HOMOTOPY DECOMPOSITION OF A GROUP OF SYMPLECTOMORPHISMS OF S2 x S2 S'ILVIA ANJOS AND GUSTAVO GRANJA Abstract.We continue the analysis started by Abreu, McDuff and Anjos [Ab, AM , McD, An] of the topology of the group of symplectomorphisms of S2x S2 when the ratio of the area of the two spheres lies in the interva* *l (1, 2]. We express the group, up to homotopy, as the pushout (or amalgam) of cer* *tain of its compact Lie subgroups. We use this to compute the homotopy type o* *f the classifying space of the group of symplectomorphisms and the correspondi* *ng ring of characteristic classes for symplectic fibrations. 1.Introduction Let M~ denote the symplectic manifold (S2 x S2, !~ = ~oe0 oe0) where 1 ~ 2 R and oe0 is the standard area form on S2 with total area equal to 1. It is known that any symplectic form on the manifold S2 x S2 is, up to scaling by a constant, diffeomorphic to !~ (see Lalonde-McDuff [LM ]). Let G~ denote the group of symplectomorphisms of M~. By now, much is known about the topology of the group G~ and, more generally, about symplectomorphism groups of ruled surfaces. Gromov [Gr ] first showed tha* *t, when ~ = 1, G~ deformation retracts onto the subgroup of standard isometries Z=2 n (SO(3) x SO(3)). He also showed that this would no longer be the case if ~ > 1, and, confirming this, McDuff [McD1 ] constructed an element of infini* *te order in H1(G~). Later, Abreu [Ab ] computed the rational cohomology of G~ when 1 < ~ 2 and his methods were extended by Abreu and McDuff [AM , McD ], who completely described the rational homotopy type of G~ and BG~ for all values of ~ (see also [McD2 ] for more information on the integral homotopy type of G~, including the fact that the topology of the group changes precisely when ~ cros* *ses an integer). In [An ], the first author computed the homology Pontryagin ring of G~ with field coefficients when 1 < ~ 2 and used this to determine the homoto* *py type of the space G~. All these results were derived from a study of the action* * of G~ on the contractible space J~ of compatible almost complex structures on M~. Our aim in this paper is to continue the analysis of the case 1 < ~ 2 by describing the homotopy type of G~ as a topological group. To explain our result, recall from [Ab ] that, for 1 < ~ 2, the space J~ is* * a stratified space with two strata U0 and U1. Each of these contains an integrable almost complex structure Ji, with isotropy group Ki G~ where K0 = SO(3) x SO(3) and K1 = S1xSO(3). Following Gromov, Abreu showed that the inclusions ____________ Date: March 7, 2003. Key words and phrases. Symplectomorphism group, amalgam, homotopy decomposit* *ion. Mathematics Subject Classification 2000: 57S05,57R17,55R35. Supported in part by FCT through program POCTI and grant POCTI/1999/MAT/3308* *1. Supported in part by FCT through program POCTI and grant POCTI/1999/MAT/3401* *5. 1 2 S'ILVIA ANJOS AND GUSTAVO GRANJA of the orbits G~=Ki Ui are weak equivalences. K0 and K1 intersect in K01 = SO(3) which sits inside K0 as the diagonal and inside K1 as the second factor. Thus, there is a commutative diagram of group monomorphisms (1) SO(3)_______//SO(3) x SO(3) j|| |i0| fflffl| fflffl| S1 x SO(3) __i1_____//_G~ where denotes the inclusion of the diagonal, j the inclusion of the second fa* *ctor, and i0, i1 the inclusions in G~ of the subgroups K0, K1. Let P denote the pushout of the diagram of topological groups (2) SO(3)_______//SO(3) x SO(3) j|| fflffl| S1 x SO(3) This is also known as the amalgam (or amalgamated product, or free product with amalgamation) of the groups SO(3) x SO(3) and S1 x SO(3) over the common subgroup SO(3) and it is characterized as the initial topological group admitti* *ng compatible homomorphisms from the diagram (2). We will review this construction in section 2. The universal property of the pushout gives us a canonical contin* *uous homomorphism (3) P -! G~. We need to describe another map derived from (1). We will write hocolim(X -f Y -g! Z) for the homotopy pushout (or double mapping cylinder) of the maps f, g. This is the quotient space of X q (Y x [0, 1]) q Z by the equivalence relation generated by (y, 0) ~ f(y); (y, 1) ~ g(y). Applying* * the classifying space functor to (1)we obtain a canonical map i B Bj j (4) hocolim BSO(3) x BSO(3) - BSO(3) -! BS1 x BSO(3) ! BG~. Finally, let Met(S2 x S2) denote the space of metrics on S2 x S2. The usual retraction [MS , Proposition 2.50 (ii)] Met(S2 x S2) -r!J~ is equivariant with respect to the action of the group of symplectomorphisms. Therefore, letting K < G~ denote a compact subgroup, the fact that the fixed point space Met(S2 x S2)K is convex implies that J~K is contractible. It follo* *ws that we can pick a path (5) [0, 1] -fl!J~K01 HOMOTOPY DECOMPOSITION OF A GROUP OF SYMPLECTOMORPHISMS OF S2x S2 3 with fl(0) = J0 and fl(1) = J1 and this is unique up to homotopy. A choice of p* *ath determines a G~-equivariant map (unique up to G~-equivariant homotopy) i ß ß j (6) hocolim G~=(SO(3) x SO(3)) - 0G~=SO(3) -!1G~=(S1 x SO(3) ! J~ where ß0, ß1 denote the canonical projections. We can now state our main result: Theorem 1.1. Let 1 < ~ 2. Then the following equivalent statements hold: (i)The homomorphism (3)is a weak equivalence. (ii)The map (4)is a weak equivalence. (iii)The G~-equivariant map (6)is a weak equivalence. Statement (i) is an immediate consequence of the observation that the homology computations in [An ] say that, for k a field, the Pontryagin ring H*(G~; k) is the pushout in the category of k-algebras of the diagram obtained by applying H*(-; k) to (2), together with Theorem 3.8 below which says that the functor H*(- : k) preserves certain pushouts. The equivalence of statements (i)-(iii) * *is an easy consequence of Theorem 3.8 together with Theorem 3.10 which says that under the same hypotheses, the classifying functor also preserves pushouts. From (ii), it is easy to compute the ring H*(BG~; Z) of integral characteristic clas* *ses of symplectic fibrations with fibre M~ (see Corollary 4.5). A better proof of Theorem 1.1 (which would, in particular, provide an alter- native proof of the main results of [An ]) would be to deduce (iii) directly fr* *om an analysis of the stratification of J~. Unfortunately, we were unable to do th* *is. The description of the stratification of J~ by McDuff in [McD ] yields a homoto* *py pushout decomposition of J~ as hocolim(U0 NU1 \ U1 ! NU1) where NU1 denotes a tubular neighborhood of U1 in J~ which fibers over U1 as a disk bundle. These three spaces have the required homotopy types but we do not know whether NU1 can be chosen so as to be invariant under the action of G~. More precisely, (iii) would follow if we could choose a tubular neighborhood NU* *1, a path fl as in (5), and t0 2]0, 1[ satisfying: o G~ . fl(t) NU1 for t > t0, o fl(t) 62 U1 for t < t0. (ii) and (iii) are the statements one should try to generalize to the cases w* *hen ~ > 2, although in those cases it might be necessary to use a topological cate- gory to index the homotopy colimit decomposition. See [MW , Appendix D] for a relationship between stratifications and homotopy colimit decompositions. It is an easy consequence of [Se, Theorem 8, p. 36] that any compact subgroup of the amalgam P is subconjugate in P to either K0 or K1. In view of Theorem 1.1 (ii), it is natural to ask whether this is also the case in G~. Yael Karsho* *n has proved in [Ka ] that the answer is yes if one assumes that the subgroup in ques* *tion is a torus (her result is not circumscribed to the case 1 < ~ 2). Finally, Theorem 1.1 suggests it might be a good idea to explore analogies wi* *th infinite dimensional algebraic groups (cf. [Mi , Ki]). In particular, compare T* *heorem 1.1(ii) with the homotopy decomposition of the classifying space of a rank 2 Ka* *c- Moody group of indefinite type as the homotopy pushout of a diagram of compact subgroups [Ki, Theorem 4.2.3] (see also [ABKS , BrK ]). 4 S'ILVIA ANJOS AND GUSTAVO GRANJA Organization of the paper. In section 2 we begin by recalling the constructions of pushouts in various categories and introducing necessary notation. In sectio* *n 3, under the assumption that the homomorphisms involved induce injections on ho- mology, we compute the Pontryagin ring and the homotopy type of the classifying space of an amalgam of topological groups whose connected components are com- pact Lie groups. In section 4, we deduce Theorem 1.1 from the results of sectio* *n 3 and [An ]. We then use it to produce an interesting fiber sequence involving BG~ and compute the ring H*(BG~; Z) of characteristic classes for symplectic fibrat* *ions with fiber M~. 2. Pushouts and amalgams. In this section we begin by reviewing the categorical notions of pushout and coequalizer. We then recall the construction of the pushout in the categories * *of groups, topological groups and k-algebras, and introduce some notation that will be necessary in the next section. Pushouts and coequalizers. Everything in this subsection is standard basic cat- egory theory. See [Ma ], for instance. Let B0, B1, B2 be objects in a category C and f1, f2 morphisms in C. Given a diagram in C, (7) B0 __f1_//B1 |f2| fflffl| B2 a cone on this diagram consists of an object C together with morphisms g1 and g2 such that f1 B0 ____//_B1 |f2| |g1| fflffl|gfflffl|2 B2 _____//_C commutes. A pushout of (7) is a cone (C, g1, g2) with the property that if (D, * *h1, h2) is any other cone, then there is a unique morphism ' : C ! D making the followi* *ng diagram commute: f1 B0 ____//_B1 00 f2|| g1||000 fflffl|gfflffl|h10002 B2 _____//_PP00PPCA PPPh2PA'00 PPPPA 00 PP(,,0(P__A D This universal property characterizes the pushout up to unique isomorphism in C (when the pushout exists). The pushout is usually denoted by B1 qB0 B2. HOMOTOPY DECOMPOSITION OF A GROUP OF SYMPLECTOMORPHISMS OF S2x S2 5 Given a pair of arrows _d0__// (8) A _____//B, d1 a cone on (8)is an arrow ffl : B ! C such that ffld0 = ffld1. A coequalizer of * *(8)is a cone ffl : B ! C with the property that given any other cone ffl0: B ! C0, ther* *e is a unique map ' : C ! C0 making the following diagram commute __d0_// ffl A _____//B____//AC d1 AAffl0A| AA '| A__fflffl| C0 Again this universal property characterizes the coequalizer up to unique isomor- phism. Pushouts and coequalizers are instances of a more general construction, that of colimit of a diagram, which is defined by a similar universal property. Example 2.1. The main examples we have in mind are the following: (a) C is the category of topological spaces. If A is a topological group acting* * on the right on the space Y and on the left on the space X then the coequalizer of the two maps d0, d1 : X x A x Y ! X x Y defined by d0(x, a, y) = (xa, y) and d1(x, a, y) = (x, ay) is the quotient of X x Y by the action (a, x, y) ! (xa, a-1y). We write X xA Y for this space. (b) C is the category of vector spaces over a field k. If A a k-algebra, V a l* *eft A-module and W a right A-module, the coequalizer of the action maps d0, d1 : V k A k W ! V k W defined as above is the tensor product V A W . If k is a field, the universal property of the coequalizer together with the * *Künneth theorem give us a canonical map (9) H*(X; k) H*(A;k)H*(Y ; k) ! H*(X xA Y ; k). The main theorem of this paper follows immediately from the fact that a similar canonical map is an isomorphism and the proof will consistently exploit such ma* *ps. Pushouts of groups. Suppose C is the category of groups. A good reference for everything in this subsection is [Se]. Let S denote the set of finite sequences x1x2. .x.n where xi2 B1 or xi2 B2 and consider the equivalence relation ~ on S generated by (i)x1. .x.n~ x1. .^.xi.x.n.if xi= 1, (ii)x1. .f.1(a) . .x.n~ x1. .f.2(a) . .x.nfor a 2 B0, (iii)x1. .x.ixi+1. .x.n~ x1. .(.xixi+1) . .x.nwhen xi, xi+1 both belong to B1 or B2. S has an associative unital product defined by concatenation and it is easy to * *check that this descends to the set P = S= ~ of equivalence classes and that P togeth* *er with the canonical maps B1 ! P and B2 ! P is the pushout of (7). 6 S'ILVIA ANJOS AND GUSTAVO GRANJA If the maps f1 and f2 are monomorphisms then the pushout of the diagram (7) is also called the amalgam1 of B1 and B2 over B0. In this case, there is a usef* *ul description of the elements of the pushout, called the Normal Form Theorem. To explain this, we will start by introducing some notation. Let i = (i1, . .,.in* *) 2 {0, 1, 2}n. Then we define __ (10) Bi= Bi1x . .x.Bin Note that there is a canonical map __ ß (11) Bi-! P __ determined by multiplication. Bn-10acts on the right on Bi by (a1, . .,.an-1) . (b1, . .,.bn) = (b1a1, a-11b1a2, a-12b2a3, . .,.a-1n-1* *bn). We will write (12) Bi= Bi1xB0 . .x.B0Bin __ for the quotient of Bi by this action. Note that we can express the quotient Bi* *as the coequalizer of __ n-1 _d0_//__ B ix B0 _d1_//_Bi where d0 is given by the action, and d1 is the projection on the first factor. * *The canonical map ß of (11)factors through this quotient and we will denote the re- sulting map also by ß. Let B0i= Bi\ B0 for i = 1, 2, and for i = (i1, . .,.in) a sequence of alterna* *ting 1's and 2's, let B0i= B0i1xB0 B0i2xB0 . .x.B0B0in denote the quotient of B0i1x . .x.B0inby the action of Bn-10. Theorem 2.2. [Se, Theorem 2, page 4] If f1, f2 are monomorphisms, the canonical maps ß determine a bijection B0 q qiB0i~-!P where i runs over all sequences of alternating 1's and 2's. Pushouts of topological groups. We now consider the case when the Bi are topological groups and the homomorphisms between them continuous. We need to give a topology to the space P defined above so that P together with the maps Bi! P is the pushout in the category of topological groups. In order to do this, it is necessary to work in a category of compactly gener* *ated spaces. Any will do for our purposes, so we will work with the simplest, namely that of Vogt [Vo, Example 5.1]. A space X is said to be compactly generated if a subset U X is closed iff for every compact Hausdorff space K and continuous map g : K ! X, g-1(U) is closed. Given an arbitrary topological space X we can refine the topology in the obvious way so that it becomes compactly generat* *ed. Denoting this space by kX, the natural transformation kX ! X ____________ 1It is also called amalgamated product of B1 and B2 over B0, and free produc* *t of B1 and B2 with amalgamation. HOMOTOPY DECOMPOSITION OF A GROUP OF SYMPLECTOMORPHISMS OF S2x S2 7 determined by the identity maps is a weak equivalence [Vo, Proposition 1.2 (h)]. The (categorical) product in the category of compactly generated spaces does not in general agree with the product in spaces; it is necessary to apply the funct* *or k to the usual product. In all that follows we will write x for the product in * *the category of compactly generated spaces. By a topological group we mean a group object in the category of compactly generated spaces. Any topological group in * *the usual sense determines such a group object by applying k to the multiplication2. We will write Pn P for the image in P of the Biwith i 2 {0, 1, 2}n and P0 for the image of B0. We * *will also write ffn and fin for the two sequences in {0, 1, 2}n of alternating 1's a* *nd 2's and (13) Qn = Bffnq Bfin. Similarly we define __ __ __ (14) Q n= Bffnq Bfin. __ We will set Q0 = Q0 = B0. With the notation above, we first give Pn the quotient topology determined by the canonical map (15) Qn -ß!Pn. __ Note that this is the same as to give Pn the topology_induced by Qn, since Qn h* *as the quotient topology_determined_by the projection Qn ! Qn. The inclusion Qn ! Qn+1 defined by (x1, . .,.xn) 7! (x1, . .,.xn, 1) induces a continuous map Pn -in!Pn+1. We give P the topology of the union of the Pn's. The product ~ : P x P ! P clearly restricts to a product ~n,m : Pn x Pm ! Pn+m . Consider the commutative diagram __ __ _~n,m __ Q nx Qm ____//_Qn+m ßxß || |ß| fflffl|~n,m fflffl| Pn x Pm _____//_Pn+m where the map __~n,mis defined by: æ __~ (x1, . .,.xn, y1, . .,.ymi)fxn 2 Bi, y1 2 Bj withi 6= j, n,m(x, y) = (x1, . .,.xny1, . .,.ymo,t1)herwise. Both vertical maps are quotient maps [Vo, Corollary 3.8] so we conclude that the maps ~n,m are continuous. Since in the category of compactly generated spaces, colimits commute with products [Vo, Proposition 3.7 (b)], it follows that the m* *ul- tiplication P x P -~! P ____________ 2This is not necessary for groups of diffeomorphisms of compact manifolds wh* *ich are metrizable and hence compactly generated. 8 S'ILVIA ANJOS AND GUSTAVO GRANJA is continuous. Similarly one checks that the inverse map on P is continuous. It* * is now easy to check that the topological group P together with the maps Bi ! P has the required universal property and hence is the pushout in the category of topological groups. Pushouts of k-algebras. Let k be a field. There is an entirely analogous descri* *p- tion of the pushout of (7)in the category of associative k-algebras. We will so* *on want to compare the construction for algebras and the homology of the analogous construction for topological groups and to make the notation more clear we will decorate the algebraic constructions with a superscript alg. Let Salgdenote the tensor algebra generated by the k-vector spaces Balg1and Balg2. Thus, as a vector space, Salgis the direct sum k Balg1 Balg2 (Balg1 Balg1) (Balg1 Balg2) . . . where denotes the tensor product of k-vector spaces. Then the pushout of (7)in k-algebras is P alg= Salg=I together with the canonical maps Balgi! P alg, where I is the ideal generated by (i)x1 . . .xn - x1 . . .^xi . . .xn if xi= 1, (ii)x1 . . .f1(a) . .x.n- x1 f2(a) . .x.nfor a 2 Balg0, (iii)x1 . . .xi xi+1 . . .xn - x1 . . .(xixi+1) . . .xn when xi, xi+1 both belong to Balg1or Balg2. In some circumstances it is possible to give a more precise description of the pushout. To do this, we need some notation. Let i = (i1, . .,.in) 2 {0, 1, 2}n.* * Then we define __alg alg alg (16) B i = Bi1 . . .Bin Note that there is a canonical map __algß alg (17) Bi -! P determined by the multiplication on P alg._ Given a sequence i 2 {0, 1, 2}n, let i2 {0, 1, 2}2n-1 denote the sequence obt* *ained from i_by inserting 0's between each two entries of i. For example, if i = (i1,* * i2, i3) then i= (i1, 0, i2, 0, i3). Then we define maps __alg_d0_////_alg B_i __d1_B i by (setting f0 = id) d0(x1 a1 x2 . . .xn)= x1fi1(a1) x2fi2(a2) . . .xn d1(x1 a1 x2 . . .xn)= x1 fi2(a1)x2 . . .fin(an-1)xn. and we define Balgito be the coequalizer (in k-vector spaces) of these maps. Th* *at is, (18) Balgi= Balgi1 Balg0Balgi1 Balg0. . .Balg0Balgin The canonical map ß of (17)factors through Balgiand we will denote the resulting map also by ß. Moreover we will write as in (13) (19) Qalgn= Balgffn Balgfin. HOMOTOPY DECOMPOSITION OF A GROUP OF SYMPLECTOMORPHISMS OF S2x S2 9 and (20) Pnalg P alg for the image of Qalgnin P alg. Remark 2.3. If the k-algebras Balgiare the homology Pontryagin rings of topolog- ical groups Bi, then they have added structure: they are Hopf algebras (see [MM* * ]). In particular there are antiautomorphisms c : Balgi! Balgi induced by the inverse map on the group, as well as augmentations (i.e. maps of k-algebras) Balgiffl-!k induced by the map to the trivial group. Let Bn0algdenote the k-algebra Bn0alg= Balg0 . . .Balg0 __alg augmented in the obvious way. For i = (i1, . .,.in) 2 {0, 1, 2}n, B i is a ri* *ght Bn-10alg-module under the action (b1 . . .bn) . (a1 . . .an-1) = (b1a1 c(a1)b1a2 . . .c(an-1)bn) where we have omitted the maps fi from the notation. Moreover, it is easy to see that __ B algi Bn-10algk = Balgi. As in (9) it is then an immediate consequence of the Künneth theorem and the universal property of a coequalizer that there is a canonical map (21) BalgiOE-!H*(Bi; k) Moreover, writing P algfor the pushout of the Pontryagin rings, the diagram OE Balgi____//H*(Bi; k) | | | | fflffl| fflffl| P alg____//H*(P ; k) commutes. We will need the following analog of Theorem 2.2 for pushouts of k-algebras, * *due to P.M. Cohn. Theorem 2.4. [Co3 , Proof of Theorem 3.1] If f1, f2 are monomorphisms, and there exist right Balg0-modules B0algisuch that Balgi= fi(Balg0) B0algias right Bal* *g0- modules. Then regarding B0algias a Balg0-bimodule via the isomorphism B0algi' Balgi=Balg0, the canonical maps ß determine an isomorphism of Balg0-bimodules Balg0 iB0algi~-! n 0Pnalg=Pnalg-1 where i runs over all sequences of alternating 1's and 2's. 10 S'ILVIA ANJOS AND GUSTAVO GRANJA For future reference, we also note that there are exhaustive filtrations Balgi . . .PnalgBalgi . . .P alg of P algby right Balgi-modules and that (cf. [Co3 , (32), p. 61]) we have canon* *ical isomorphisms (22) iB0algi Balg0Balgj~-! n 0PnalgBalgj=Pnalg-1Balgj of right Balgj-modules, where i runs over all sequences of alternating 1's and * *2's not ending in j. Remark 2.5. Theorem 2.4 also holds under the assumption that the fiare monomor- phisms and B0algi= Balgi=fi(Balg0) are flat right Balg0-modules (cf. [Co2 , (1.* *6)-(1.8) p. 436]) but the above formulation is sufficient for our purposes. 3. Amalgams of topological groups. In this section, under the assumption that the homomorphisms involved induce injections on homology, we compute the Pontryagin ring of an amalgam of topolog- ical groups whose connected components are compact Lie groups. Using this and Puppe's theorem, we obtain a homotopy decomposition of the classifying space. Although the results are stated for the case when the free products involved * *have only two factors, it is clear that all the statements hold (with the same proof* *) for an arbitrary number of factors. Homology of the pushout. In order to compute the homology of the pushout of a diagram of topological groups we require some assumptions. Definition 3.1. We will say that a pushout diagram (7)of topological groups is homologically free if it satisfies the following conditions: (i)The connected components of the Bi are compact Lie groups, (ii)The continuous homomorphisms f1, f2 are monomorphisms. (iii)The homomorphims H*(fi; k) are injective for every field k. The first condition of the previous definition can certainly be weakened. We make this assumption because it suffices for the application we have in mind and it considerably simplifies the point set topology involved. Lemma 3.2. Under the assumptions (i)-(ii) of Definition 3.1, and with the nota- tion of (15)we have (a) Pn is a Hausdorff space. (b) (Qn, ß-1(Pn-1)) ßn-!(Pn, Pn-1) is a relative homeomorphism. Proof.(a) is easy and (b) follows easily from compactness and the Normal Form Theorem 2.2. The following lemma will be used often in the rest of this section. Lemma 3.3. Let X ! B be a principal fibration with fiber A and k be a field. If the inclusion of the fiber induces an injection H*(A; k) ! H*(X; k) then (a) H*(X; k) is a free (and hence flat) right H*(A; k)-module, (b) The canonical map H*(X; k) H*(A;k)k ! H*(B; k) is an isomorphism. HOMOTOPY DECOMPOSITION OF A GROUP OF SYMPLECTOMORPHISMS OF S2x S211 Proof.Since the inclusion of the fiber A ! X induces an injection on connected components, it suffices to consider the case when A is connected. It follows th* *at the action of ß1(B) on H*(A; k) is trivial. The homology Leray-Serre spectral seque* *nce is a spectral sequence of right H*(A; k)-modules. Since H*(A; k) ! H*(X; k) is * *an injection, spectral sequence collapses. The E2 and hence the E1 terms are free H*(A; k)-modules on H*(B; k). It follows that H*(X; k) is a free H*(A; k)-module and the projection induces an isomorphism H*(X; k) H*(A;k)k ! H*(B; k) as required. Remark 3.4. The first part of Lemma 3.3 is also a consequence of [MM , Theorem 4.4]. In order to compute the homology of the pushout, we need one more lemma. Let denote the partially ordered set {0, 1, 2} where the order is defined by * *0 1 and 0 2 (1, 2 are incomparable). Given n > 0, n is again a poset. The order relation is (i1, . .,.in) (j1, . .,.jn) () ik jk for eachi. For j i 2 n there are obvious inclusions Bj- ! Bi. Lemma 3.5. Suppose the diagram (7)satisfies conditions (i) and (ii) of Definiti* *on 3.1. Let i 2 n and n be such that 3 (a) j i for every j 2 , (b) j 2 and k j ) k 2 . Then the canonical map colimj2 Bj! Bi is a closed cofibration. Proof.If__ = {j} consists of a single element, this follows from the fact that Bj Bi has a Bn-10-equivariant tubular neighborhood, from which we can obtain a neighborhood deformation retraction of Bj Bi. The result now follows by induction using the union theorem for cofibrations [Li, Corollary 2] (whichSstates that if A X, B X and A \ B X are closed cofibrations then A B X is again a closed cofibration). For the rest of this section we set Balgi:= H*(Bi; k). Corollary 3.6. If the pushout diagram (7) is homologically free, then for any i 2 n the canonical map BalgiOE-!H*(Bi; k) is an isomorphism. ____________ 3These conditions ensure that the colimit is just the union of the images of* * the spaces Bjin Bi. 12 S'ILVIA ANJOS AND GUSTAVO GRANJA Proof.It is easy to check that, under our assumptions, the inclusion of the fib* *er in the total space of the principal Bn-10-fibration __ B i! Bi is an injection. The result now follows from immediately from Lemma 3.3 and Remark 2.3. Note that if the diagram (7)is homologically free, then the inclusions Bj Bi induce inclusions on homology with field coefficients, i.e. the canonical maps Balgj-! Balgi are injective. This follows from the obvious fact that __alg __alg Bj -! Bi __alg is injective, together with the fact that the H*(Bn-10; k)-modules B i are flat* * by Lemma 3.3(a). More generally, the same argument implies that the canonical map (23) colimj2Balgj-! Balgi is injective, when n satisfies the two conditions of Lemma 3.5. Lemma 3.7. Suppose the pushout diagram (7)is homologically free. Let n satisfy the two conditions of Lemma 3.5. Then the canonical map colimj2 Balgj-! H*(colimj2 Bj; k) is an isomorphism. S Proof.Suppose 0= {i} and let = {j 2 : j i}. Then we have a pushout square (24) colimj2Bj _____//colimj2Bj j|| || |fflffl fflffl| Bi _________//colimj2B0j By Lemma 3.5 the map j is a cofibration. If we assume that the result is true for then j induces an injection in homology by (23)and so the Mayer-Vietoris sequence for (24)splits. This implies that the result holds for 0, and hence i* *t is true in general, by induction. Theorem 3.8. If the pushout diagram (7)is homologically free then the canonical map P alg= H*(B1; k) qH*(B0;k)H*(B2; k) -! H*(P ; k) is an isomorphism. Proof.Since homology commutes with sequential colimits of T1 spaces along closed inclusions, it suffices to show that the canonical map Pnalgfln-!H*(Pn; k) HOMOTOPY DECOMPOSITION OF A GROUP OF SYMPLECTOMORPHISMS OF S2x S213 is an isomorphism for each n. This is obvious for n = 0. Assume it is true for n m and consider the following diagram 0 _______//_Pmalg_______//_Pmalg+1______//Pmalg+1=Pmalg__//_0 flm|| flm+1|| OEm+1|| fflffl| fflffl| fflffl| H*(Pm ; k)____//H*(Pm+1 ; k)__//_H*(Pm+1 , Pm ; k) It is enough to see that the induced map OEm+1 is an isomorphism since it then follows that _n is surjective and then, by the 5-lemma, that flm+1 is an isomor* *phism. To simplify notation, write Wn = ß-1(Pn-1) Qn ; Wnalg= ß-1(Pnalg-1) Qalgn. It is clear from the Normal Form Theorem 2.2 that Wn = colimj where |Yi| = |Xi| = i + 2, |T | = 2, and |Z| = 5. Proof.Since H*(BSO(3) x BSO(3); Z) = Z[X1, X2, Y1, Y2, Z]= < 2X1, 2Y1, 2Z, Z2 > this follows easily from the Mayer-Vietoris sequence . .-.! H*(BG~; Z) -! H*(BSO(3) x BSO(3); Z) H*(BS1 x BSO(3); Z) -OE!H*(BSO(3); Z) -! . . . which identifies H*(BG~; Z) with the kernel of the map OE. HOMOTOPY DECOMPOSITION OF A GROUP OF SYMPLECTOMORPHISMS OF S2x S217 In particular, the map H*(BS1xBSO(3)xBSO(3); Z) ! H*(BG~; Z) induced by (30)is surjective. Remark 4.6. Using the computation of H*(BG~; Q) by Abreu and McDuff [AM ], Januszkiewicz and Kedra [JK ] have shown that all the characteristic classes in H*(BG~; R) come from integrating monomials on the Chern classes of the vertical tangent bundle and the coupling class4 over the fibers. In more detail, the fibration (31) M ,! MhG~ := M xG~ EG~ ! BG~ yields a vertical tangent bundle T MhG~ := T M xG~ EG~ ! MhG~. Since J~ is contractible, this bundle has (up to homotopy) a canonical complex structure. Integrating monomials on the Chern classes ck and the coupling class over the fiber of (31)yields elements in H*(BG~; R). If ff : T ! G~ is a torus action on M~, the commutativity of the diagram B~*ff H*(MhT) oo_____H*(MhG~) |ßT*| |ßG~*| fflffl|B*ff fflffl| H*-2n(BT ) oo___H*-2n(BG~) together with the fact that ßT*can be computed by localization methods gives in* *for- mation on the corresponding classes in H*(BG~). For example, Januszkiewicz and Kedra's calculations [JK , Proposition 4.1.1] say that the obvious map BG~ - ! BS1 derived from (30) classifies the class -ßG~*(c31)=8 2 H2(BG~; Z). Acknowledgments. The first author would like to thank Joe Coffey for useful conversations. The second author would like to thank Jer^ome Scherer for pointi* *ng out that Whitehead's theorem is an immediate consequence of Puppe's theorem (cf. proof of Theorem 3.10), and also Kasper Andersen and Jesper Grodal for useful discussions. References [Ab] M. Abreu, Topology of symplectomorphism groups of S2xS2, Invent. Math., 1* *31 (1998), 1-23. [AM] M. Abreu and D. 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