Infinitely many non-isotopic real symplectic forms on S 2 × S 2
aa r X i v : . [ m a t h . S G ] S e p INFINITELY MANY NON-ISOTOPIC REAL SYMPLECTIC FORMS ON S ˆ S GLEB SMIRNOV
Abstract.
Let p S , ω q be a symplectic sphere, and let τ : S Ñ S be an anti-symplectic involution of p S , ω q . We consider the product p S , ω q ˆ p S , ω q endowed with the anti-symplectic involution τ ˆ τ ,and study the space of monotone anti-invariant symplectic forms on this four-manifold. We show thatthis space is disconnected. In addition, during the course of the proof, we produce a diffeomorphismof Gr p , q which induces the identity map on all homology and homotopy groups, but which is nothomotopic to the identity.
1. Introduction.
Is there a closed four-manifold X and a cohomology class ξ P H p X ; R q such thatthe space Ω ξ of symplectic forms of class ξ is connected? This uniqueness problem up to isotopy forcohomologous symplectic forms is completely open in dimension four, though disconnected examplesare known in higher dimensions, see Problem 2 and Theorem 9.7.4 in [5]. This short note concernsthe following modified version of this problem.A real symplectic 4-manifold is a triple p X, σ, ω q , where ω is a symplectic form on X and σ : X Ñ X is an involution which is anti-symplectic σ ˚ ω “ ´ ω. Pick a class ξ P H p X ; R q such that σ ˚ ξ “ ´ ξ and let R Ω ξ denote the space of those symplectic formson X which are σ -anti-invariant and are in the class ξ . A natural question to ask is: are there anyexamples of disconnected spaces R Ω ξ ?Let us introduce a couple of simple examples of real symplectic manifolds. Consider the real ruledquadric X defined as X : “ x P CP | x ` x “ x ` x ( . (1.1)The set of real points of X is a -torus in RP which is doubly ruled by real projective lines.Being a smooth projective variety, the surface X inherits a K¨ahler form ω from the ambient space p CP , Ω st q . Here Ω st stands for the Fubini-Study -form. After rescaling ω , we assume it is monotonemeaning that r ω s “ ´ K X P H p X ; Z q , where K X is the canonical class of X . The complex conjugation σ : CP Ñ CP , σ p x i q “ ¯ x i is ananti-symplectic involution w.r.t. to Ω st . If a projective hypersurface is cut out by a real polynomialthen it is preserved by σ and hence itself carries an anti-symplectic involution, namely p σ | X q ˚ ω “ ´ ω .In this anti-invariant setting, we answer to the question of uniqueness in the negative proving Theorem 1.
The space of monotone anti-invariant symplectic forms on X has infinitely many con-nected components. A weaker statement applies to the quadric X defined as X : “ x P CP | x ` x ` x ` x “ ( . (1.2)In this case we have: A somewhat unfortunate term, but let’s follow [11, 3].
Theorem 2.
The space of monotone anti-invariant symplectic forms on X has at least two connectedcomponents. To prove both theorems, concrete representatives of different connected components of R Ω p´ K X q (for abbreviation, we let R Ω K stand for R Ω p´ K X q ) will be presented; these forms will be related by adiffeomorphism and so belong to the same connected component of the moduli space of σ -anti-invariantforms. It is not uninteresting to compare Theorems 1 and 2 with the recent results of Kharlamov andShevchishin (see [3]), who study real symplectic 4-manifolds up to the equivalence relation generatedby deformations and diffeomorphisms. In particular, Theorem 1.1 of [3] states that if a real rational symplectic 4-manifold p X, σ, ω q is σ -minimal, then it is a real K¨ahler surface. We do not discuss thenotion of σ -minimality but note that a minimal surface (e.g. S ˆ S ) is also σ -minimal.Although there are countably many non-isomorphic complex structures on S ˆ S , we stick to theone coming from the product CP ˆ CP , for it is the only complex structure which admits a K¨ahlerform in the anti-canonical class. A classical result is that there are exactly four different types ofanti-holomorphic involutions on CP ˆ CP (see Lemma 1.16 in [4], in this note we only discuss thetypes Q , and Q , .) Given an underlying real algebraic surface p X, σ q and a class ξ P H p X ; R q , thespace of σ -anti-invariant K¨ahler forms in class ξ is convex. It then follows from Moser’s argumentthat, up to isotopy, there exists exactly one K¨ahler form in a given cohomology class. The classicalMoser’s trick considers a family of symplectic forms ω t such that the cohomology class of ω t is constantand provides a family of diffeomorphisms ϕ t such that ϕ ˚ t ω t “ ω . The argument is easily adapted tothe case of anti-invariant forms: if the family ω t consists of σ -anti-invariant forms, then the derivedisotopy ϕ t commute with σ .Therefore, be given a pair of rational symplectic 4-manifolds p X, σ, ω q and p X, σ , ω q with X dif-feomorphic to S ˆ S and with both ω and ω being monotone. Then p X, σ, ω q is diffeomorphic to p X, σ , ω q iff σ is diffeomorphic to σ . In other words, the natural mapping D iff K p X, σ q Ñ R Ω K , f Ñ f ˚ ω is surjective. Here D iff K p X, σ q stands for the subgroup of those diffeomorphisms which preserve K X P H p X ; Z q and are σ -equivariant. Therefore, the moduli space R Ω K { D iff p X, σ q consists of a single point,yet, according to our claim, the space R Ω K itself may have infinitely many connected components.Another problem worth considering in this anti-invariant setting is to construct a pair of symplecticforms which are not deformation equivalent. Two ( σ -anti-invariant) symplectic forms ω and ω are saidto be deformation equivalent if there exists a ( σ -equivariant) diffeomorphism ϕ such that ϕ ˚ ω and ω are connected by a path of ( σ -anti-invariant) symplectic forms. In general, without taking in accountany involutions, inequivalent symplectic forms in dimension 4 have been obtained by McMullen andTaubes [6] and later by Smith [9] and Vidussi [10]. It is not immediately clear how to construct similarexamples in the presence of an anti-holomorphic involution. Note however that according to [3], onecannot produce such examples in the realm of real rational symplectic 4-manifolds. Acknowledgements.
The author thanks Sewa Shevchishin for valuable conversations about thework in this paper and the referee for the prompt reply and useful comments. This work is funded bythe ETH Zurich Postdoctoral Fellowship program.
2. Proof of Theorem 1.
We now describe X in a way that visibly exhibits its complex structure.There is a projective transformation sending equation (1.1) to y y “ y y , NFINITELY MANY NON-ISOTOPIC REAL SYMPLECTIC FORMS ON S ˆ S and transforming σ : x i Ñ ¯ x i into σ : p y , y , y , y q Ñ p ¯ y , ¯ y , ¯ y , ¯ y q . The rational functions z “ y y , w “ y y define a biholomorphism from X onto CP p z q ˆ CP p w q . In the inhomogeneous coordinates p z, w q ,the map σ takes the form σ p z, w q “ ` ¯ z ´ , ¯ w ´ ˘ . (2.1)Finally, the form ω on X splits into a product form ω CP ‘ ω CP .We now introduce an invariant which is capable to distinguish between some connected componentsof the space R Ω K of anti-invariant monotone ( ξ “ ´ K X ) symplectic forms on X . To understand thisinvariant, it is the easiest to start with the product K¨ahler form ω . We let L denote the fixed pointset of σ , which is the product of the two copies of RP defined, respectively, by | z | “ and | w | “ .Pick a point p on L and observe that there is but one smooth complex sphere passing through p foreach of the generators H p X ; Z q – Z p A q ‘ Z p B q . Denote these spheres by C A and C B , respectively.Notice that the curves γ A “ C A X L, γ B “ C B X L are transversally intersecting simple closed curves in L , which form a basis for H p L ; Z q . Our invariantassociates to ω P R Ω ξ the class r γ A s P H p X ; Z q . There is no natural choice for orientaion of γ A , sowe orient it somehow. To see γ A is indeed an invariant for connected components of R Ω K , we use thefollowing observation of Gromov: Theorem 3 (Gromov, 2.4.A in [1]) . Let p X, ω q be S ˆ S endowed with a product monotone form.Then, every ω -compatible almost-complex structure J defines two tranversal fibrations of X into J -holomorphic spheres and these fibratons continuously (even smoothly) depend on J . Therefore, for every ω -compatible almost-complex structure J there is but one J -holomorphic sphere C A in class A passing through p .We let J ω denote the space of ω -compatible almost-complex structures, where one can find thesubspace of σ -anti-invariant ( σ ˚ ˝ J “ ´ J ˝ σ ˚ ) structures, denoted by R J ω . We have already seenthat C A intersects L by a simple closed curve in the integrable case, but we wish to see this for anarbitrary J P R J ω . Since σ is anti-holomorphic, it must send the curve C A to another J -curve in class A . Since both C A and σ p C A q pass through the point p , they must coincide. As such, the restriction of σ : C A Ñ C A is an anti-holomorphic involution of C A that has a fixed point; it has, therefore, a fixedsmooth circle, which exhausts the fixed points. We conclude that for every J P R J ω the sphere C A inclass A intersects L by a closed simple curve γ A . The class r γ A s P H p L ; Z q does not depend on J , forthe space R J ω is connected (see Lemma 1 below.) Nor it depends on the isotopy class of ω , as longas our isotopy is σ -equivariant: given two forms at the same connected component of R Ω K , we useMoser’s trick to obtain a family of σ -equivariant diffeomorphisms between them, thus identifying thecorresponding spaces of almost-complex structures, spaces of holomorphic spheres etc. Lemma 1.
Let p X, σ, ω q be a real symplectic manifold, and let R J ω be the space of ω -compatiblealmost-complex structures which are anti-invariant under the anti-symplectic involution. The space R J ω is non-empty and connected, and in fact it is contractible by Proposition 1.1 in [11] .Proof. Let R σ be the space of Riemannian metrics on X which are invariant under the anti-symplecticinvolution. Clearly, the space R σ is convex and hence, contractible. There is a natural embedding i : R J ω Ñ R σ defined by J i ÝÑ ω p¨ , J ¨q for J P R J ω . GLEB SMIRNOV
We will prove that R J ω is a retract of R σ . Since R σ is connected, this would imply that R J ω isconnected too. For every g P R σ there is a unique field of endomorphisms A g of T X such that ω p¨ , ¨q “ g p A g ¨ , ¨q . Since ω is σ -anti-invariant and g is σ -invariant, it follows that A g and σ anti-commute, i.e. A g ˝ σ ˚ “ ´ σ ˚ ˝ A g . (2.2)Furthermore, as ω is skew-symmetric, so is A g , i.e. A tg “ ´ A g , where A tg is the adjoint for A g w.r.t. g .Set J g : “ p´ A g q ´ A g , (2.3)where the square root in the right-hand side of (2.3) is well defined because p´ A g q is g -self-adjointand positive-definite. In fact, self-adjoint positive operators have a unique self-adjoint positive squareroot, and this is the root we pick for p´ A g q . Also, since p´ A g q and σ commute, it follows that p´ A g q ˝ σ ˚ “ σ ˚ ˝ p´ A g q and p´ A g q ´ ˝ σ ˚ “ σ ˚ ˝ p´ A g q ´ . (2.4)Here we have used that σ is an involution. Using (2.2) and (2.4), we see that J g and σ anti-commute.Moreover, J g satisfies J g “ ´ id . (2.5)To prove (2.5), it suffices to check that p´ A g q ´ ˝ A g “ A g ˝ p´ A g q ´ . (2.6)One should pass to the complexification of T X to see this. Also, using (2.6), it is straightforward tocheck that J g is ω -compatible. We conclude that J g P R J ω . Define u : R σ Ñ R J ω as u p g q : “ J g . It is easy to see that u ˝ i “ id , so u is a retraction. (cid:3) To prove our theorem, we shall construct a sequence of forms ω k P R Ω K whose invariants r γ kA s P H p L ; Z q are pairwise distinct. To this end, we find a diffeomorphism f : X Ñ X such that:(1) We wish f to satisfy f ˝ σ “ σ ˝ f , so that f ˚ ω would be anti-invariant. The condition also impliesthat f keeps L invariant.(2) The restriction of f to L is smoothly isotopic to (a power of) the Dehn twist along the curve γ B ,so, in particular, we would have f ˚ r γ A s “ r γ A s ` m r γ B s ‰ r γ A s in H p L ; Z q – Z for some m P Z .(3) And, finally, we wish f ˚ : H p X ; Z q Ñ H p X ; Z q to be the identity isomorphism. This is neces-sary for using the invariant r γ A s , as it was defined with the class A .Once f is found, it can be used to obtain infinitely many non-isotopic forms ω k by setting ω k : “ f k ˚ ω .This finishes the proof.To construct f , we first do it on the subset CP p z q ˆ K ε Ă X, K ε “ w P CP p w q | ´ ε ă | w | ă ` ε ( with the formula f p z, w q : “ p z e m Arg p w q , w q for some m P Z , which can be seen as a mapping from K ε to D iff ` p CP p z qq , the group of orientation-preserving diffeo-morphisms of CP p z q . We ask whether or not this mapping can be extended to a mapping defined overthe whole sphere CP p w q . When m is even, the answer is yes since: K ε is homotopy equivalent to S and π p D iff ` p S qq “ Z . We thus constructed f as an orientation-preserving fiberwise diffeomorphismof the fibration CP p z q ˆ CP p w q Ñ CP p w q . NFINITELY MANY NON-ISOTOPIC REAL SYMPLECTIC FORMS ON S ˆ S Whether f is smoothly isotopic to the identity? If we do not impose the condition σ ˝ f “ f ˝ σ onthe isotopy, then the answer is yes since: π p D iff ` p S qq “ . Therefore, all the forms f k ˚ ω are in thesame connected component of Ω K but not of R Ω K .
3. Proof of Theorem 2.
We employ a similar model for X . Namely, X is again the product CP p z q ˆ CP p w q , but now σ is given by σ p z, w q “ ` ´ ¯ z ´ , ´ ¯ w ´ ˘ . (3.1)Just like in (2.1), the action of σ is componentwise, but in contrast to (2.1), the new involution hasno fixed points. Using Cartesian coordinates p x , y q on S ˆ S Ă R ˆ R , we describe σ as σ p x , y q “ p´ x , ´ y q . Consider the diffeomorphism f : X Ñ X given by f p x , y q : “ p´ x ` x x , y y y , y q . Here x x , y y stands for the Euclidean inner product of x and y . This map does nothing on thesecond sphere and does the reflection of the first sphere with the respect to the axis passing throughthe antipodal points y and ´ y . As f and σ commute, we obtain a descendant self-mapping g of Z : “ X { σ and the commutative diagram X f ÝÝÝÑ X §§đ p §§đ p Z g ÝÝÝÑ Z, (3.2)where p : X Ñ Z is the covering map, which identifies p x , y q with p´ x , ´ y q . Geometrically, Z – Gr p , q , the Grassmannian of two-planes in R , and X – S ˆ S is the corresponding Grassmannianof oriented planes.It is interesting to look at the algebraic properties g . Taking π k and H k of (3.2), we obtain com-mutative diagrams of abelian groups. Since f is homotopic to the identity (not equivariantly!), wehave f ˚ “ id : π k p X q Ñ π k p X q and f ˚ “ id : H k p X q Ñ H k p X q for all k . (3.3)Since p : X Ñ Z is a connected covering we also have isomorphisms p ˚ : π k p X q Ñ π k p Z q for k ‰ . (3.4)From (3.3) and (3.4) we obtain g ˚ “ id : π k p Z q Ñ π k p Z q for all k ‰ . (3.5)To prove that g ˚ “ id for k “ , observe that π p Z q “ Z and that the only automorphism of Z is theidentity.Recall that H p Z ; Z q “ Z , H p Z ; Z q “ Z , H p Z ; Z q “ Z ‘ Z , H p Z ; Z q “ Z , H p Z ; Z q “ Z . (3.6)Here the first and the last equality follow from the connectivity of Z . The group H is easy to recoversince we know that π p Z q “ Z , whereas the group H is recovered via Poincar´e duality. What is leftto compute is H . Recall that the Schubert cell decomposition of Gr p , q consists of one 0-cell, one1-cell, two 2-cells, one 3-cell, and of a single 4-cell. Thus, the dimension of H is not greater than 2.To show that H is exactly Z ‘ Z , we explicitly describe two non-homologous cycles in Z . GLEB SMIRNOV
Let us introduce the diagonal sphere ∆ : “ tp x , y q P S ˆ S | x “ y u . The sphere ∆ is invariant w.r.t σ , as σ p x , x q “ p´ x , ´ x q . Let Q denote p p ∆ q , which is an embedded RP in Z . The map s : X Ñ ∆ , s p x , y q : “ p x , x q fits into the diagram X s ÝÝÝÑ ∆ §§đ σ §§đ σ X s ÝÝÝÑ ∆ , and hence induces a map Z Ñ Q which we denote by the same letter s . Note that s : Z Ñ Q is a fiberbundle over Q . This bundle has a tautological section, given by Q itself. Let F be any fiber of s . Weclaim that the classes r F s , r Q s P H p Z ; Z q are non-zero and are not equal to each other. Indeed, since Q , being a section, intersects F at exactly one point, we have r F s ¨ r Q s “ . This implies that both F and Q are not homologically trivial. They are also not homologous to eachother, as for if they were that would imply the equality r F s ¨ r Q s “ r F s . However, the cycle F , beinga fiber, must have self-intersection number 0.We claim that g ˚ “ id : H k p Z ; Z q Ñ H k p Z ; Z q for all k . This is obvious for k ‰ for dimensionreasons. To prove that for k “ , observe that g and s commute and that g keeps Q fixed. Hence, wemust have g ˚ r F s “ r F s , g ˚ r Q s “ r Q s .A slightly more subtle computation reveals: H p Z ; Z q “ Z , H p Z ; Z q “ Z , H p Z ; Z q “ Z , H p Z ; Z q “ , H p Z ; Z q “ Z , (3.7)with H p Z ; Z q generated by r F s . Clearly, the action of g on H k p Z ; Z q is also trivial for all k .We conclude that g induces the identity morphisms on all homology and homotopy groups. Never-theless, we claim that g : Z Ñ Z is not homotopic to the identity. To see this we need to introducethe mapping torus T g : “ Z ˆ r , s{ tp z, q „ p g p z q , qu . We will show now that the Stiefel-Whitney class w p T g q doesn’t vanish; by contrast, the mappingtorus T id has vanishing w . Therefore, g is not homotopic to the identity. Lemma 2. w p T g q ‰ . Proof.
Remember that g | Q “ id . Therefore, there is a natural embedding Q ˆ S “ Q ˆ r , s{ tp q, q „ p g p q q , qu into T g .We refer to this embedded copy of Q ˆ S as R . It suffices to prove that the restriction of w p T g q on R does not vanish. Let us look at the restriction of the tangent bundle to T g on R . It splits as N R { T g ‘ T R , where N R { T g is the normal bundle to R in T g and T R is the tangent bundle to R . As R “ Q ˆ S , wehave T R “ T Q ‘ R , (3.8)where T Q is the pull-back of the tangent bundle to Q under the projection R Ñ Q , and R is a trivialline bundle.Now, we will prove the formula N R { T g “ N Q { Z b L S . (3.9) NFINITELY MANY NON-ISOTOPIC REAL SYMPLECTIC FORMS ON S ˆ S Here L S is the non-trivial (non-orientable) line bundle over S , and N Q { Z is the normal bundle to Q in Z . In the right-hand side of the formula both factors are interpreted as the pull-backs w.r.t. theprojections R Ñ Q and R Ñ S .Observe that, although g | Q “ id , the derivative d g : N Q { Z Ñ N Q { Z is not the identity butd g “ ´ id : N Q { Z Ñ N Q { Z . Now (3.9) follows from a general fact. Suppose we have a topological space X with a vector bundle I : E Ñ X , and that we form a twisted bundle I ö over X ˆ S with the total space E ˆ r , s{ tp e , x, q „ p´ e , x, qu , where a triple p e , x, q consists of x P X ˆ t u and e P I ´ p x q , and likewise for p e , x, q . Then it is easyto see that I ö “ I b L S , and as N R { T g “ N ö Q { Z , formula (3.9) follows.Now, we will establish an isomorphism N Q { Z – T Q . (3.10)This is again a general fact. Suppose we are given a manifold Y and a free involution τ : Y Ñ Y . Letus consider the diagonal ∆ “ tp y , y q P Y ˆ Y | y “ y u . Set Z : “ Y ˆ Y { „ , Q : “ ∆ { „ , where p y , y q „ p τ p y q , τ p y qq .There is a canonical way (there are two, choose either) to identify T ∆ with the normal bundle to ∆ in Y ˆ Y . This identification is equivariant w.r.t. the involution τ ˆ τ : Y ˆ Y Ñ Y ˆ Y , and hence givesrise to an identification between T Q and N Q { Z .Combining (3.8) with (3.9) and (3.10) yields N R { T g ‘ T R – T Q ‘ R ‘ p T Q b L S q . (3.11)We wish to compute the class w of this bundle. To this end, we apply the Whitney sum formula toobtain w p T Q ‘ p T Q b L S qq “ w p Q q w p T Q b L S q ` w p Q q w p T Q b L S q , (3.12)where w i p Q q “ w i p T Q q are the Stiefel-Whitney classes Q (more precisely, their pull-backs to R underthe projection R Ñ Q ), and w i p T Q b L S q are the classes of the bundle T Q twisted by L S . Incomputing w we have ignored the R component of (3.11) because characteristic classes are invariantunder taking the sum with a trivial bundle.In general, if L is a line bundle and E is an arbitrary vector bundle, then there are closed expressionsfor both the first and the top classes of E b I in terms of w i p E q and w i p L q . They are as follows: w n p E b L q “ w n p E q ` w n ´ p E q w p L q ` w n ´ p E q w p L q ` . . . , w p E b L q “ w p E q ` n w p L q , (3.13)where n “ rk E . The proof of both formulas is a simple application of the splitting principle and isomitted. Since rk T Q “ , we get w p T Q b L q “ w p Q q ` w p Q q w p L S q , w p T Q b L q “ w p Q q . (3.14)Since dim Q “ , we have w p Q q w p Q q “ . Substituting (3.14) into (3.12) we obtain w p Q q w p L S q .Recall that Q – RP and R – Q ˆ S . The class w p RP q generates H p RP ; Z q , while the class w p L S q generates H p S ; Z q . Clearly, their product does not vanish. (cid:3) Lemma 3. w p T id q “ . GLEB SMIRNOV
Proof.
Since T id – Z ˆ S , the Whitney formula yields w p T id q “ w p Z q . One of the definitions of w says w p Z q ¨ r Z s “ χ p Z q mod . Since there is a double covering S ˆ S Ñ Z , it follows that χ p Z q “ χ p S ˆ S q “ . (cid:3) Remark.
Suppose that X is a simply-connected -manifold. Then a classical result of Quinn [7]states that a homeomorphsim f : X Ñ X is homotopic to the identity iff it induces the identitymorphism on homology. The theorem’s generalization to the non-simply-connected case is unknown,not even for the simplest case of the group Z . Suppose that Z is a closed -manifold with π p Z q – Z .Of course, for a homeomorphism g : Z Ñ Z to be homotopic to the identity, it is first necessary tohave g acting identically on all π k and H k . However, this is not sufficient. For example, a simpleobstruction is that any self-homeomorphism of Z that is homotopic to the identity must have a lift (tothe universal cover) that is also homotopic to the identity, but even that in general is not sufficient,as the example above shows.Thus, although f is isotopic to the identity as a diffeomorphism of X , it is not isotopic to the identitywithin D iff p X , σ q . Lemma 4.
Suppose that h : p X , σ, ω q Ñ p X , σ, ω q is a symplectomorphism such that h ˝ σ “ σ ˝ h andsuch that h ˚ “ id : H p X ; Z q Ñ H p X ; Z q . Then h is isotopic to the identity within D iff p X , σ q .Proof. Let J be the underlying complex structure of the K¨ahler surface X . Since h is a symplec-tomorphism and J P R J ω , it follows that h ˚ J P R J ω . Since R J ω is connected, there is a path J p t q P R J ω , t P r , s such that J p q “ J and J p q “ h ˚ J . Recall that every structure J p t q P R J ω gives rise to a pair p F tA , F tB q of transversal fibrations of X into J p t q -holomorphic spheres. Since σ isanti-holomorphic for each J p t q , these fibrations are invariant w.r.t. σ in the following sense: if C is afiber of F tA , then so is σ p C q , and likewise for F tB . Following [1, 2.4.A ], one constructs a path of dif-feomorphisms α p t q P D iff p X , σ q with α p q “ id such that α p t q sends p F A , F B q to p F tA , F tB q . Composing h with α p t q , we can assume henceforth that h preserves p F A , F B q . This is a very restrictive conditionimplying that h is componentwise in the sense that h “ h ˆ h : S ˆ S Ñ S ˆ S , where both diffeomorphisms h i : S Ñ S must be orientation-preserving and must also commute withthe antipodal map τ : S Ñ S , τ p x q “ ´ x . Here the condition h ˚ “ id has been used twice, onceto ensure h does not interchange the fibrations F A and F B , and once to ensure that h preserves thenatural complex orientation of their fibers. What is left is to show that the group D iff ` p S , τ q ˆ D iff ` p S , τ q , (3.15)is connected. Here D iff ` p S , τ q stands for the subgroup of those diffeomorphisms of S which areorientation-preserving and τ -equivariant.A more general statement is that D iff ` p S , τ q is homotopy equivalent to SO p q . Note that everyelement of D iff p S , τ q induces a self-diffeomorphism of RP – S { τ , and we have a surjective homo-morphism D iff p S , τ q Ñ D iff p RP q , (3.16) NFINITELY MANY NON-ISOTOPIC REAL SYMPLECTIC FORMS ON S ˆ S which is not injective, since id : RP Ñ RP has exactly two lifts, the trivial lift id and the non-triviallift τ . However, if we replace D iff p S , τ q by D iff ` p S , τ q , then (3.16) becomes an isomorphism. One canprove that D iff p RP q is homotopy equivalent to SO p q by following the steps of the proof of Smale’stheorem (see [8]), as in [2, Appendix A]. A shorter proof is as follows. It is a classical result thatthe -sphere S admits a unique complex structure. That is, the group of diffeomorphisms of S actstransitively on the space of complex structures on S . From here, it is easy to show that D iff ` p S , τ q acts transitively on the collection of τ -anti-invariant complex structures on S , and that we have afibration Aut p CP , τ q Ñ D iff ` p S , τ q Ñ J τ , where Aut p CP , τ q is the subgroup of those biholomorphisms of CP which are τ -equivariant, and J τ is a connected component (there are two of them!) of the space of τ -anti-invariant complex structureson S . The space J τ is contractible, while Aut p CP , τ q , being a Lie group, contracts on its maximalcompact subgroup, which is SO p q . Hence, the group D iff ` p S , τ q itself contracts onto SO p q . (cid:3) We are now in a position to show ω and f ˚ ω are at different components of R Ω K . Suppose they arenot. Then we could join them with a path of monotone anti-invariant forms. By Moser’s trick, thatwould imply that f is isotopic to a symplectomorphism within D iff p X , σ q , and that, by the previouslemma, would imply that f is isotopic to the identity within D iff p X , σ q . This is a contradiction. References. [1] M. Gromov. Pseudoholomorphic curves in symplectic manifolds.
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