View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Pergamon TopologyVol. 34, No. I, Pp. 155-176,1995 Copyright0 1QQ4Elsevie? Scicnac Ltd Printedin Gnat Britain.All ri&ts rtscrval 0@4%938395 s9.50 + 0.00 0040-9383(94)EOO15-C SYMPLECTIC FIXED POINTS, THE CALABI INVARIANT AND NOVIKOV HOMOLOGY LB HONG VAN and KAORU ONO? (~~ee~u~d 31 March 1993) 1. INTRODUCTION A SYMPLECTIC structure o on a manifold M provides a one-to-one correspondence between closed l-forms and infinitesimal automorphisms of (~,~), i.e. vector fields X satisfying _Yxw = 0. An infinitesimal automorphism X is called a Hamiltonian vector field, if it corresponds to an exact l-form, A symplectomorphism rp on M is called exact, if it is the time-one map of a time-dependent Hamiltonian vector field. In fact, one can find a periodic Hamiltonian function such that o, is the time-one map of the Hamiltonian system. For each symplectomorphism cp isotopic to the identity through symplectomo~hisms, one can assign a “cohomology class” Cal(rp), which is called the Calabi invariant of cp. Banyaga [3] showed that rp is exact if and only if Cal(p) = 0. The Arnold conjecture states that the number of fixed points of an exact symplectomorphism on a compact symplectic manifold can be estimated below by the sum of the Betti numbers of M provided that all the fixed points are non-degenerate. Arnol’d came to this conjecture by analysing the case that 9 is close to the identity (see Cl]). If cp is the time-one map of a time-independent Hamiltonian vector field corresponding to a Morse function f which is C2-small, the fixed points coincides with the critical points of fand the validity of the conjecture follows directly from Morse theory. There are many partial results to the Arnol’d conjecture. Great progress was made by Floer, who combined the variational approach (see [4]) and Gromov’s theory of pseudo- holomorphic curves to prove the Arnol’d conjecture for monotone symplectic manifolds [S]. He developed an analogue of Morse theory for the action functional on the loop space and this led to the notion of Floer homology. The Arnold conjecture is derived from the fact that the Floer homology group is isomorphic to the ordinary homology group of M. Recently, Hofer and Salamon ES] defined Floer homology groups for a wider class of symplectic manifolds (which are called weakly monotone symplectic manifolds). An almost complex structure J on M is calibrated by o, if (5>9) = w(& JV) (1.1) defines a Riemannian metric on M. Such a J is unique up to homotopy and we denote by c1 = cl(M)the first Chern class of the almost complex manifold (M,f). A symplectic manifold (M,co) is called monotone, if c1 is a positive multiple of o on n2(M).The condition of weak monotonicity [S] implies the non-existence of J-holomorphic spheres with negative t On leave from Department of Math~nti~, Faculty of Science, Ochano~u University, Otsuka, Tokyo 112, Japan. 15.5 156 L2 HBng Van and K. Ono Chern number for a generic J. Hofer and Salamon computed the Floer homology and verified the Arnol’d conjecture in the case that (M, u) is either monotone, or cl (Q(M)) = 0, or the minimal Chern number is at least half of the dimension of M. Later, the second author defined the “modified Fleer homology groups” and verified the conjecture for weakly monotone symplectic manifolds [lo]. However we do not know whether the Floer homology groups defined by Hofer and Salamon and the modified Floer homology groups coincide or not. In this note, we consider an analogue of the Arnol’d conjecture for non-exact symplec- tomorphisms isotopic to the identity through symplectomorphisms. In the case of non- exact symplectomorphisms, the fixed point set may be empty. For example, an irrational rotation on an even-dimensional torus with the standard symplectic structure preserves the symplectic form, and has empty fixed point set. For this reason, we have to consider Novikov homology instead of ordinary homology. The aim of this note is to show the following theorem. MAINTHEOREM. Let (M, w) be a closed symplectic manifold of dimension 2n which satis$es the following condition Cl In*(M) = ~%2(M), 1 z 0 and if A < 0, the minimal Chern number N satisfies N > n - 3. Suppose rp is a symplectomor- phism on (M, w) which is isotopic to the identity through symplectomorphisms. If all thejxed points of cp are non-degenerate, then the number ofJixed points of cp is at least the sum of the Betti numbers of the Novikov homology over Zz associated to the Calabi invariant of cp. It is well-known that Novikov homology groups are isomorphic for almost all cohomol- ogy class in H’(M;R) and the rank of these groups is minimal in; the class of Novikov homology groups associated to all the cohomology class in H’(M;R) (see Appendix C). Hence we get an estimate of the number of fixed points in terms of Novikov homology for generic l-forms. To prove the Main Theorem, we reduce the problem to one concerning the l-periodic solutions of a periodic Hamiltonian system and define Floer homology in this setting. The argument in [S, 81 shows that the Floer homology groups are isomorphic under the deformations preserving the Calabi invariant. However, in order to compute the Floer homology, we also have to consider deformations which change the Calabi invariant. To apply the weak compactness argument, it is necessary to estimate of the energy functional for solutions of the “chain homomorphism” equation. This is done for specific deformations with the help of a variant of the Palais-Smale condition (Section 5). This note also contains three Appendices. The first one concerns the classification of loops in a symplectic manifold under symplectomorphisms. We prove that two embedded contractible loops are congruent under a time-one map of a time-dependent Hamiltonian flow if and only if their Poincare integral invariants coincide, The second contains a proof of a fact which is needed in the computation of the Floer homology groups (see also [14]). The third is a note on Novikov homology theory. Since it seems difficult to find a reference containing proofs, we give proofs for the sake of completeness. 2. THE CALABI INVARIANT AND A VARIATIONAL APPROACH Given a symplectic form w on M, there is an isomorphism 0 from the space of vector fields to the space of differential forms on M: qV)(w) = -o(V, w). (2.1) SYMPLECTIC FIXED POINTS 157 Let g1 be an element of the identity component BiflU M of the symplectomorphism group and {gt} a path connecting the identity element and gl. We define a vector field B,(gJ by a w?t)(4 = (-g gt> (gt-- ‘4 * Clearly, the cP(D,(gJ) are closed differential forms. Recall that g1 is said to be an exact symplectomorphism if all the Q(Q(gJ) are exact l-forms. TheJlux homomorphism @ from the universal cover -to H’(M,R) is defined as follows (see [3]): This homomorphism was first considereeabi and we call the image Q(J) the Calabi invariant Cal(g) of an element fi = gi E Diflz M. Passing to the group m the Calabi invariant of a symplectomorphism on M is an element of the quotient H’(M, R)/T, where F is the image of the subgroup q(Diflz M) which is identified with the kernel of the projection from -to Diffi M) under the homomorphism a. It is known ([3], see also Lemma 2.1) that F is a discrete subgroup in H’(M,R) if and only if the subgroup of exact symplectomorphisms is closed in Difli M. Kiihlerian manifolds, or more generally, any symplectic manifold M such that the multiplication by w”- ’ induces an isomorphism from H’(M,R) to H2”-‘(M,R), are such examples [3]. DEFORMATIONLEMMA 2.1. Let [g] EH’(M, R) be the Culabi invariant of an element gI. Then there exists a smooth path { gt} in Di#E M, joining the identity element Id and gl, and u periodic Hamiltonian Ht on M such that @(D,(g,)) = 8 + dHi,for all t. Proof: First, we show that we can choose path gt connecting the identity element and g1 such that each 8, = @(D,(g,)) has cohomology class [0] and & = el. Put 1 V(x) = D&)(x) dt . s 0 Let G,(x) be the one-parameter subgroup of symplectomorphisms generated by the vector field V(x): G, = exp tV. Then the Calabi invariant for the path G_,ogt is zero. It is known that there exists a smooth path pt in the subgroup of exact symplectomorphisms, Diff_ M, such that PO = Id and pl = G-,*gl (see [3]). By reparametrizing the parameter t, we may assume that pt is constant around 0 and 1. Now let us consider the path g; = G, *pr, so we have gb = Id and g; = gl. We obtain @(D,(G,“~J) = Q((G,) *(D,(P,)) + Q(V) . (2.2) From (2.2) we obtain that the form cD(D,(g;)) = 8, lies in the same cohomology class [e] for all t. Let pr be as above. Then Q(D,(p,)) is the differential dH, of a smooth function H, on M. Now, the periodicity condition e. = fI1 is equivalent to the following: dH,, + O(V) = G;(dH,) + Q(V).