On the Symplectomorphism Group of Product of Complex Projective Spaces 1

Abstract Seidel proved in [1] that the homomorphism βk : πk(Symp(M,ω)) → πk(Diff(M)) induced by the inclusion is not surjective for (M,ω) = (CPN1 × CPN2 , ωN1 × ωN2) and each odd k ≤ max{2N1 − 1, 2N2 − 1}, where ωn is the standard symplectic form on CP n with ∫ P 1 ωn = 1. In this note, this profound result is generalized to the case (M,ω) = (CPN1 × · · · × CPNm , ωN1 × · · · × ωNm) with arbitrary m ≥ 2.

In particular, β k is not surjective.This is the first example of symplectic manifolds in which the homeomorphism β k : π k (Symp(M, ω)) → π k (Diff(M)) is not surjective for dimension greater than 4. The case that m = n = k = 1 can be derived from a result of Gromov [3, 0.3.C].This note shows that Seidel's method can still deal with symplectic manifolds (M, ω) = (CP The main result, Theorem 3.2, gives the estimate of rank(cokerβ d−1 ) via the Betti numbers for every even d ≥ 2. In particular, Corollary 3.
2 Deformations of the Q-algebra H * (M ) For reader's convenience we review some necessary definitions and properties in [1].All algebras are finite-dimensional, commutative (in the graded sense), and have a unit.Definition 2.1 Let R be a graded F-algebra over a field F. A deformation of R of dimension d > 0 consists of: (i) a graded F-algebra R, (ii) an element t ∈ R d with t 2 = 0, such that the sequence 0 → R/t R → R → R/t R → 0 is exact, where the second arrow is ×t, (iii) a homomorphism of graded algebras j : R → R which is surjective with kernel t R.
One can also define the notion of isomorphisms between d-dimensional deformations of R. Denote by Def d (R) the set of isomorphism classes of all ddimensional deformations of R. It has a natural structure of an F-vector space.In Remark 2.2 of [1], the author also gave the following equivalent definition of Def d (R).Definition 2.2 If Z d (R) denotes the space of F-bilinear maps ψ : R × R → R of degree −d which are graded symmetry and satisfy So if odd dimension parts of R are 0, then Def d (R) = 0 for all odd d.Definition 2.3([1, Sec.6])Let R 1 and R 2 be two graded F-algebras and let R 1 and R 2 be their deformations of the same dimension, respectively.For the graded tensor product of R 1 and R where acts on R i by multiplication with t i , t := t 1 ⊗ 1 = 1 ⊗ t 2 and j := j 1 ⊗ j 2 .The triple ( R, t, j) is called the exterior product of R 1 and R 2 .
This notion can be generalized to the case of finitely many deformations of the same dimension.Definition 2.4([1, Sec.6])A deformation of R is called split if it is isomorphic to an exterior product of finitely many deformations of the same dimension.
Denote by Def s d (R) the set of all spit deformations of R. Definition 2.5(semi-split) Following the notations in Definition 2.3, we say a deformation ( R, t, j) of R to be semi-split with respect to R s (s = 1 or 2) if there is a deformation ( R s , t s , j s ) of R s and a morphism fs : R s → R which lie over the natural inclusion f s : R s → R. Theorem 2.6([1, Lemma 2.3]) Let M be a compact smooth manifold, and let π : E → S d be a smooth fibre bundle with fibre M. For some fixed b 0 ∈ S d let i : M → E b 0 be a diffeomorphism.Let denote the generator of H d (S d ) and According to [page 138, 4], if i * is onto then M is said to be totally nonhomologous to zero in E with respect to the field F. Lemma 2.7 Let π : E → S d be as in Theorem 2.6.Suppose that d > 0 is even and that the cohomology H * (M; F) has no nontrivial element of odd degree.Then i * is onto.Proof.This easily follows from spectral sequences.In fact, the second page of the spectral sequence is zero except (2n, d) position.Then all elements are not changed under the differential d r for r = d + 1.But on the (d + 1)-th page, the differential d d+1 = 0 since the cohomology lies in even degree.As in [Theorem 5.8,4] we may express i * by d r , and deduce that i * is onto. 2 In the following we always assume Then for each even d > 0 and a smooth fibre bundle π : E → S d with fibre M Theorem 2.6 and Lemma 2.7 show that (H With the similar arguments to Sections 2, 3 in [1], we may get the following theorem, which classifies deformations of R. Theorem 2.8 For each even , and isomorphic to ).Since d ≤ 2N s +2 for some s, for each such s we may write a s as by comparing their degrees) and hence can be written as the form in (2.1) if 2N s + 2 ≥ d.It follows from these and Definition 2.1(ii) that there exist the following relations as in (2.2) for each s with 2N s + 2 ≥ d, Using these it is easy to construct a homomorphism f from R a 1 ,• Step 3.
) be two deformations of R as in Step 1, and let ψ be an isomorphism from the former to the latter.We claim: If d > 2 we have rs = 0 and hence For each s with 2N s + 2 ≥ d we may write By (2.2) we have (ũ s ) Ns+1 + tã s = 0 and (ũ s ) By the exact sequence in Definition 2.1(ii) we deduce that ψ(ã s ) − ã s + (N s + 1)β s (ũ s ) Ns = 0 and hence

can be written as a (graded) symmetric, bilinear map ψ
The proof of this result was completed with the parametrized Gromov-Witten invariants.Using this theorem Seidel obtained: Theorem 2.10([Prop.6.5, 1]) Suppose that (M , ω ) is a compact manifold whose cohomology ring H * (M ; Q) is generated by H 2 (M ; Q), and that ω (H s 2 (M ; Z)) ⊂ Z.Let (E, π, i, Ω) be a symplectic fibre bundle with fibre (M, ω) = (M , ω ) × (CP n , ω n ) over S d for some even d > 0. Then the deformation of R = H * (M; Q) given by H * (E; Q) is semi-split with respect to H * (M ; Q).
Corresponding to split theorem [Prop.6.2, 1] we have Theorem 2.11 Let (E, π, i, Ω) be a symplectic fibre bundle over S d with fibre

it is the exterior product of certain deformations of H * (CP
Clearly, H 2 (M s ; Q) generates the ring H * (M s ; Q), and ω s (H s 2 (M s ; Z)) ⊂ Z.By Theorem 2.6 and Lemma 2.7 both H * (M; Q) and H * (M s ; Q) have d-dimension deformations.In particular, (H * (E; Q), t, j = i * ) is a d-dimension deformation of H * (M; Q).Now Theorem 2.10 tells us that H * (E; Q) is semi-split with respect to R s := H * (M s ; Q).Hence there exists a deformation ( R s , t s , j s ) and a morphism fs : R s → H * (E; Q) covering the inclusion f s : R s → H * (M; Q).
Suppose that fs (r s ) = 0. Since f s is injective and j fs = f s j s , we have j s (r s ) = 0.It follows from Definition 2.1(iii) that rs = t s r s for some r s ∈ R s .Note that t fs (r s ) = fs (t s ) fs (r s ) = fs (t s r s ) = fs (r s ) = 0.By the exact sequence in Definition 2.1(ii) we deduce that fs (r s ) = 0.This implies that r s = t s γs for some γs ∈ R s again.Hence rs = t s r s = (t s ) 2 γs = 0, and fs is injective.
Identifying R s and its image via fs and checking the proof of Proposition 6.5 in [1], the deformation can be chosen good enough, for example, we can require that R s is generated by t, ũ1 , From the proof of this theorem one may see that the conditions P 1 ω Ns = 1, s = 1, • • • , m, are necessary for us using Theorem 2.10.

Results and proofs
Denote by F (S d , M) the set of all isomorphism classes of smooth fibre bundles over S d with fibre M. It is a group under the operation of fibre connected sum, and is isomorphic to π d−1 (Diff(M)).Fix a base point b 0 ∈ S d .We have an explicit isomorphism obtained by the clutching construction, where φ : ) provided that d > 0 is even and that H * (M; F) has no nontrivial element of odd degree.Theorem 2.6 defines a homeomorphism Corresponding to [Prop.3.1, 1] we have: is surjective.
Proof.We only need to prove that each 0 Since the proofs are similar, without loss of generality we may assume j = 1.We also write p = d/2 for conveniences.By Theorem 2.8 . Moreover, the first condition implies that the Chern classes of it can be written as where u 2 and are the generators of H 2 (CP N 2 ; Q) and and denote it by η i 2 .Then we have also For a s ∈ {3, • • • , m}, if i s ≥ 1 we may use the obstruction theory to choose a complex line bundle L s → CP Ns such that c 1 (L s ) = u s , and then set ξ is to be the sum of i s line bundles L s .We get that c j (ξ is ) = τ j u j s with τ j ∈ Z ∀j, and specially , and denote it by ξ is .Let ε denote the trivial complex line bundle over This shows that the deformation of R determined by P(ζ) as in Theorem 2.6 is isomorphic R b,0,...0 , where . The desired conclusion follows immediately.If i > p we proceed with the following: As in the above construction of b in (3.6) we may obtain an element Since b 2q (CP n ) = 1 for q = 0, 1, • • • , n, and b q (CP n ) = 0 for other q, dim R 2N = dim H 2N (M) = 1 and c j (ζ) = 0 for other j by (3.2) and (3.3),where γ i 2 +p is as in (3.2).Note that we can always write we may view P(ζ) as a bundle over S d with fiber CP N 1 × • • • × CP Nm .So the deformation of R determined by P(ζ) as in Theorem 2.6 sits in the image of α d−1 id É .From the Leary-Hirsch theorem we know that H *