Dedicated to Professor Friedrich Hirzebruch

TOPOLOGY OF SYMPLECTOMORPHISMGROUPS AND PSEUDOHOLOMORPHIC CURVES

HÔNG-VÂN LÊ^{∗} MAX-PLANCK-INSTITUT FÜR MATHEMATIK GOTTFRIED-CLAREN-STR. 26 53225 BONN

Abstract. In this note we study the topology of symplectomorphismgroups using pseudoholomorphic curves and symplectic bundles. As application we show the non-triviality of certain homotopy groups of some symplectomorphismgroups.

1. Introduction.

Given a symplectic manifold (M^{2n}, ω), one of the basic mathematical objects associated to (M^{2n}, ω) is its automorphismgroup Sym(M, ω). There are many important questions related to this object: how this automorphismgroup acts on M and its natural geometrical objects (e.g. the loop space, the tangent bundle of M, the space of compatible almost complex structures)?, how to describe representations of this automorphismgroup into other (finite or infinite dimensional) space, how to compute the algebraical cohomology group of Sym(M, ω) and etc. On the other hand, since the group Sym(M, ω) can be equipped with a natural smooth (C^{∞}) topology, we would like also to know the homotopy type of this automorphismgroup. This note is devoted to the topological properties of the group Sym(M, ω). Namely we are interested in the following questions:

1) How big is the rank of the homotopy group π_{i}(Sym_{smooth}(M, ω)⊗ Q)?

2) What are the characteristic classes of Sym_{smooth}(M, ω), that is, the cohomology ring of the classifying space BSym_{smooth}(M, ω).

Like diffeomorphismgroups, the group symplectomorphisms of a compact symplectic manifold has infinite dimension. One naturally wishes to apply the methods and ideas in the study of topology of diffeomorphismgroups to the symplectic case. We know three approachs in

Date: 14 August 1997.

∗ supported by DFG/Heisenberg-Programm, present address: Universität Leipzig, Mathematisches Institut, Augustusplatz 10-11, D-04109 Leipzig, Germany.

2 V. H. LÊ

the study of topology of diffeomorphism groups: the geometro- homotopical approach (using gluing, bordism, cocordance,...), the geometroalgebraical approach (construction of concrete cycles or cocycles by invariant differential forms, by the imbedding of certain compact Lie groups, or by considering the action of the group of symplectomorphisms on the associated geometrical objects,..), and the geometroanalytical approach. Our note is influenced by the seminal Gromov’s paper [Gr] and inspired by the talk of John Lott [L 1] on the third approach.

This note consists of three parts. In the section 2 we recall the definition of symplectic bundle and extend the notion of stable holomorphic map to a family version. In section 3 we show that the basic properties of the moduli space of stable maps also hold in the parametrized (family) version. As an immediate consequence we construct parametrized Gromov-Witten invariants for symplectic bundles. In section 4 we apply this construction to problems 1, 2 mentioned above. We associate to an element in π_{i}(Sym(M, ω)) a symplectic bundle over S^{i+1} which is the sum of two trivial symplectic bundles over disk D^{i+1} glued along the boundary ∂D^{i+1} by this element in π_{i}(Sym(M, ω)). By computing a parametrized Gromov-Witten invariant for a symplectic bundle over S^{2} we reinterpret a result by Gromov [G, Theorem 2.4.C_{2}] on the existence of an element of infinite order in the symplectomorphism group of S^{2} × S^{2}, see Theorem 4.2. We also prove the following theorem on higher homotopy group of the group of symplectomorphisms of a symplectic manifold of dimension greater than 4.

Theorem 4.4 . Let (M, ω) be a symplectic manifold (S^{2} × S^{2} × N^{2k}, ω_{1} ⊕ ω_{2} ⊕ ω_{3}) and let A_{i} denote the image of i-th sphere S^{2} in M. Suppose that ω_{1}(A_{1}) > ω_{2}(A_{2}). Then we have rk(π_{1}(Sym(M, ω)) ⊗ Q) ≥ 1 and rk(π_{3}(Sym(M, ω)) ⊗ Q) ≥ 2.

In the section 4 we also construct characteristic classes of the group Sym(M, ω) by dualizing the construction of Gromov-Witten invariants. We compare this construction with a recent construction of the characteristic classes by Kontsevich.

Acknowledgement. I am indebted to John Lott for his wonderful lecture [Lott1], his preprint [Lott2] and his help in the reference in the fields. I thank Maxim Kontsevich for his beautiful lecture [K], for discussion and explanation of his construction. I thank Kaoru Ono for sharing some his independent ideas in this problem and Tabachnikov for giving his intersting paper [T]. After completing a first version of this note I learn from Kaoru Ono some his related (not yet published) results and related (not yet published) results by Abbreu, McDuff and Seidel.

2. Symplectic bundles and vertical stable holomorphic maps

2.1. \underline{Symplectic bundles and their fiberwise compatible almost} \underline{complex structures}. We recall that a fibration F → E →^{j} B is called a symplectic fibration if the fiber F is diffeomorphic to a symplectic manifold (F, ω) and the transition function takes its value in the group Sym(F, ω). Let us denote by J (B, F) the associated bundle over B whose fiber is the space J (F) of smooth compatible almost complex structures on (F, ω) which is compatible to ω. Since the fiber J (F) is contractible, the space of sections J(E) : B → J (B, F) is also contractible, in particular non-empty.

In what follows we are interested in defining invariants which detect the non-triviality of symplectic bundles. Associated to any symplectic bundle F → E →^{j} B there is a (co)homological bundle H_{∗}^{(∗)}(F) → E_{∗}^{(∗)} →^{j_{∗}^{∗}} B, (where, in what follows, the coefficients of cohomology groups are in R or Z, and the coefficients of homology groups are in Z). Clearly all the invariants of the associated (co)homological bundles are also invariants of symplectic bundles. In particular if a symplectic bundle is trivial then the associated (co)homological bundle is trivial. We call a bundle with the trivial associated homological bundle homological simple. We also observe that, for a symplectic bundle E there is always a section s^{[ω]} : B → E^{∗} which takes a constant value [ω] ∈ H^{2}(F, R) and there is also a section s^{c_{1}} : B → E^{∗} which takes a constant value c_{1}(F, ω) ∈ H^{2}(F, Z).

2.2. \underline{Vertical stable holomorphic maps.} First we introduce a notion of a stable vertical map (cf. [LT]). Let (Σ, x⃗) be a semi-stable (or prestable) curve i.e. a connected union of Riemann surfaces with possible singularity of ordinary double points and with the set of marked points x⃗ outside the singularity set (se [Ko1] and [KM], [BM])for a more precise definition). A map u from a semi-stable curve (Σ, x⃗) to E is called a vertical stable map if j(u) = pt ∈ B and u is a stable map (i.e. the components of Σ which is mapped to a point must be stable).

Let J(E) be a vertical compatible almost complex structure. A map u : (Σ, x⃗) → E is called a vertical stable J(E)-holomorphic map if the composition map j ∘ u sends (Σ, x⃗) to a point pt ∈ B and u is a stable holomorphic map from (Σ, z⃗) to j^{−1}(pt) = (F, J(E)_{|F}).

4 V. H. LÊ

The equivalence between two stable maps is defined via the biholomorphism between their domains which make the diagram involving the mappings commutative. ( cf[Ko1], [FO], [LT].)

To define the moduli space of vertical stable holomorphic maps we assume first, for the sake of simplicity and a later application, that the fibration is homologically simple (e.g. if the base B is simply connected .^{1}) In this case for a class A ∈ H_{2}(F, Z) there is also a constant section s_{A} : B → E_{∗} whose value is A. Then we denote by CM_{g,m}(E, J(E), s_{A}) the moduli space of vertical stable maps of (Σ_{g}, x⃗), where |x⃗| = m and the image u realizes the homology class s_{A}(j(u)) ∈ H_{2}(F, Z):

CM_{g,m}(E, J(E), s_{A}) = {(b ∈ B, u ∈ CM_{g,m}(F = j^{−1}(b), J(E)_{|F} , A))}.

If E is not homological simple, we still can define a notion of a moduli space of vertical stable holomorphic maps by considering a multi-valued section s_{A} : B → E_{∗} which is a locally constant and takes a value A at a reference fiber j_{∗}^{−1}(b_{0}) ⊂ E_{∗}. We note that the pairing < s^{[ω]}(b), s_{A}(b) > as well as the pairing < s^{c_{1}}(b), s_{A}(b) > are constant functions on B. Using the Gromov compactness theorem it is easy to see that there is only finite number of values of s_{A} in each fiber H_{2}(F = j^{−1}(b), Z) such that there is a J-holomorphic curve realizing class s_{A}(b) for some J ∈ J (F) . (Thus the compactness of the moduli space of vertical stable holomorphic maps can be obtained as in the case of homological simple symplectic bundles). The moduli space of vertical stable J(E)-holomorphic curves is a subspace of the space Map_{g,m}(E, j, s_{A}) of equivalence classes of stable maps of genus g with k marked points and realizing class s_{A} in each fiber F.

\underline{2.3. Examples of symplectic fibrations.} There are several ways to construct symplectic fibrations.

The first way is the induction method. Suppose that a group G acts symplectically on a symplectic manifold (M, ω). Then we can associate to each principal G-bundle E over B a symplectic bundle E ×_{G} (M, ω). This symplectic bundle is non-trivial if and only if the image of the homotopy class λ ∈ [B, BG] defining the G-bundle is non-trivial in [B, BSymp(M, ω)].

The second way is the pull-back method. Suppose we are given a symplectic bundle E over a base E. Then any map f from B′ to B pulls the bundle E back to a symplectic bundle E′ ove B′.

The third way is the reduction method. We begin with a differentiable fibration F → E → B with F being a symplectic manifold and ask if this fibration also admits a structure of a symplectic fibration. Of course if the inclusion of Sym(F, ω) to Diff^{+}(M) is a homotopy equivalence then it does (e.g. if dim M = 2). Otherwise we can use the following criterium in [McD-S 1].

1Symplectic fibration over S^{1} is classified by the group π_{0}(Sym(M, ω)). Using Floer homology we can detect non-trivial elements in this group [Li].

Lemma 2.1. [McD-S1, p. 193]Let π : E → B be a locally trivial fibration with connected base and ω ∈ Ω^{2}(E) be a symplectic form such that the fibers are all symplectic submanifolds of E. Then π : E → E admits the structure of a symplectic fibration which is compatible with ω.

The fourth way to construct symplectic fibrations is the gluing method. Suppose that we are given two symplectic bundles E_{1} and E_{2} over bases B_{1} and B_{2} respectively. Suppose that the restriction of E_{1} over the boundary ∂B_{1} is isomorphic to the restriction of E_{2} to the boundary ∂B_{2}. Then we can glue the bundle E_{1} with E_{2} along the boundary (E_{i})_{|∂B_{i}}. In particular when the restriction of E_{i} to ∂B_{i} is trivial then the glued bundle is defined uniquely by a map ∂B_{1} → Sym(F, ω). If B_{i} is closed we can define the operation of (direct) fiber connected sum by throwing away from E_{1} and E_{2} the restriction of E_{i} over a small disk B_{i} and identifying E_{i}| ∂B_{i} with the help of trivial map ∂B_{i} ↦ Id ∈ Symp(F, ω).

Remark 2.2. Each element g ∈ π_{k}(Sym(M, ω)) defines a symplectic bundle E with the fiber (M, ω) over a sphere S^{k+1} by gluing two trivial symplectic bundles (M × D^{k+1}) along the boundary M × S^{k} by a representative of g. (We can also think in terms of classfying spaces and homotopy exact sequences.) Conversely any symplectic bundle over S^{k+1} is defined by such a method. Clearly (symplectic) bundles over S^{k} is homologically trivial if k ≥ 2.

3. Parametrized Gromov-Witten invariants.

\underline{3.1. Geometric picture}. We consider the case 3g−3+m ≥ 0 and we assume that the base B is compact and oriented. Let CM_{g,m} denote the Deligne-Mumford moduli space of stable curves, and E^{⊗m} - the Whitney sum of m copies of E. As in the non-parametrized case (see [KM, 2.4] and [F-O], [LT]) there is a projection-evaluation map

Π = pr × ev_{g,m,s_{A}} : CM_{g,m}(E, J(E), s_{A}) → CM_{g,m} × E^{⊗m}, (3.1)

where pr associates each vertical stable holomorphic map u to its stabilized domain (stabilization means consecutive contraction of nonstable components, which must be a CP^{1} without proper singular point) - an element in CM_{g,m}. The evaluation map ev associates each stable holomorphic map u to the evaluation of u at m marked points. Denote by

6 V. H. LÊ

Π_{∗}(CM_{g,m}(E, J(E), s_{A})) the induced class in H_{∗}(CM_{g,m} × E^{⊗m}, Q). The map Π induces a map in cohomologies

I_{g,m,s_{A}}^{E} : H^{∗}(E^{⊗m}, Q) → H^{∗+μ}(CM_{g,m}, Q)  (3.2))

by

I_{g,m,s_{A}}^{E}(γ) = P D(γ\Π_{∗}(CM_{g,m}(E, J(E), s_{A})))  (3.2.1)

The parametrized Gromov-Witten invariants, as in a non-parameterized case, are the collection of maps I_{g,m,s_{A}}^{E} defined in (3.2.1). The grading μ equals, as in non-parametrized case, < 2c_{1}(F), A > +2(g − 1) dim F.

Remark. As in the non-parametrized case we can also interpret the the parametrized Gromov-Witten invariants of homologically trivial symplectic bundles in term of “counting holomorphic curves”. First we consider the case of Gromov-Witten invariants of codimension 0 (which are most important in our applications) i.e. those elements I_{g,m,s_{A}}(γ) of maximal degree in H^{∗}(CM_{g,m}, Q). It is known (by using intersection theory cf.[KM]) that the definition of codimension zero G-W classes is equivalent to the definition of counting holomorphic curves passing through certain cycles in our fiber bundle E (or E^{⊗n}). Namely

< I_{g,m,s_{A}}^{E}(γ), [CM_{g,m}] >= ∫_{CM_{g,m}} P D(γ\Π_{∗}(CM_{g,m,s_{A}}(E)) = ∫_{Π_{∗}(CM_{g,m,s_{A}}(E))} j_{1}^{∗}(γ),

(3.3)

where j_{1} denotes the projection map to the factor CM_{g,m}. We represent γ as (the sum of) an element P D(i_{∗}Γ_{1}) ∧ ⋯ ∧ P D(i_{∗}Γ_{m}) ∧ j^{∗}(P D(Γ_{B})), where Γ_{i}, i = 1, m, is a cycle in the fiber F and Γ_{B} is a cycle in the base B. Then the RHS of (3.3) can be expressed as the number of the vertical stable holomorphic curves of genus g and m marked points which realizes fiberwise the class s_{A} and satisfies the following properties: the image of the i-th marked point is lying on (a fixed representative of ) the cycle Γ_{i} × B of the fiber as well the projection of the marked points ia lying on (fixed representative of) cycle Γ_{B}.

The case of Gromov-Witten invariant of positive codimension can be treated analogously (see e.g. [R-T] for the non-parametrized case with a weakly monotone symplectic structure). Unfortunately we don’t have a good geometric way to describe the group H_{∗}(E, Z) if the bundle E is not homologically trivial.

\underline{3.2 Technical (analytical) tool}. To define rigorously the parametrized Gromov-Witten invariants for all compact symplectic bundles we need to show that the moduli space (CM_{g,m}(E, J(E), s_{A})) is compact; it carries a fundamental class, and moreover, the map I_{g,m,s_{A}}^{E} does not depend on the choice of J(E). The compactness is automatically satisfied because we have the compactness fiberwise and we assume that the base is compact. To show that the moduli space carries a fundamental class and our definition does not depend on the choice of J(E) we use the same technique as in non-parametrized case. (We also note that to construct the Gromov-Witten invariant in non-parametrized case we need to construct a bordism between the moduli spaces correspondent to different almost complex structures J_{1} and J_{2}. This bordism is a version of the moduli space of parametrized vertical stable pseudoholomorphic curves considered in the present note). For the sake of completeness we shall review a needed technical construction here. This construction is due to Fukaya and Ono [F-O] and Li and Tian [L-T]. They developed a theory of Kuranishi structures (or differently speaking, a theory of the Euler class of generalized Fredholm orbibundles) to construct Gromov-Witten invariants for compact symplectic manifolds. (Their construction is in spirit close to a construction of Gromov-Witten invariants in algebraic geometry). First we observe that the moduli space of vertical stable holomorphic maps is the zero set of the Cauchy-Riemmann operator defined over the orbibundle Map_{g,m}(E, j, s_{A}). The main observation now is that the zero set of the Cauchy-Riemmann operator may be not smooth but the Euler class of the Cauchy-Riemann operator ( a section of a certain orbibundle) is well defined over Q. To show this we perturb the (vertical) Riemann-Cauchy operator with the help of a multi-section of the related Banach bundle such that the perturbed (now a multi-)section is transversal to the zero section of the associated Banch bundle. Hence follows, that the moduli space of the (vertical) stable holomorphic maps defines a fundamental class over Q. (If the symplectic manifold is weakly-monotone then it suffices to take a single perturbation and henceforth the moduli space defines a fundamental class over Z.)

This construction consists of the following steps

\underline{Step 1}: Set-up. Our original configuration space Map_{g,m}(E, j, s_{A}) is the space of vertical stable maps u from semi-stable curves (Σ, x_{1}, ⋯, x_{k}) to E such that the image of u realizes the class s_{A} (cf. [LT]). Clearly Map_{g,m}(E, j, s_{A}) is a fibration of Banach orbifold (see [LT]) over B, therefore it possesses also a structure of a Banach orbifold.

Next we define a bundle E over F̄(E, s_{A}, g, k) whose fiber over point (b, u) consists of (0, 1)-forms over the domain Σ_{u} of map u taking values in u^{∗}(T F (j^{−1}(b))).

Clearly the moduli space of vertical stable holomorphic maps is the zero set of the Cauchy-Riemann section ¯∂_{J(E)} of the bundle E:

¯∂_{J(E)}(x, u)_{|y} = ¯∂_{J}u_{|y}

for all y ∈ Σ_{u}.

\underline{Step 2}. We assume that our space is equipped with a sufficient good Sobolev norm or a C^{l}-norm. In order to show that our moduli space of vertical stable holomorphic curves carries a fundamental class it suffices to show that the bundle E with its Cauchy-Riemmann section ¯∂_{J(E)} is a generalized Fredholm orbifold bundle (see Theorem 1.2 in [LT], or that is to say that the moduli space possesses a Kuranishi structure, see [FO]). Thus the following Proposition (cf. Proposition 2.2 [L-T]) gives us the desired fundamental class.

Proposition 3.2.1.(i) The section ¯∂_{J(E)} gives rise to a generalized Fredholm orbifold bundle with the natural orientation and of index 2c_{1}(s_{A}) + 2k + (2n − 6)(1 − g).

(ii) The homotopy class of ¯∂_{J(E)} does not depends on the choice of J(E).

Proof. (i) (We follow the argument in [LT]). We need to construct a covering U_{δ}(u_{b}, Σ, x⃗) of the zero section of ¯∂_{J(E)}, a smooth approximation E_{i} over a local uniformization Ũ_{δ}(u_{b}, Σ, x⃗) of our chosen covering, and a smooth (equivariant) resolution (F_{i}, ψ_{i}) for the smooth approximation (Ũ_{δ}, E_{i}).

We construct a covering U_{δ} of the zero set as in [LT]. At a point (u_{b}, Σ, x⃗) a neighborhood

U_{δ}(u_{b}, Σ, x⃗) = Map_{δ}(D_{δ}(b), W_{0}, K),

is defined via parameters: D_{δ}(b) - a small neighborhood of b in B, W_{0} - a neighborhood of the stabilization Stab(Σ, x⃗) of (Σ, x⃗) in CM_{g,m} and K - a compact subset in the universal family Ũ of curves over the local uniformization W̃ of W, where W - a small neighborhood of the pair of a curve (Σ, x⃗, z⃗) and point b in the product of CM_{g,m+l} with D_{δ}(b), where z⃗ are l marked points added to bubbling components.

More precisely δ and D_{δ}(b) are chosen so small such that Lemma 3.1 in [L-T] holds.

Next we construct a smooth approximation E_{i} as follows. Let S be a finitely dimensional subspace in Γ_{l−1}^{0,1}(Ũ, T F × D_{δ}(b)) (l ≥ 2). Let E_{S} be a bundle (of finite rank) over Map_{δ}(D_{δ}(b), W, K) whose fiber is S.

The following Lemma is just a copy of Proposition 3.4 [LT].

Lemma 3.2.2. Suppose that the restriction S_{|u} to a holomorphic stable map u is transversal to L_{u}. Then by shrinking W if necessary, if δ is sufficient small and K is sufficient large, the preimage (¯∂)_{J_{E}})^{−1}(E_{S}) is a smooth submanifolds, which contains u, in Map_{δ}(D_{δ}(b), W, K) and of dimension 2c_{1}(V)(A) + 2(n − 3)(1 − g) + 2k + dim S. Moreover, E_{S} ↦ (¯∂_{J_{E}}(E_{S}) is a smooth bundle.

This Lemma shows us that (E_{S}, Ũ(u_{b}, Σ, x⃗) is a smooth approximation of the local uniformization of our covering U_{δ}(u_{b}, Σ, x⃗).

At the final step we construct a resolution (F_{i}, ψ_{i}) inductively as in

[LT] using the compactness of ¯∂_{J(E)}^{−1}(0).

The part (ii) can be proved in the same way.

3.3. Properties of parameterized Gromov-Witten invariants

The following theorem is obvious because our construction of parametrized Gromov-Witten invariants is preserved under symplectic bundle isomorphisms.

Theorem 3.3.1. The parametrized Gromov-Witten invariant are invariants of symplectic bundles.

Remark 3.3.2. The bordism type invariants of the moduli space of pseudo holomorphic curves may have more informations than the Gromov-Witten invariants. Such an examples of finer Gromov-invariants of symplectic manifolds can be found in [McD 3].

In order to distinguish symplectic bundles by using Gromov-Witten invariants we need to compute those for trivial symplectic bundles E = B × F. As we have seen, it is convenient to switch time to times from cohomology classes to homology classes by using Poincare duality. As in (the Remark in) 3.1 we denote cohomology classes by small greek letters and their Poincare dual by capital Greek letter. Thus the map I_{g,m,s_{A}}^{E} has also arguments (Γ_{1}, ⋯, Γ_{n}). By Kuenneth’s formula the algebra H^{∗}(B × (F)^{m}, Q) equals (H^{∗}(F, Q))^{m} ⊗ H^{∗}(B, Q). Let α_{i} ∈ H^{∗}(F, Q) and β ∈ H^{∗}(B, Q).

Proposition 3.3.3. The Gromov-Witten invariants of a trivial symplectic fibration equals

I_{g,m,S_{A}}^{E}(α_{1} ⊗ ⋯ ⊗ α_{m} ⊗ β) = I_{g,m,A}^{M}(i^{∗}(α_{1}) ⊗ ⋯ ⊗ i^{∗}(α_{m})) ∫_{B} β. (3.4)

Proof. We choose a vertical compatible almost complex structure J(E) as a fiber-wise constant compatible almost complex structure J ∈ J (F). Clearly the moduli space of vertical stable maps CM_{g,m}(E, B × J, s_{A}) is the direct product B × CM_{g,m}(J, A). Denote by Π^{pt} the mapping in (3.1) with base B = pt. By (3.2.1) the LHS of (3.4) equals

P D(α_{1}⊗⋯ α_{m}⊗β\Π_{∗}[B×CM_{g,m}(J, A)]) = P D(α_{1}⊗⋯⊗β\([B]×Π_{∗}^{pt}[CM_{g,m}(F, A)])

(3.5)

Clearly the RHS of (3.4) equals the RHS of(3.5). □

10 V. H. LÊ

Now let us compute the Gromov-Witten invariants of a pull-back symplectic bundle. Let us consider the case when the base B_{1} is kcovering of the base B_{2} and a symplectic bundle E_{1} over B_{1} is induced by a symplectic bundle E_{1} by the covering map. Clearly a section s_{A} in the associated bundle over B_{2} can also be lifted to a section s_{A} of the associated bundle over B_{1}. We get immediately the following

Proposition 3.3.4. We have

I_{g,m,s_{A}}^{E_{1}} = kI_{g,m,s_{A}}^{E_{2}}.

At this moment we would like to introduce a class of parametrized Gromov-Witten invariants. We call an invariant I_{g,m,s_{A}}^{E} a natural parmetrized Gromov-Witten invariant if the moduli space CM_{g,m}(F, A) is empty (e.g. by dimension reason). Now we shall compute natural parametrized Gromov-Witten invariants for a direct fiber connected sum of two symplectic homological trivial bundles.. To get a good expression of natural parametrized Gromov-Witten invariants we use again the geometric way in terms of “fiberwise counting J(E)- holomorphic curves” as I_{g,m,s_{A}}^{E}(Γ_{1}, · · · · · · , Γ_{m}, e_{B}^{dim B}).

Proposition 3.3.5.Let E = E_{1}#E_{2} is a fiber connected sum of symplectic bundles E_{1} and E_{2}. Then we have the following formula for natural parametried Gromov-Witten-Invariants.

I^{E} = I^{E_{1}} + I^{E_{2}}.

Proof of Proposition 3.3.5. By a homotopy argument we see that there is a fiberwise compatible almost complex structure J(E_{i}) such that its restriction to D(ε) × M equals the product of D(ε) and a compatible almost complex structure J on M. In the procedure of a construction of a smooth resolution for ¯∂_{J_{(E_{i})}} we can use a smooth resolution for the zero set of ¯∂_{J_{(}E_{i})} restricted to D(ε)×M, in particular we can use a smooth resolution for ¯∂_{J}. Now in the next step to construct a fundamental class for the moduli space of vertical holomorphic maps we use a perturbation which extends a regular perturbation for ¯∂_{J} over D(ε)×M. The naturality of the Gromov-Witten invariants implies that the moduli space of vertical stable holomorphic curves on E equals the union of the 2 corresponding moduli spaces. Hence follows the statement. □

4. Homotopy groups of symplectomorphismgroups.

In this section we combine Remark 2.2, Propositions 3.3.3, 3.3.4, 3.3.5 and other ideas to estimate the rank of homotopy groups of symplectomorphismgroups. The following Proposition is an immediate consequence of Remark 2.2 and Proposition 3.3.5.

Proposition 4.1. Natural parametrized Gromov-Witten invariants define elements in Hom(π_{i}(BSym(M, ω), H^{∗}(CM_{g,m}, Q)).

As an application of Proposition 4.1 (or Proposition 3.3.4) we reinterpret a result by Gromov [Gr, Theorem 2.4.C_{2}].

Let (M, ω) = (S^{2}×S^{2}, ω_{1}⊕ω_{2}) be a product of symplectic manifolds. We denote by A_{i} the generators of H_{2}(M, Z) realizing by the i-th sphere S^{2}.

Theorem 4.2. If ω_{1}(A_{1}) > ω_{2}(A_{2}), then there is a S^{2}×S^{2}-symplectic bundle over S^{2} such that a natural parametrized Gromov-Witten invariant is non-trivial. In particular the rank of π_{1}(Sym(S^{2} × S^{2}, ω)) is at least 1.

Proof. According to Proposition 4.1 it suffices to find a symplectic bundle E over S^{2} with fiber (S^{2} × S^{2}, ω_{1} ⊕ ω_{2}) and a natural parametrized Gromov-Witten invariant whose value on E is non-trivial. We shall construct the bundle E by finding its associated element g ∈ Symp(S^{2} × S^{2}, ω_{1} ⊕ ω_{2}). The existence of such element g was shown by Gromov [G], and in what follows we shall give a complete proof of this existence. First we need the following lemma (compare with [G, 2.4.B′_{3}]).

Lemma 4.3. The subspace J_{0} of compatible almost complex structures J on S^{2} × S^{2} for which there exists a J-holomorphic sphere in a class A_{1} − lA_{2}, 1 ≤ l, is a non-empty closed subset of codimension 2 in J (S^{2} × S^{2}, ω). For each J ∈ (J (S^{2} × S^{2}, ω) \ J_{0}) and for each point in S^{2} × S^{2} there is a unique J-holomorphic sphere realizing class A_{i} and passing through x.

Proof. The subspace J_{0} is non-empty because the sphere (x, −x) is symplectic. To prove the closedness of J_{0} we first notice that the energy of a holomorphic sphere in class A_{1} − lA_{2} is less than ω_{1}(A_{1}). Thus we can apply the Gromov compactness argument to see the following fact. If at an almost complex structure J_{0} which is a limit point of a sequence of almost complex structures in J_{0} a holomorphic sphere A_{1} − lA_{2} bubbles, then there most be J_{0}-holomorphic spheres of type k_{i}A_{1} − l_{i}A_{2} such that ∑ k_{i} = 1 and ∑ l_{i} = l. Since there is always a J_{0}-holomorphic sphere in class A_{2}, taking into account of positivity

12 V. H. LÊ

of intersection in dimension 4 we conclude that the bubbling spheres must be of type A_{1} + l_{i}A_{2} and l_{j}A_{2} such that ∑ l_{i} + l_{j} = −l. Since l_{j} must be positive we conclude that there must be a bubble of type A_{1} − l′A_{2}, l′ ≥ 1, that proves the closedness of J_{0}.

The codimension of J_{0} is at least 2 by the same argument as in [MD 1] or [LO 1,2]. (Namely for a fixed l we consider the universal moduli space of the pairs (J, J−holomorphic sphere in class A_{1} − lA_{2}). The Fredholm index of the projection of this moduli space on the first factor is equal 4 + 2(1 − l) − 6 ≤ −2.) To prove that the codimension is precisely 2 we use the uniqueness of J-holomorphic sphere in class A_{1} −A_{2} if it exists. (cf. with the argument in [HLS]. It follows that the kernel of the projection of the moduli space on the first factor equals zero.)

Now let us prove the last statement. Since A_{2} is a minimal class in sense of Gromov for each J ∈ J (S^{2} × S^{2}, ω) there is a unique J-holomorphic sphere realizing A_{2} and passing through x. Finally to show the existence of J-holomorphic sphere in class A_{1} for J ∈ (J \J_{0}) we use again the bubbling analysis. Indeed if a bubble happens there must be a J_{0}-holomorphic sphere in a class A_{1} − lA_{2} for 1 ≤ l. □

Now let us find an element g ∈ Sym(S^{2} × S^{2}, ω_{1} ⊕ ω_{2}) by studying the action of the group of symplectomorphisms on J \ J_{0}. Since J_{0} is a closed subset of codimension 2 we can choose a small disk D^{2} in J (S^{2} × S^{2}) such that this disk intersects at J_{0} at exactly one interior point. By a result of Gromov (see also Appendix 1, Proposition A.1) there is a loop g_{t} in the group Sym(S^{2} × S^{2}, x) such that the image g_{t}(J_{0}) is homotopic to loop ∂D^{2}. Let D_{∗}^{2} be a disk in J (S^{2} × S^{2}) with boundary g_{t}(J_{0}) which intersects with J_{1} at exactly one point (this disk D_{∗}^{2} can be chosen as the sum of the disk D^{2} and a homotopy cylinder between ∂D^{2} and g_{t}(J_{0})).

Now we construct our bundle E by gluing two trivial S^{2} × S^{2} bundles over a disk B^{2} using this loop g_{t}. We denote by e^{0} the cohomology class dual to the fundamental (homology) class of the bundle E^{⊗3}. We claim that the parametrized Gromov-Witten invariant I_{0,3,A_{1}−A_{2}} is natural and its value I_{0,3,A_{1}−A_{2}}^{E}(e^{0}) is 1. The naturality follows from the fact that, c_{1}(A_{1} − A_{2}) = 0 and hence the moduli space CM_{0,3}(J, A_{1} − A_{2}) is empty for generic J. To compute this invariant for our bundle E explicitly we construct a generic fiberwise compatible almost complex structure J(E) on E as follows. (Note that we in the case of weakly monotone symplectic manifolds and we are working with Gromov-Witten invariants of genus 0, therefore it suffices to perturb J to get the fundamental class of the correspondent moduli space). We observe that the standard product complex structure J_{0} on S^{2} × S^{2} is a (A_{1} − A_{2})-regular. Then we take J(E) being the gluing of the product structure J_{0} × B^{2} and D_{∗}^{2} along the boundary ∂D^{2} (on which they agree with the gluing map g_{t}). By a perturbation of J(E) around the singular point (which does not change the intersection number of the perturbed disk D_{∗}^{2} with the singular set J_{0}) we can assume that this almost complex structure J(E) is (A_{1} −A_{2}) regular. By the construction the parametrized moduli space CM_{0,3,A_{1}−A_{2}}(E, J(E)) consists of one point, corresponding to the singular value in J_{0}. According to (3.2.1) the value I_{g,m,A_{1}−A_{2}}^{E}(e^{0}) = 1. By Proposition 4.2 the non-trivial natural parametrized Gromov-Witten invariant defines a non-trivial element in Hom(π_{2}(BSym(M, ω), Q). Hence π_{1}(Sym(M)) = π_{2}(BSym(M)) contains an element of infinite order. (We can also use Proposition 3.3.4 to conclude that g^{n} ≠ 0 for all n). □

Now we are going to prove the following

Theorem 4.4. Let (M, ω) be a symplectic manifold (S^{2} × S^{2} × N^{2k}, ω_{1} ⊕ ω_{2} ⊕ ω_{3}) where (N^{2k}, ω_{3}) is a compact symplectic manifold. Let A_{i} denote the image of the i-th sphere S^{2} in M. Suppose that ω_{1}(A_{1}) > ω_{2}(A_{2}). Then we have rk(π_{1}(Sym(M, ω)) ⊗ Q) ≥ 1 and rk(π_{3}(Sym(M, ω)) ⊗ Q) ≥ 2 .

Proof. To prove the first statement we shall construct a symplectic bundle E with fiber (M, ω) over S^{2} and compute a natural parametrized Gromov-Witten invariant of E. Namely we consider the moduli space of vertical holomorphic mappings f : S^{2} → E whose image realizes class A := A_{1} − A_{2}.

The dimension computation shows us the moduli space of vertical stable holomorphic curves in class A on a (M, ω)-bundle over S^{2} has dimension 2k. Thus we can “count” the number of vertical pseudoholomorphic spheres in class A which intersect the cycle Γ representing by S^{2} × S^{2}. We claim that this number of the the trivial bundle S^{2} × (M, ω) equals zero. More precisely the corresponding Gromov-Witten invariant is natural. To show it we consider a vertical almost complex structure J^{pr} on the trivial bundle (M, ω) × S^{2} such that on each fiber (M, ω) we have J^{pr} = J_{0} × J^{N}, where J_{0} is the standard product complex structure on S^{2} × S^{2} and J^{N} is an almost complex structure on N^{2k}, ω_{3}). Clearly any J^{pr}-sphere is also a J_{0}-sphere. Hence the moduli space of J^{pr}-sphere in class A_{1} − A_{2} is empty.

Now we construct a symplectic (M, ω)-bundle E over S^{2} by gluing two trivial symplectic (M, ω)-bundles over disk D^{2} via the map g ×Id : S^{1} →∈ Symp(S^{2} × S^{2}) × Id ⊂ Symp(M, ω), where g is constructed in

14 V. H. LÊ

the proof of Theorem 4.2 (see also Proposition A.1 in the Appendix). To compute the Gromov-Witten invariant I_{0,3,A_{1}−A_{2}}^{E}(P D[N]) we construct a fiberwise compatible almost complex structure J(E) by gluing two generic fiber-wise compatible almost complex structures on the restriction of E to each disk D^{2}. The first one (on the first disk D^{2}) is defined as follows: J(z) = (J_{0} × J^{s}), where J_{0} is the standard product symplectic structure on S^{2} × S^{2} and J^{s} is a singular almost complex structure in J_{0}(S^{2} × S^{2}, ω_{1} ⊕ ω_{2}) (see the Proof of Theorem 4.2). The second fiber-wise compatible almost comple structure is defined as follows: J(z) = (ϕ(z)×J^{s}), where ϕ is the diffeomorphism from D^{2} to the disk D_{∗}^{2} constructed in the proof of Theorem 4.2. We claim that the constructed vertical almost complex is A-generic. Clearly outside the only singular point in S^{2}, where the vertical almost complex structure take value (J^{2} × J^{2}), the moduli space of J(E) holomorphic curves realizing A is empty. At the singular point the moduli space is diffeomorphic to N^{2}, namely it consists of maps u_{y} = (u_{1}, y), where f_{1} is a holomorphic sphere in S^{2} × S^{2} and y is a point in N^{2k}. To show the regularity of the solution we need to prove that the linearization map D ¯∂_{J} : T_{u}Map_{g,m}(E, j, A) → Ω^{0,1}(S^{2}, u^{∗}(T M)) is surjective. Since u = (u_{1}, y) the surjectivity of the linearization map follows from the surjectivity of the linearization map considered in the proof of Theorem 4.2. This proves the first statement in Theorem 4.4.

Now let us prove the second statement. It suffices to show that the subgroup SO(3) × SO(3) ⊂ Symp(M, ω) realizes two linearly independent elements in π_{3}(Sym(M, ω)). Clearly the group Sym(M, ω) acts also on the frame E = Sp(4 + 2k, R)-bundle over M. We choose a point (x_{1}, x_{2}, y) ∈ (S^{2} × S^{2} × N^{2k}) and a tangential vector (v_{1}, v_{2}, 0) at this point (v_{i} ≠ 0). The evaluation map at this tangential vector maps the subgroup SO_{3} × SO_{3} injectively. It is easy to check that the image of this subgroup realizes two linearly independent elements in π_{3}(E). Hence follows statement. □

There are also a dual way to see parametrized Gromov-Witten invariants. We recall that the parametrized Gromov-Witten invariants are defined via the fundamental class of the moduli space of stable curves, the slant product (and Poincare duality). Since we have two kinds of slant product we also have the following map

CGW_{g,m,s_{A}}^{E} : H^{∗}(CM_{g,m}, R) → H^{∗+μ}(E^{⊗m}, R)  (4.1)
δ ↦ P D(δ\Π_{∗}(CM_{g,m,s_{A}}(E, J(E)))

which pushes forward to the following map

CGW_{g,m,s_{A}}^{B} : H^{∗}(CM_{g,m}, R) → H^{∗+μ−2nm}(B, R).  (4.2)

We call the image of the map CGW - the Gromov-Witten characteristic classes.

Theorem 4.5. The Gromov-Witten characteristic classes are invariants of symplectic bundles. All the Gromov-Witten characteristic classes with positive degree of a trivial symplectic bundles are zero.

Proof. The first statement is obvious (cf. Theorem 3.3.1). To prove the second statement we compute the Gromov-Witten characteristic classes on trivial bundle M × B. Let [T] be a cycle in B. As before we denote also by j the projection E^{⊗m} → B.

CGW_{g,m,s_{A}}^{B}(δ)[T] =< P D(δ\Π_{∗}(B × CM_{g,m}(J, A)), j^{−1}([T]) >
=< δ\(B × Π_{∗}CM_{g,m}(J, A)), j^{−1} ^{∗}P D_{B}([T]) >
= ∫_{B} P D_{B}([T])· < δ ⊗ 1^{0}, Π_{∗}(CM_{g,m}(J, A)) > .  (4.3)

Clearly (4.3) equals zero if dim[T] ≥ 1 ist. □

The word “characteristic” is explained by the following functoriality of these classes. In particular we see that the Gromov-Witten characteristic classes are cohomology classes of the classfying space BSym(M, ω).

Theorem 4.6. Let E be a symplectic bundle over B and f^{∗}(E) be the induced symplectic bundle by a map f : B_{1} → B. Then the map f also induces the Gromov-Witten characteristic classes of the bundle E.

Proof. The map f alo induces a homomorphism in cohomology f̃_{∗} : H_{∗}(CM_{g,m} × f^{∗}(E^{⊗m})) → H_{∗}(CM_{g,m} × E^{⊗m}). On the other hand by the construction we have

CM_{g,m,s_{A}}(f^{∗}(E)) = f^{∗}(CM_{g,m,s_{A}}(E))
f̃_{∗}(Π_{∗}(f_{∗}(CM_{g,m,s_{A}}(E))) = Π_{∗}(CM_{g,m,s_{A}}(E)).

Hence we can see immediately that

CGW_{g,m,s_{A}}^{B_{1}} = f^{∗} ∘ CGW_{g,m.s_{A}}^{B}
□

It is an easy exercise to interpret Theorem 4.2 and Theorem 4.4 in term of Gromov-Witten characteristic classes.

Before closing this section we would like to make a remark. In his talk [Kon 2] Kontsevich gives a construction of characteristic classes of symplectic bundles using graph homology. We believe that there is a connection of his construction and ours. Kontsevich considers two

16 V. H. LÊ

infinite-dimensional Lie algebras a_{n} and c_{n}. The last one is a subalgebra of the algebra h_{n} of Hamiltonian vectors fields in R^{2n} with polynomial coefficients. Kontsevich proved that [Kon 3] the stable homology of c_{n} equals the graph homology and the stable homology of a_{n} equals homology of the moduli space of smooth algebraic curves with unlabeled points. Now using the Fuks method [F] ( generalizing a construction of Bernstein-Rosenfeld) Kontsevich construct a homomorphism from the cohomologies of the algebra h_{n} to the cohomologies of a symplectic fibration E over a base B^{2n}. This homomorphism descends to a cohomology of the base B. This homorphism is characteristic (in the sense of Theorem 4.6). Thus we can use graph homology to construct characteristic classes of symplectic fibrations. On the other hand there is also a map c_{n} → a_{n}. Thus it is also possible to using cohomology classes of M_{g,m} to construct characteristic classes of symplectic fibrations.

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[BH] W. Balcerak and B. Hajduk, Homotopy type of automorphism groups of manifolds, Colloq. Math., 1981, v. 45, 1-33.

[C] E. Calabi, On the group of automorphism of a symplectic manifold, in: Problem in analysis, 1-26, Princeton Univ. 1970.

[F-O] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, preprint 1996.

[F] D. B. Fuks, Cohomology of infinite dimensional Lie algebra, Consultant Bureau, New York- London, 1986.

[G] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inv. Math. 82 (1985), 307-334.

[HLS] H. Hofer, V. Lizan, Sikorav, On genericity for holomorphic curves in 4-dimensional almost complex manifolds, Preprint 1994, Univ. Toulouse III.

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[Kon 2] M. Kontsevich, Symplectic fibration, graph homology and Rosansky-Witten invariants, talk at Bonn 1997.

[Kon 3] M. Kontsevich, Formal (non)-commutative symplectic geometry, The Gelfand Seminar, 1990-1992, Birkhäuse, 1993, 173-178.

[KM] M. Kontsevich and Iu. Manin, Gromov-Witten classes, quantum cohomology and enumerative geometry, Comm. Math. Physics, 1994, 525-562.

[Li] W. Li Casson -Lin’s Invariant of a knot and Floer homology, preprint 1996.

[LO 1] H. V. Le and K. Ono, Perturbation of holomorphic curves, Int. J. Math., 1996, 771-774.

[LO 2] H. V. Le and K. Ono, Cuplength estimate for symplectic fixed points, in book: Contact and symplectic geometry, Publication of Newton Institute, Cambridge Uni. Press, 1996, 268-295.

[Lott 1], J. Lott, To see the group of diffeomorphisms by using analysis, talk at Bonn 1996.

[Lott 2], J. Lott, Diffeomorphisms, analytic torsion ans noncommutative geometry, preprint MPI 96-98.

[LT] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, preprint 1996.

[Mi] J. Milnor, Remarks on infinite-dimensional Lie groups, In: Relativity, groups and topology II, (Edited by B. Dewitt and R. Stora) North-Holland, 1984 , 1007-1058 (385).

[MD 1] D. McDuff, Blow ups and symplectic embedding in dimension 4, Topology 1991, v. 30, 409-421.

[MD 2]D. McDuff, The structure of rational and ruled symplectic 4-manifold, J. of AMS, 1990, v.3, 679-712.

[McD 3]. D. McDuff, Examples of symplectic structures, Inv. Math., 1987, 89, 13-36.

[McD-S 1] McDuff and Salamon, Introduction in Symplectic Topology, Oxford Sci. Publ. 1995.

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18 V. H. LÊ

5. Appendix

We recall the following Gromov result [Gr]

[Gr, 2.4.A_{1}] The group Symp(S^{2} × S^{2}, ω) is contractible to a Z_{2} extension of SO_{3} × SO_{3} if ω is a monotone form.

[Gr, 2.4.C_{2}] If ω is not monotone then π_{1}(Sym_{ω}S^{2} × S^{2}) contains an element of infinite order.

Gromov gaves a sketch of his proof in [Gr]. Using a part of his argument we interpret his result [Gr, 2.4.C_{2}] in term of Gromov-Witten invariant (Proposition 4.2). For the case of completeness we shall prove the following weaker statement which we used in our proof. Let us use the same notation J_{0} as it is defined in the proof of Lemma 4.3. Denote by J_{0} the standard complex structure on S^{2} × S^{2}.

Proposition A.1. There is a loop g_{t} in Sym(M) such that the loop g_{t}(J_{0}) bounds a diks D^{2} which intersects with J_{0} at exactly one interior point.

Proof. Fix a point v_{0} ∈ S^{2} × S^{2}. Clearly the group Diff(S^{2} × S^{2}) contains a subgroup SO_{2}×SO_{2} which consists of fiberwise rotations (w. r. to metric g_{0} = (J_{0}, ω)) keeping v_{0} invariant. First we shall construct a map α from J^{reg} = J \ J_{0} to the factor-group Diff(M, x_{0})/(SO_{2} × SO_{2}).

Let J ∈ J^{reg}. Since S^{2}×S^{2} is fibered by 2 families of J_{0} holomorphic spheres in class A_{1} and A_{2}, for a fixed point v_{0} = (s_{0}, s_{0}) ∈ S^{2} × S^{2} we can construct a map α as follows.

Let A_{1}(J) and A_{2}(J) be two J-holomorphic spheres through point v_{0}. Let f_{i} be a biholomorphic map from A_{i}(J) to A_{i}(J_{0}) which can be defined uniquely up to P SL(2, C). Requiring f(s_{0}) = v_{0} imposes other two constrains parameters. Further we note that (J, ω) and (J_{0}, ω) define 2 compatible metrics g and g_{0} on S^{2} × S^{2}. Hence on each sphere A_{i}(J) (resp. A_{i}(J_{0})) there exists uniquely a south poles ( once s_{0} is supposed to be the north pole). Thus we have still 2 free parameters for f_{i} after requiring that the south pole on A_{i}(J) must go to the south pole on A_{i}(J_{0}). We can impose the last condition on f_{i} that the preimage of the hemisphere on A_{i}(J_{0}) has a half of the area of the sphere A_{i}(J). Now it is easy to see that f_{i} is defined uniquely up to the rotation subgroup S^{1} ∈ Iso(S^{2}, g_{0}). Next we extend the map f_{i} to a diffeomorphism of (S^{2} × S^{2}, v_{0}) in the following way. Note that each element x ∈ S^{2} × S^{2} is the intersection point of two J-holomorphic spheres through the point (s_{i}) ∈ A_{i}(J). So we define α(J)(x) to be the intersection point of two J_{0}-holomorphic spheres through f_{i}(s_{i}). Note that the map α(J) is defined up to the subgroup SO_{2} × SO_{2}.

Lemma A.2. The map α factors through a map

α̂ : J_{0} → Sym_{ω}(V, v_{0})/(SO_{2} × SO_{2})  (A.1)

Proof([Gr]). Since the map α(J) sends the A_{i}(J) spheres to A_{i}(J_{0}) spheres, and the fibration A_{1}(J_{0}) is symplectically orthogonal to the fibration A_{2}(J_{0}) we observe that the form α(J)^{∗}(ω) is J-positive. Now we apply the Moser stability theorem to the family tω+(1−t)α(J)^{∗}(ω) and get immediately our Proposition A. 2. □

Continuation of the proof of Proposition A.1. In this step we consider the natural evaluation map

ev_{J_{0}} : Sym(S^{2} × S^{2}, ω) → J^{reg}(S^{2} × S^{2}, ω)
g ↦ g^{∗}(J_{0})

Since the standard product structure J_{0} is SO_{2}×SO_{2}-invariant the evaluation map ev_{J_{0}} descends to a map ēv_{J_{0}} from the quotient Sym(S^{2} × S^{2}, v_{0})/(SO_{2} × SO_{2}).

It is easy to check

Lemma A.3 [Gr]. The composition α̂ ∘ ēv_{J_{0}} is the identity map. From Lemma A.3 and considering an exact homotopy sequence we

conclude that the map ev_{J_{0}} is surjective on the homotopy groups. Now statement in Proposition A.1 follows immediately. □

email: lehong @ mathematik.uni-leipzig.de