it has dimension − m . The Thom sp e ctrum T h ( V ∗ ) it defined a s follow: the n th space is T h ( V ∗ ) n := Th( V n ), t he Thom space of V n (the Thom space of the empt y v ector bundle o v er the empty space is the one-p oint space). The structural map e n : Σ Th( V n ) → Th( V n +1 ) is defined b y the v ector bundle map ǫ n . The P on tryagin-Thom construction [20 ] g ives the in terpretation o f the homotopy groups π n ( T h ( V ∗ )) (whic h agree with π n (Ω ∞ T h ( V ∗ )) for n ≥ 0). The group π n ( T h ( V ∗ )) is iso- morphic to the group of b ordism classes of pa ir s ( M , φ ), where M is a closed smo oth n − d - dimensional ma nif o ld and Φ : ν M → V ∗ is a map of stable v ector bundles. Because the susp ension sp ectrum Σ ∞ X + is the Thom sp ectrum of the trivial stable vector bundle, whose n th t erm is X × R n , it follows that π n (Σ ∞ CP ∞ + ) is the group o f b ordism class of triples ( M , b, E ), M a closed smo oth n -manifo ld, b a framing of M and E → M a complex line bundle. Consider the following stable v ector bundle − L o f dimension − 2. Its 2 n th term is the orthogonal complemen t L ⊥ n of the tautolo gical line bundle L n ⊂ CP n − 1 × C n . Its 2 n + 1st term is L ⊥ n ⊕ R . The Madsen-Tillmann sp ectrum MTSO (2) is just the Thom sp ectrum of this stable ve ctor bundle. It follows that π n ( MTSO (2)) is the group o f bo rdism classes of triples ( M , W, f ), M a closed smo oth n + 2-manifold, W → M a complex line bundle and f : R n ⊕ W ∼ = T M is isomorphism of stable v ector bundles. Eq uiv alen tly , w e can sa y that w e hav e triples ( M , W , g ) with a stable v ector bundle isomorphism g : ν M ⊕ W ∼ = R − n instead. Given the description of the homotopy gro ups as b ordism groups and the maps in the fib er se- quence 2.0.1, it is quite easy to deriv e the g eometric meaning o f the maps in the long exact homotop y sequence. Theorem 2.1.1. ( J. Giansir acusa, unpublishe d) The m aps in the long exa ct homotopy se quenc e of 2 .0.1 ar e the fol lowing. • The c onne cting h o momorphism π n +1 ( S − 1 ) → π n ( MTSO (2)) maps a fr ame d manifol d ( M n +2 , b ) to ( M , C , b ⊕ id C ) ( se c ond description). • The homomorph i s m π n ( MTSO (2)) → π n (Σ ∞ CP ∞ + ) is r epr esente d by the fol low i ng pr o c e dur e. L et ( M n +2 , L, f ) a s ab ove b e given. Cho ose a se ction s of L which is tr ans v e rse to the zer o se ction. Then s − 1 (0) is an n -man i f o ld a nd f d efines a fr aming of it. 2 The a uthor wan ts to thank Jeff Giansir acusa for explaining this interpretation to him. 6 • The cir cle tr ansfer π n (Σ ∞ CP ∞ + ) maps a triple ( M , b, E ) to the total sp ac e S ( E ) of the spher e bund le of E , endo w e d w i th the induc e d fr am ing. W e use Theorem 2.1.1 to give a description of the second and of the fourth homotop y group of MTSO (2). There is a map of sp ectra λ : MTSO (2) → Σ − 2 MU . It arises from the interpretation o f − L a s a stable complex v ector bundle of real dimension − 2. The effect of λ on homoto p y gro ups is the follo wing. If ( M ; L ; f ) represen ts an elemen t in π n ( MTSO (2)), then f induces a stable bundle isomorphism ν M ∼ = R n − L , whic h giv es a stable complex structure. W e can th us view M as a stable complex manifold and hence as an elemen t in π n +2 ( MU ). Recall also the Conner-Flo yd map µ : MU → K [4]. Bott p erio dicit y giv es an isomorphism β : π 2 n ( K ) → Z . Then β ◦ µ sends [ M ] ∈ π 2 n ( MU ) to the T o dd gen us Td( M ) := h td( T M ) , [ M ] i ∈ Z . The first comp onen ts of the T o dd class td( V ) o f a complex v ector bundle a r e [11]: td 1 ( V ) = 1 2 c 1 ( V ), td 2 ( V ) = 1 12 ( c 2 ( V ) + c 2 1 ( V )) , td 3 ( V ) = 1 24 ( c 2 ( V ) c 1 ( V )) . Theorem 2.1.2. The c omp osition π 2 ( MTSO (2)) → π 4 ( MU ) → π 4 ( K ) ∼ = Z is an i s o- morphism. Pro of: W e already men tioned that π 2 ( MTSO (2)) ∼ = Z , but we need a generator. Lo ok at the sequence 0 / / π 2 ( MTSO (2)) / / π 2 (Σ ∞ CP ∞ + ) / / π 2 ( S − 1 ) / / 0 0 / / Z / / Z ⊕ Z / 2 / / Z / 24 / / 0 . Generators of π 2 (Σ ∞ CP ∞ + ) are the class of S 2 with the Ho pf bundle and the canonical framing (order ∞ ) and the torus with t he Lie framing and the t r ivial line bundle (order 2). The image of a generator of π 2 ( MTSO (2)) in π 2 (Σ ∞ CP ∞ + ) is t w elv e times the first generator plus the second one. It follows that a g enerato r of π 2 ( MTSO (2)) can b e described by the follo wing data. It also follow s t ha t these data exist. • M is a closed smo oth 4-manifold. • L → M is a complex line bundle. • There is a stable v ector bundle isomorphism C ⊕ L ∼ = T M . • There is a section s in L , transv erse to the zero section, suc h that the zero set S := s − 1 (0) is an orien ted surfa ce with self-in tersection n um b er 12 . W e claim that the T o dd gen us o f suc h a manifold is necessarily equal to 1 . The T o dd class of a 4 - manifold like M is 1 12 ( c 2 ( T M ) + c 1 ( M ) 2 . The third condition ab o v e ensures that c 2 ( T M ) = 0 and hence td( T M ) = 1 12 c 1 ( L ) 2 Th us Td( M ) = 1 12 h c 1 ( L ) 2 ; [ M ] i = 1 12 12 b y the fourth condition. A similar arg umen t shows 7 Prop osition 2.1.3. The c omp os i tion Z ∼ = π 4 ( MTSO (2)) → π 6 ( MU ) → π 6 ( K ) ∼ = Z is zer o. W e can also find a map π 4 -isomorphism Ω ∞ MTSO (2) → K ( Z ; 4). This needs some preparation. The ta utological line bundle L → CP ∞ defines a sp ectrum cohomology class in K 0 ( CP ∞ + ) a nd thus a sp ectrum map Σ ∞ CP ∞ + → K . It induces a map L : Ω ∞ Σ ∞ CP ∞ + → Ω ∞ K = Z × B U . It is a theorem by Segal [21] tha t there exists a splitting Φ : Z × B U → Ω ∞ Σ ∞ CP ∞ + , suc h that L ◦ Φ ∼ id (it is not an infinite lo o p map). Moreov er, the fib er of L has finite homot o p y gr oups. Be cause π 4 (Ω ∞ Σ ∞ CP ∞ + ) ∼ = Z and b ecause π 4 ( B U ) ∼ = Z , it follows that L is a π 4 -isomorphism. Moreo v er an insp ection of 2.0.1 sho ws that ω : MTSO (2) → Ω ∞ Σ ∞ CP ∞ + is a π 4 -isomorphism as we ll. Prop osition 2.1.4. L et a, b ∈ Z . T he effe ct of the c ohomolo gy class ac 2 1 + bc 2 , c onsider e d as a map B U → K ( Z ; 4) , on π 4 is given by multiplic ation by b (up to sign ; b oth homotopy gr oups ar e infinite cyclic). In p articular, c 2 : B U → K ( Z ; 4) is a π 4 -isomorphism. Pro of: The quaternionic Hopf bundle on S 4 giv es a map S 4 → B S U (2) → B U , whic h is a generator o f π 4 ( B U ). The v alue of the c haracteristic class ac 2 1 + bc 2 on this bundle is clearly b . Th us we ha v e constructed a π 4 -isomorphism Ω ∞ MTSO (2) → K ( Z ; 4). It should b e re- mark ed that for n ≥ 3, there is no π 2 n -isomorphism B U → K ( Z ; 2 n ). This is a consequence of Bott ´ s divisibilit y theorem. Note that the comp osition B Γ ∞ → Ω ∞ MTSO (2) → Ω ∞ CP ∞ + → Z × B U is a virtual v ector bundle. It is w orth to describe this v ector bundle in more surface-theoretic terms. This is done in [3], p.6 9. W e only state the r esult. Endo w the univ ersal surface bun- dle π : B Γ 1 g → B Γ g with a complex structure as ab ov e. Then we consider the virtual v ector bundle whose fib er ov er x ∈ B Γ g is giv en b y H 0 ( C x , ω C x ) − H 0 ( C x , ω 2 C x ), where C x := π − 1 ( x ), ω C x is the canonical in v ertible sheaf on the Riemann surface C x . This bundle is classified by the map ab ov e. The homotopy-theoretic significance of the π 4 -isomorphism ab o v e is explained b y the fol- lo wing lemma. Lemma 2.1.5. L et X b e a simple sp a c e. Then ther e exists a π n -isomorphism X → K ( π n ( X ); n ) if and only if the Postnikov invari a nt k n is trivial. Pro of: The homopty class a n : X → τ ≤ n X has the follo wing univ ersal prop ert y . If Y is an n - co connected space and f : X → Y a homoto p y class, t hen there exists a unique homotop y class g : τ ≤ n X → Y such that g ◦ a n = f . Thu s the existence of a π n -isomorphism f : X → K ( π n ( X ); n ) implies the existence of a π n -isomorphism g : τ ≤ n X → K ( π n ( X ); n ). In o t her words, the fibration p n : τ ≤ n X → τ ≤ n − 1 X is trivial. The classifying map k n of p n is then also trivial. Conv ersely , if k n = 0 , then p n is trivial and there is a π n -isomorphism g : τ ≤ n X → K ( π n ( X ); n ) and the comp osition g ◦ a n is the desired π n -isomorphism. In o ur case, it follows that τ ≤ 4 B Γ + ∞ ≃ τ ≤ 3 B Γ + ∞ × K ( Z ; 4 ). T o determine t he homoto p y t yp e τ ≤ 4 B Γ + ∞ , we need to determine the P ostnik o v inv arian t k 3 : τ ≤ 2 B Γ + ∞ → K ( Z / 24; 4). This will o ccup y the rest of the pap er. 8 3 The icosahedral group and an in teresti n g acti o n of it on a sur face Consider a regular icosahedron I in Euclidean 3 - space, cen tered at 0 and suc h that all v erticaes lie on S 2 . It has 20 faces (whic h a re triangles), 12 v ertices ( a t ev ery v ertex, ex- actly 5 edges a nd 5 faces meet) and 30 edges. Let G ⊂ S O (3) b e the symmetry group of the icosahedron. W e choose a v ertex, an edge and a face of the icosahedron and denote the isotropy subgroups by G 2 , G 3 and G 5 , resp ectiv ely . These groups are cyclic of or der 2 , 3 , 5, resp ectiv ely and w e c ho ose generators y 2 , y 3 , y 5 of these groups. The group G a cts transitiv ely o n the v ertices as we ll as on the edges a nd f aces of the icosahedron and thus it has order 60. It is we ll- known that G is isomorphic to the groups P SL 2 ( F 5 ) and A 5 (the alternating gro up). In particular, G is p erfect. Using the action of G on the icosahedron, it is easy to see t ha t { y 2 , y 3 , y 5 } generates G . Let ˆ G t he preimage of G under the 2 - fold cov ering S 3 → S O (3). The group ˆ G is also p erfect and its center is the same as t he kernel of the map φ : ˆ G → G and con tains exactly one nontrivial elemen t h . It can b e sho wn that the extension Z / 2 → ˆ G → G is the uni- v ersal cen tral extension of G . The gro up ˆ G is t he binary i c osahe dr al gr oup . By the wa y , ˆ G ∼ = SL 2 ( F 5 ). W e c ho ose preimages x i of the g enerato r s y i . These set { x 2 , x 3 , x 5 } generates ˆ G . The quotien t ˆ G \ S 3 is the famous P oincar´ e homology 3-sphere. Our a nalysis of G and ˆ G b egins with t he description of the Sylo w-subgroups. Clearly G (3) ∼ = h y 3 i ∼ = Z / 3 is a 3- Sylo w subgroup and G (5) ∼ = h y 5 i ∼ = Z / 5 is a 5-Sylo w subgroup of G . In ˆ G , h x 2 3 i and h x 2 5 i are Sylow subgroups. The subgroup G (2) has order 4 and is th us ab elian. If it w ere cyclic, there would b e z ∈ G with z 2 = y 2 , whic h is ob viously imp ossible. Th us G (2) ∼ = Z / 2 × Z / 2 =: V 4 , the g enerato r s b eing y 2 and another elemen t whic h fixes a p erp endicular edge of I . The preimage ˆ G (2) = φ − 1 ( G (2) ) ⊂ ˆ G is a Sylow-2-subgroup and it con tains a unique inv olution, b ecause S 3 has a unique in v olution. It follows that ˆ G (2) is conjuga t e to Q 8 = { ± 1 , ± i, ± j, ± j } ⊂ S 3 ⊂ H . W rit e V 4 = { 0 , z 1 , z 2 , z 1 + z 2 } . The homomorphism Q 8 → V 4 whic h is defined by defined b y − 1 7→ 0, i 7→ z 1 and j 7→ z 2 , is an ab elianization. Prop osition 3.0.1. Ther e is an is o morphism of rings H ∗ ( B Q 8 ; Z ) ∼ = Z [ a, b, u ] /I , wher e I is the ide al gene r ate d by 8 u , 2 a , 2 b , ab , a 2 and b 2 . T he cl a ss u has de gr e e 4 and it is the s e c- ond Chern class o f the r epr esen tation Q 8 → SU(2) , while a and b ar e the first Chern cla s s es of the r epr e sentations Q 8 → V 4 → Z / 2 ⊂ S 1 (use two differ ent pr oje ctions V 4 → Z / 2 ). The c ohomolo gy ring H ∗ ( B ˆ G ; Z ) is i s o morphic to Z [ u ] / (120 u ) , w her e u ∈ H 4 ( B ˆ G ; Z ) is the se c ond Chern class o f the r epr esentation ˆ G → S 3 = SU(2 ) . Pro of: Because b oth groups act fr eely on S 3 , their cohomology is p erio dic of p erio d 4 and the m ultiplication with u is a p erio dicit y isomorphism in b oth cases (use the Gysin sequence ). By Poincar ´ e dualit y , the closed orien ted manifold S 3 /Q 8 has H 2 ( S 3 /Q 8 ) ∼ = V 4 , 9 H 3 ( S 3 /Q 8 ) ∼ = Z . All other cohomolog y in po sitive degrees is trivial. No w consider the comm utativ e diagram of fibrations S 3 / /   S 3 /Q 8   E S 3 / /   B Q 8   B S 3 B S 3 . The cohomology ring of B S 3 is w ell know n: it is the p olynomial ring , generated b y the second Chern class of B S 3 . The upp er horizontal map has degree 8 , and t hese fa cts together with the Leray-Serre sp ectral sequence quic kly give the first part o f the prop osition. The calculatio n for ˆ G is analogous and uses that S 3 / ˆ G is an in tegral homology sphere. . It follows from this computation that the restriction maps H 4 ( B ˆ G ) → H 4 ( B Q 8 ) ∼ = Z / 8, H 4 ( B ˆ G ) → H 4 ( B Z / 3) ∼ = Z / 3 and H 4 ( B ˆ G ) → H 4 ( B Z / 5) ∼ = Z / 5 are all surjectiv e. Th us their sum H 4 ( B ˆ G ) → H 4 ( B Q 8 ) ⊕ H 4 ( B Z / 3) ⊕ H 4 ( B Z / 5) ∼ = Z / 8 ⊕ Z / 3 ⊕ Z / 5 ∼ = Z / 120 is an isomorphism. The last fact ab out these groups we shall need concerns the represen tations o f Q 8 . The ab elian gro up V 4 has four nonisomorphic onedimensional irreducible represen tations and their pullbacks t o Q 8 are still no nisomorphic (they ha ve differen t c haracters). There are precisely 5 conjugacy classes in Q 8 , and a w ell-know n formula (see [2 2]) tells us that there m ust b e another, necessarily 2 - dimensional irreducible represen tation of Q 8 . Clearly , the represen tation U := ( Q 8 → S U (2)) is suc h a represen tation. It is the only one o n whic h the cen tral elemen t in Q 8 do es not act as the iden tity , but b y − 1. It follows : If W is a Q 8 -represen tation, on whic h the central elemen t a cts by − 1, then V is a direct sum of copies of U . 3.1 Surfaces with an action of the icosahedral group No w w e construct certain actions of the binary icosahedral group on Riemann surfaces whic h in the end will giv e in teresting elemen ts in π 3 ( B Γ + g ). The construction is based on an easy lemma. Let O ( k ) b e the k th tensor p ow er of the Hopf bundle on CP 1 . Recall that the global holo mo r phic sections of O ( k ) can b e iden tified with the v ector space of homogeneous p o lynomials of degree k on C 2 . F urther, O ( k ) is an SL 2 ( C )-equiv aria n t bundle ov er the SL 2 ( C )-space CP 1 . If k is o dd, then the central elemen t − 1 ∈ SL 2 ( C ) acts as − 1 on O ( k ) (i.e. on a n y fib er of the bundle). If k is ev en, then − 1 acts trivially o n O ( k ), i.e. the action descends t o a n action of P SL 2 ( C ). Lemma 3.1.1. L et G ⊂ P SL 2 ( C ) b e a fi n ite sub gr oup and let ˆ G ⊂ SL 2 ( C ) b e the exten- sion of G by Z / 2 . L et m ∈ N b e p ositive. L et s b e a G -invariant holomo rphic se ction of O (2 m ) havi ng only simp le zer o es. Then ther e exists a c onne cte d Riemann surfac e F with a ˆ G -action and a two-sh e ete d ˆ G -e quivariant br anche d c overing f : F → CP 1 , which br anche d pr e cisely over the zer o es of s . If m is o dd, then the c entr al elemen t h ∈ ˆ G is the hyp er el liptic in v olution on F , if m is 10 even, then h acts trivial ly on F . Pro of: Let S ⊂ O (2 m ) b e the g r aph of the section s . It is a surface of g en us 0 and it is stable under the G -action on O ( 2 m ). Let q : O ( m ) → O (2 m ) b e the squaring map, let F := q − 1 ( S ) and let f := q | F . Clearly , F has a ˆ G -action a nd f is equiv a r ia n t. Also, a generic p o in t of S has exactly 2 preimages, b eing p ermute d b y the in v olution − 1 on the fib ers on O ( m ). W e need to sho w that F is a smo oth connected Riemann surface. The smo othness of F is equiv alen t to the condition that the zero es are simple (the lo cus { ( x, y ) ∈ C 2 | y 2 = x m } is smo oth if and only if m = 1), and the connectivit y is clear since s has precisely 2 m > 0 zero es. W e hav e seen that the an tip o dal map of O ( m ) induces t he hy p erelliptic in v olution. F rom no w on, let G a gain b e the icosahedral group. Example 3.1.2. Let z 1 , . . . , z 30 ∈ S 2 b e the midp oin ts of the edges of the icosahedron and consider them as p oints on CP 1 after the c hoice of a confo rmal map S 2 ∼ = CP 1 . The precise v alue of the p oin ts do es not play a significan t role in this discussion. No w we ta ke holomorphic sections s i , i = 1 , . . . , 30 of the Hopf bundle O (1) (alias square ro ot o f the tang en t bundle of CP 1 ) having a simple zero at z i and b eing nonzero elsewhe re. Suc h sections exist and are unique up to m ultiplication with a complex constan t. Set s := s 1 ⊗ . . . ⊗ s 30 ∈ H 0 ( CP 1 , O (30)). F or g ∈ ˆ G , there exists a c ( g ) ∈ C × with g s = c ( g ) s , b ecause g s has the same zero es as s . The map c : g 7→ c ( g ) is a homomorphism ˆ G → C × . Since ˆ G has no Ab elian quotien t, c is constan t. Th us, s is an in v arian t section. If w e apply the construction of the lemma to s , w e o bta in a surface of genus 14 (b y the Riemann-Hurwitz formu la) with a ˆ G -action. Prop osition 3.1.3. The numb er of fixe d p oints of the eleme nts x 2 , x 3 , x 5 ∈ ˆ G in Example 3.1.2 is 2 , 0 , 0 , r e s p e ctively. A l l p owers x 2 r 3 , r 6≡ 0 (mo d 3) a nd x 2 r 5 , r 6≡ 0 (mo d 5) have pr e cisely 4 fix e d p oints. Pro of: Since h is the hy p erelliptic in volution, it has precisely 2 g + 2 = 3 0 fixed p oin ts, namely the branc h p oin ts whic h lie ov er the midp oints of the edges of the icosahedron. There are t w o fixed p oin ts of y 2 on P 1 , and they are tw o opp o site branc h p o ints (i.e. midp oin ts of edges). Th us, x 1 has exactly 2 fixed p oints on F . y 3 fixes precisely tw o opp osite midp oin ts of faces. Hence all fixed p oints of x 3 lie ov er these t w o p o ints. But if o ne of them w ould b e a fixed p oint of x 3 , then this m ust also b e a fixed p oin t of x 3 3 = h . This is impo ssible, since h has already the 30 fixed p oin ts mentioned ab o ve and since no nontrivial automor phism of a R iemann surface of gen us g ≤ 2 can ha ve more t ha n 2 g + 2 fixed p oints ([5 ], p.257). Thu s x 2 is fixed-p oint-free. The same arg ument sho ws that x 3 is also fixed-p oint free. The second sen tence f o llo ws from the same argument. 11 Of course, o ne can apply a similar construction to the midp oin ts of the f a ces or to the ver- tices of the icosahedron. The results are surfaces of low er g en us and elemen ts in π 3 ( B Γ g ) + of low er o rder. 3.2 The pro of of Prop osition 1.0.5 Consider the surface F of gen us 14 with the action o f ˆ G constructed in the last section. Let ρ : ˆ G → Γ 14 b e defined b y this action. The action is holomorphic b y construction and therefore, t here is an induced linear action ˆ G y H 1 ( F ; O ). The cohomology class B ρ ∗ ζ 2 of 1.0 .5 is the same a s the second Chern class of this linear ˆ G -represen tation, whic h will b e briefly denoted b y c . By the Dolb eault t heorem, the following isomorphism of complex ˆ G -represen tations holds: H 1 ( F ; O ) ⊗ R C ∼ = H 1 ( F ; C ) . (3.2.1) According to the structure of H 4 ( B ˆ G ) describ ed ab o v e, w e need t o sho w: The restriction of c to H 4 ( B Z / 3) has order 3, the restriction t o H 4 ( Z / 5) is trivial and the restriction to H 4 ( B Q 8 ) has order 8. Denote b y L r b e t he irreducible represen tation 1 7→ exp( 2 π ir p ) of Z /p on C . The Lefsc hetz fixed p oint form ula, applied to the result of Prop osition 3.1.3, allow s us to determine the decomp osition of the represen tation of Z / 3 ∼ = ˆ G (3) = h x 2 3 i on the 28-dimensional space H 1 ( F ; C ). The result is H 1 ( F ; C ) ∼ = 8 C ⊕ 10 L 1 ⊕ 10 L 2 . By 3.2.1, it f o llo ws tha t H 1 ( F ; O ) ∼ = 4 C ⊕ 5 L 1 ⊕ 5 L 2 as Z / 3-mo dules. The Chern p olynomial of L n is 1 + nv ; v ∈ H 2 ( Z / 3) an appro priate generato r . It follows that c | B Z / 3 = v 2 . The result for the subgroup Z / 5 follow s b y an analogous arg umen t. The decomp osition of H 1 ( F ; O ) as a Z / 5-mo dule is 2 C ⊕ 3( L 1 ⊕ L 2 ⊕ L 3 ⊕ L 4 ). W e hav e seen that the cen tral elemen t h ∈ Q 8 ⊂ ˆ G a cts as a h yp erelliptic inv olution. Th us it acts o n H 1 ( F ; O ) by − 1. As w e hav e seen, this implies that H 1 ( F , O ), as a Q 8 -mo dule, decomp oses in to sev en copies of the tw o dimensional represen tation U . The first Chern class of U is zero, while the second is a g enerator of H 4 ( B Q 8 ). Th us c | B Q 8 is sev en times a generator and thus again a generator. This finishes t he pro of of Prop osition 1.0.5 4 Conclus ion Lemma 4.0.1. The or der of k 3 in H 4 ( K ( Z ; 2); Z / 24) is the quotient 24 ♯ T ors H 4 ( B Γ ∞ ; Z ) . Pro of: The triviality of k 4 sho ws that H 4 ( B Γ ∞ ; Z ) = H 4 ( τ ≤ 4 B Γ ∞ ; Z ) = H 4 ( τ ≤ 3 B Γ ∞ × K ( Z ; 4); Z ) = H 4 ( τ ≤ 3 B Γ ∞ ) ⊕ Z . No w consider the pullbac k-diagram of homotopy-fibrations 12 K ( Z / 2 4; 3) / /   K ( Z / 2 4; 3)   τ ≤ 3 B Γ + ∞ / /   ∗   K ( Z ; 2) k 3 / / K ( Z / 2 4; ) (4.0.2) and t he asso ciated Leray-Serre sp ectral sequences in homolo gy . By the univ ersal co efficien t theorem, ♯ T ors H 4 ( B Γ ∞ ; Z ) = ♯H 3 ( B Γ + ∞ ; Z ). Also, the order of k 3 in H 4 ( K ( Z ; 2); Z / 24) is equal to the order of the image of k 3 ∗ : H 4 ( K ( Z ; 2); Z ) ∼ = Z → H 4 ( K ( Z / 24; 4); Z ) ∼ = Z / 24. One deduces a comm utative square H 3 ( K ( Z / 24; 3); Z ) H 3 ( K ( Z / 24; 3); Z ) H 4 ( K ( Z ; 2); Z ) k 3 / / d 3 O O H 4 ( K ( Z / 24; 4); Z ) , d 3 O O where the d 3 on the righ t-ha nd- side is an isomorphism. This sho ws that ♯ ( H 3 ( τ ≤ 3 B Γ + ∞ ; Z )) = ♯ ( Z / 24 / im k 3 ) = 24 ord k 3 . T o conclude our argumen t, we consider the fibration τ ≥ 3 B Γ + ∞ ι 2 / / B Γ + ∞ a 2 / / τ ≤ 2 B Γ + ∞ = K ( Z ; 2) . (4.0.3) The Lera y-Serre sp ectral sequence yields a short exact sequence 0 / / H 4 ( K ( Z ; 2)) a ∗ 2 / / H 4 ( B Γ + ∞ ) ι ∗ 2 / / H 4 ( τ ≥ 3 B Γ + ∞ ) / / 0 . (4.0.4) The torsion subgroup T ors H 4 ( τ ≥ 3 B Γ + ∞ ) is isomorphic to Z / 24 b ecause of Theorem 1.0.1. Because γ 2 = γ 2 1 − 2 ζ 2 is a torsion elemen t and b ecause ι ∗ 2 γ 1 = 0, w e see tha t ι ∗ 2 ζ 2 ∈ T ors H 4 ( τ ≥ 3 B Γ + ∞ ). By obstruction t heory , there is a unique (up to homotopy ) lift ˜ B ρ : B ˆ G → τ ≥ 3 B Γ + ∞ of B ρ : B ˆ G → B Γ + ∞ (recall that H ∗ ( B ˆ G ) = 0 f or 0 < ∗ < 4). But b y Prop osition 1 .0 .5, ˜ B ρ ∗ ι ∗ 2 ζ 2 = B ρ ∗ ζ 2 has order 24. Thus ι ∗ 2 ζ 2 is a generator of T ors H 4 ( τ ≥ 3 B Γ + ∞ ). The f unctor whic h assigns to an ab elian group A it s torsion subgroup T ors A is left- exact. Th us T ors H 4 ( B Γ + ∞ ) → T ors H 4 ( τ ≥ 3 B Γ + ∞ ) is injective . The elemen t γ 2 ∈ T ors H 4 ( B Γ + ∞ ) has order 12 or 24 , again b y Prop osition 1.0.5. Finally , w e sho w that ι ∗ 2 ζ 2 do es not lie in the image o f ι ∗ 2 : T or s H 4 ( B Γ + ∞ ) → T ors H 4 ( τ ≥ 3 B Γ + ∞ ). In view of the exact seque nce 4.0.4, a ll preimages o f ι ∗ 2 ζ 2 in H 4 ( B Γ + ∞ ) a re of the form nζ 2 1 + ζ 2 . If nζ 2 1 + ζ 2 is torsion, then (2 n + 1 ) ζ 2 1 w ould b e torsion (b ecause ζ 2 1 − 2 ζ 2 is torsion), whic h is absurd b ecause o f 4.0.4. This finishes the pro of of Theorem 1.0.2. Finally , w e determine the order of θ ∈ π 3 ( B Γ + ∞ ) g iv en b y B ρ as a b o v e. Conside r the Bo c kstein sequence 0 / / H 3 ( τ ≥ 3 B Γ + ∞ ; Z / 24) δ / / H 4 ( τ ≥ 3 B Γ + ∞ ; Z ) 24 / / H 4 ( τ ≥ 3 B Γ + ∞ ; Z ) 13 There is a unique β ∈ H 3 ( τ ≥ 3 B Γ + ∞ ; Z / 24) suc h that δ ( β ) = ζ 2 . It is a g enerato r and thus the map τ ≥ 3 B Γ + ∞ → K ( Z / 24 ; 3 ) a ssociated to β is a π 3 -isomorphism. Hence the following construction giv es an isomorphism ψ : π 3 ( B Γ + ∞ ) → Z / 24. Let f : S 3 → B Γ + ∞ and denote b y ˜ f : S 3 → τ ≥ 3 B Γ + ∞ the unique lift. Set ψ ([ f ]) := h ˜ f ∗ ( β ); [ S 3 ] i (tak e the Kronec k er pro duct with Z / 2 4 -co efficien ts). Lemma 4.0.5. The map M → B ˆ G and thus its plus-c onstruction S 3 → B ˆ G + induc es an isomorphism H 3 ( B ˆ G ; Z / 24) → H 3 ( M , Z / 24) . Pro of: W e hav e seen that H 3 ( B ˆ G ) ∼ = Z / 120 and using similar argumen ts as in section 3, one can see tha t H 3 ( M ; Z ) ∼ = Z → H 3 ( B ˆ G ; Z ) is surjectiv e. By the univ ersal co efficien t theorem, it fo llo ws tha t the induced map in Z / 2 4 -cohomology is an isomorphism. Consider the comp osition f : S 3 → B ˆ G + → τ ≥ 3 B Γ + ∞ . It is an easy diagra m c hase using the Bo c kstein sequences for τ ≥ 3 B Γ + ∞ and B ˆ G + to conclude that H 3 ( f ; Z / 24) is an isomorphism. This sho ws that t he class θ ∈ π 3 ( B Γ + ∞ ) is a generator, as asserted in Theorem 1.0 .6. References [1] M. F. Atiy ah; I. M. 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