Floer-Novikov cohomology and symplectic fixed points, revisited

This note is mostly an exposition of a few versions of FloerNovikov cohomology with a few new observations. For example, we state a lower bound for the number of symplectic fixed points of a non-degenerate symplectomorphism, which is symplectomorphic isotopic to the identity, on a compact symplectic manifold, more precisely than previous statements in [9, 12]. Анотація. В огляді наведено кілька версій теорії когомологій ФлоєраНовікова та доведено кілька нових фактів. Зокрема, отримано (більш точну ніж в [9, 12]) нижню межу для числа симплектичних нерухомих точок невиродженого симплектоморфізма, який є симплектоморфно ізотопний до тотожного відображення на компактному симплектичному многовиді.


INTRODUCTION
Fixed points and periodic points are the simplest objects in dynamical systems. For time periodic flows, they are identified with periodic orbits. For a time dependent Hamiltonian system on a closed symplectic manifold (M, ω), Arnold conjectured that the number of fixed points of the time-one map is at least the minimal number of critical points of smooth functions on the manifold M . In case all the fixed points are non-degenerate, he also conjectured that the number of fixed points is at least the minimal number of critical points of Morse functions on M . Motivated by these conjectures, Floer developed what is now called Floer theory [2]. In the non-degenerate case, it is now known that the number of fixed points is at least the sum of Betti numbers of M , see [3,5,7,10,11].
In [9] we considered a larger class of time dependent locally Hamiltonian systems on closed symplectic manifolds (M, ω). Recall that a vector field V on (M, ω) is said to be locally Hamiltonian, if i(V )ω is a closed 1-form on M 2n . Here i(X)ω is the interior product by the vector field X to the symplectic form ω. Note that the interior product i(¨)ω : X(M ) Ñ Ω 1 (M ) induces an isomorphism i(‚) : T M Ñ T˚M . Let tX t u tPR/Z be a 1-periodic family of locally symplectic vector fields and integrating tX t u 0ďtď1 , we obtain a symplectic isotopy tφ t u 0ďtď1 with φ 0 = id. Write η t = i(X t )ω and define the flux Flux(tφ t u 0ďtď1 ) of tφ t u 0ďtď1 by the de Rham cohomology class 1 of ş 1 0 η t dt. In [9], we extended the construction of Floer homology for non-degenerate Hamiltonian systems to Floer-Novikov homology for non-degenerate locally Hamiltonian systems. Since the former is related to Morse homology of the manifold, the latter must be related to Novikov homology for the flux instead of Morse homology. As a result, we obtained the following: We assumed that N ą n´3 when λ ă 0 due to the argument for transversality at that time. Under the same assumption, using the argument on orientation of the moduli space of connecting orbits, e.g. [3,5], see also [4], we can use the dimension of Novikov homology with F p = Z/pZcoefficients 2 for any prime number p. Using virtual technique in [5], we can show Suppose that all fixed points of the time-one map φ 1 of a symplectic isotopy are non-degenerate. Then the number of fixed points of φ 1 is bounded below by the sum of the Novikov-Betti number 3

of [η], the dimension of Novikov cohomology HN˚([η]) as a vector space over the field Λ Q
[η] , which is the Novikov field associated with the flux [η] of tφ t u.
Here we set a restriction to the class of positively or negatively monotone symplectic manifolds because of the difficulty in computing Floer-Novikov cohomology, especially, the coefficient ring of Floer-Novikov cohomology depends on Flux(tφ t u 0ďtď1 ) P H 1 (M 2n , R), see Remark 2.13 below for detailed discussion.
In [12], the second author constructed Floer-Novikov chain complexes defined over a variant of Novikov ring which is strictly contained in the one defined in [9] so that Floer-Novikov cohomology with fluxes close to each other can be compared. The following is a result in [12]. Here min-nov p (M ) is the minimum of the rank of Novikov cohomology over all de Rham cohomology classes of degree 1, see [12]. For We call min-nov p (M ) the p-th minimal Novikov number. In fact, we have a better statement. Theorem 1.4. Let tφ t u 0ďtď1 be a smooth family of symplectic diffeomorphisms on (M, ω) with φ 0 = id. If all the fixed points of the time-one map φ = φ 1 are nondegenerate, then the number of fixed points of φ is bounded from below by the sum of the Novikov-Betti numbers of the flux [η] of tφ t u 0ďtď1 . Remark 1.5. Theorems 1.3 and 1.4 hold for any closed symplectic manifold. For weakly monotone closed symplectic manifolds, the conclusion holds using the dimension of Novikov cohomology with coefficient in any field. For general closed symplectic manifolds, the same follows using the argument in [6]. Since the details are not yet written, we state the results with Q-coefficients for general symplectic manifolds.
There are several variants of Floer-Novikov complexes [9,12,13]. We explain their difference although there are certainly similar properties enjoyed by them, see e.g. Proposition 3.10, and Remarks 3.11 and 3.12. We also explain the construction of Floer-Novikov complex for any component of the loop space of M . For the first step in the argument of the comparison of Floer-Novikov complexes with sufficiently close, but different, fluxes, we formulate Lemma 3.5, which is valid regardless of contractibility of periodic orbits. In [12], we used the maximal abelian covering of M in the construction of variants of Floer-Novikov complex to prove Theorem 1.3. Using a suitable intermediate abelian covering space of M , we improve it to Theorem 1.4.
The proof of Arnold's conjecture [5,10] implies the existence of contractible 1-periodic orbits for any periodic Hamiltonian function. For a loop tφ H t u of Hamiltonian diffeomorphisms based at the identity, the free homotopy class of a loop t P [0, 1] Þ Ñ φ H t (p) does not depend on p P M , hence, contractible. In other words, the homomorphism by evaluating at p 0 P M is trivial. Although it is not the case for We observe a similar behavior for diffeomorphisms under the assumption that the Euler characteristic of M is not zero. It may be of independent interest. Proposotion 1.7. Let tφ t u 1 t=0 be an isotopy with φ 0 = id on a closed manifold M . If the Euler characteristic χ(M ) of M is non-zero, there is a fixed point p of φ 1 such that the loop t Þ Ñ φ t is null-homotopic. In particular, the homomorphism ev x˚: π 1 (Diff(M ), id) Ñ π 1 (M, x) is trivial.
As a corollary, we see that the flux group of a closed symplectically aspherical manifold is zero provided its Euler characteristic is non-zero (Corollary 4.2).
Our paper is organized as follows. In Section 2, we give a unified exposition of Floer-Novikov cochain complexes and their cohomology. We also collect some fundamental facts on Novikov rings in Section §2.2 for reader's convenience. For example, the Novikov ring Λ ω,η (R) which is the ring of coefficients of Novikov-Floer chain complexes is an integral domain if R is an  integral domain R (Proposition 2.9). Then the rank of the Floer-Novikov cohomology groups is defined as the dimension after tensoring with the field of fractions of the Novikov ring 4 . In Section 3, we recall the construction of the Floer-Novikov cochain complex and their cohomology over smaller Novikov subrings and give a proof of Theorem 1.3. Namely, we compare the ranks of the cohomology of two "close" Floer-Novikov cochain complexes through "smaller Floer-Novikov cochain complexes over smaller subrings of Λ ω,η " as in [12], the argument of which uses results and ideas in our previous work [9]. In Section 4, we prove Proposition 1.6 and Proposition 1.7. In Appendix, a correction of [12, Lemma 5.1] is included.
In this note, M denotes a closed connected symplectic manifold. We shall use cohomological convention, i.e., Floer-Novikov cohomology as in [12] although in [9] we used homological convention.
Acknowledgement. The second author is grateful for the organizers, in particular, Professor Sergiy Maksymenko, of "Morse theory and its Applications dedicated to the memory and 70th anniversary of Volodymyr Sharko" for a kind invitation to the conference. The authors are also grateful for the anonymous referee for careful reading.

REVIEW ON FLOER-NOVIKOV COHOMOLOGY
We review the construction of Floer-Novikov cohomology for locally Hamiltonian systems, cf. [9,12,13]. Let tη t u be a one-parameter family of closed 1-forms on M . Denote by X t the symplectic vector field defined by i(X t )ω = η t and by tφ t u the corresponding symplectic isotopy with φ 0 = id. Let φ be the time-one map φ 1 . Without changing the homotopy class of symplectic isotopies joining the identity and φ, we may assume that each η t represents the same de Rham cohomology class, see [9, Lemma 2.1]. We may also assume that η t is 1-periodic in t. Then we identify the set P(ω, tX t u) of one-periodic solutions of the equation with the zero-set of the following closed 1-form α tφ t u , in a formal sense, on the free loop space LM over M : Denote by [η] the flux of tφ t u. What we call Floer-Novikov theory is a semi-infinite analogue of Morse-Novikov theory on the free loop space LM .
Thus we take certain covering spaces of (a path connected component of) LM on which the lift of the closed 1-form α tη t u is exact. (There are some choices of covering spaces as we see in §2.1.) §2.1. Floer-Novikov cochain complexes. In this section, we introduce a few versions of Floer-Novikov complex. For a P π 0 (LM ) we denote by L a M the corresponding connected component of LM . Let γ a 0 P L a M . Given de Rham cohomology classes [κ] and [θ] of degree 2, and 1, respectively, we define the homomorphisms : π 1 (L a M, γ a 0 ) Ñ R as follows. Let tγ τ u be a loop in L a M with the base point γ a 0 and C(tγ τ u) the "torus" in M swept by tγ τ u. We set where I [θ] is the pairing between [θ] P H 1 (M ; R) and loops in M and e is the evaluation at t = 0, i.e., γ P LM Ñ γ(0) P M .
We consider covering spaces of LM such that the pull-back of α tφ t u becomes exact. Pick a closed 1-form η representing the flux of tφ t u. The smallest one is the covering space of L a M , which is associated with . Then its covering transformation group is isomorphic to Denote by N a the minimal non-negative integer in Im I (2),a c 1 Ă Z, (the number N = N 0 is ofter called the minimal Chern number.) To a zero of α tϕ t u , i.e., a 1-periodic orbit γ, we assign In fact, this Z/2Z-grading can be lifted to a Z/2N a Z-grading 5 . When N a ‰ 0, we can lift the Z/2Z-grading to a Z-grading after taking the covering space assocaited with We recall a description of this covering space. For γ P L a M , there is a cylinder v : Here v#(´v 1 ) is the "torus" obtained by gluing v and v 1 with orientation reversed along the boundaries γ a Y γ. The space of equivalence classes of (γ, v) becomes a covering space r L a M of L a M with the projection Then the covering space r L a M of L a M is associated with We choose a primitive function of the closed 1-form α tφ t u to be the action functional A a tφ t u : r where tf t u is a 1-periodic family of smooth functions on M such that There is no canonical grading for [γ, v], when a ‰ 0. We pick and fix a 1-periodic orbit γ˚, a cylinder w joining γ and γ˚. Then the relative index µ(γ, γ˚; w) along w is the difference of the Conley-Zehnder indices of γ and γ˚with respect to a trivialization of w˚T M as a symplectic vector bundle. Set the grading of [γ˚, v 0 ] to be k P Z such that k is even (resp. odd) if is positive (resp. negative). Then the grading of [γ, v 0 7w] is defined as µ(γ, γ˚; w) + k. By the third condition in the definition of the equivalence relation ", we obtain a well-defined grading for each critical point We define the Floer-Novikov cochain complex for tφ t u as follows. From now on, we assume that all 1-periodic orbits in L a M are non-degenerate. Let R be a ground ring. We set The grading is given by g Þ Ñ I (2),a c 1 (g). The graded module CFN˚(tφ t u, R) a is a finitely generated free module over Λ a ω,η (R), [13], see also [9,12]. The coboundary operator δ is defined by counting isolated solutions 6 u : RˆS 1 Ñ M of the following equation: for some 1-periodic solutions γ˘. Here J t , t P R/Z, is a 1-periodic family of almost complex structures compatible with ω. We call a solution u a connecting orbit joining [γ´, v´] and This is Floer-Novikov complex used in [13], which is suitable for the proof of the flux conjecture. Let π : Ă M Ñ M be a covering space of M on which π˚η is an exact 1-form, i.e., [η] P ker ( π˚: ) .
Denote by Γ( Ă M ) its covering transformation group. Here we do not assume that Ă M is the minimal covering space enjoying this property 7 and introduce Floer-Novikov complex for symplectic isotopies with flux [η] associated with the covering Ă M . Let a P π 1 (LM ). Pick p 0 P Ă M such that π(p 0 ) = γ a 0 (0). Consider triples (γ, v, p), where p P Ă M such that π(p) = γ(0) and the path ℓ v lifts to a path from p 0 to p in Ă M . Note that, for a generic we have ker I In other words, it is the covering space of L a M associated with ker I (2),a [ω] X ker I (2),a We set ) .
Then, for [η] P ker(H 1 (M ; R) Ñ H 1 ( Ă M ; R), we define the corresponding Novikov ring by This Floer-Novikov complex for tφ t u with respect to Ă M Ñ M was introduced in [12], when L a M is the space of contractible loops. i.e., the case that ker I [η] = π˚(π 1 (M η , p 0 )), we set Then we obtain the Floer-Novikov cochain complexes 8 Their cohomology groups are denoted by HFN˚(tφ t u) a , Ć HFN˚(tφ t u) a and HFN˚(tφ t u; Ă M ) a , respectively.
Hence, the covering space r L 0 M η Ñ L 0 M is the one associated with ker I (2),0 [ω] X ker I   Ă Im I (1) [η] are independent over Z. Thus we find that ker(I Hence, if the flux of tφ t u enjoys this property, the complexes CFN˚(tφ t u; M η ) a and CFN˚(tφ t u) a are the same. §2.2. Novikov rings Λ a ω,η (R), r Λ a ω,η (R). In this subsection we assume that R is a commutative unital ring. Given a group Γ and a homomorphism ϕ : Γ Ñ R, we denote by R((Γ, ϕ)) the upward completion of the group ring R[Γ] with respect to the weight homomorphism ϕ. More precisely, we define R((Γ, ϕ)) := ! ÿ λ g¨g , g P Γ, λ g P R | for all c P R there is only finite number of g such that λ g ‰ 0 and ϕ(g) ă c ) .
If Γ is a subgroup of R and ι : Γ Ñ R is the natural embedding, then we abbreviate R((Γ, ι)) to R((Γ)). We recall the following Proposition 2.4 attributed to J.-C. Sikorav.

Remark 2.5.
This proposition is shown by looking at the leading terms, i.e., the lowest order terms with respect to the weight homomorphism ϕ. It also implies that if ϕ is injective and R is an integral domain, so is R ((Γ, ϕ)). (2) Let η be a closed 1-form on M and I η : H 1 (M, Z) Ñ R the period homomorphism. Assume that R be an integral domain. The Novikov ring Λ [η] (R) = R((Γ η , I η )) is the Novikov ring associated to the cohomology class [η] P H 1 (M, R). Novikov cohain complex CN˚(η) is a complex over Λ [η] and Novikov cohomology HN˚(η) is a module over Λ [η] 9 .
In the remainder of this subsection, we assume further that R is an integral domain. Proof. Since Γ = ϕ(Γ) is a torsion free finitely generated abelian group, hence a free abelian group, we take a splitting of Denote by ϕ : Γ Ñ R the homomorphism induced from ϕ. Note that ϕ is injective. Proposotion 2.8. Assume that Γ is a finitely generated abelian group and ϕ : Γ Ñ R is a homomorphism. Then we have a ring isomorphism R((Γ, ϕ)) = R[ker ϕ](((Γ), ϕ)).

Proposotion 2.9. Assume that Γ is a finitely generated free abelian group and R is an integral domain. Then the ring R((Γ, ϕ)) is an integral domain.
Proof. Let k be the rank of ker ϕ as a finitely generated free abelian group.  Here (M, ω) is said to be semi-positive, if any A P π 2 (M ) with 3´n ď xc 1 (M ), Ay ă 0 satisfies x[ω], Ay ď 0. For the proof, we use a τ -dependent analogue of (2.7). Let tX t u and tX 1 t u be 1-periodic families of symplectic vector fields generating tφ t u and tφ 1 t u, respectively. Take a two parameter family tX τ,t u of symplectic vector fields of the form X τ,t = (1´β(τ ))X t + β(τ )X 1 t . Here β is a smooth function such that, for some d ą 0, β(τ ) = 0 for τ ă´d, and β(τ ) = 1 for τ ą d. Roughly speaking, we count isolated solutions of the following equation to construct a chain homomorphism CFN˚(tφ t u) Ñ CFN˚(tφ 1 t u): with a similar asymptotic condition as (2.8). Note that the cohomology class [i(X τ,t )ω] is independent of τ and t. This fact guarantees that the energy estimate for solutions of (2.12) holds as in Floer theory for Hamiltonian systems, see the proof of [9,Theorem 4.3]. When c P H 1 (M ; R) is sufficiently close to 0, we can pick a C 1 -small closed 1-form η representing the class c so that all 1-periodic orbits of the symplectic vector field X defined by i(X)ω = η are null-homotopic. Moreover, Floer-Novikov cochain complex can be described by Novikov complex. For (CFN(tϕ t u : Ă M ), δ), we take Ă M = M η , which is used in the construction of Novikov cochain complex of η. As a consequence, we obtain the following: Here HN˚(η; R) is Novikov cohomology for η with coefficients in R, and Λ η (R) is the Novikov ring over R associated to [η]. For component a ‰ 0, Floer-Novikov cohomologies for tφ t u with a sufficiently small flux vanishes cf. Remark 3.11. Remark 2.13. There are two issues in comparing Floer-Novikov homologies associated to symplectic isotopies whose flux are not the same. The first issue is that Novikov rings for them are not the same when c ‰ 0. The second issue is that the derivation of an energy inequality to guarantee the weak-compactness of the moduli space used for the construction of chain homomorphisms, chain homotopies under Hamiltonian deformations does not work for deformations with varying flux. The strategy in [9] is as follows. To deal with the first issue, we restrict the class of˘monotone symplectic manifolds. Then the degree 0-part of the Novikov ring r Λ ω,η is identified with Λ [η] , which depends on the positively proportional class of [η]. In general case, we compare Floer-Novikov cohomology with varying flux through the comparison of their smaller Floer-Novikov cohomology defined over smaller subring as in [12]. To deal with the second issue, we use a special deformation of tφ t u changing the flux based on Lemma 3.7 below, see Section 5 (see, the paragraph after Lemma 5.2) in [9]. This trick and related refining techniques, then work for general case. We shall discuss these technique in more details in the next section.

THE RANK OF THE FLOER-NOVIKOV COHOMOLOGY
In this section we first show that the ranks of Floer-Novikov cohomology associated to a symplectic isotopy tφ t u is well-defined. Then we shall prove the following Theorem 3.1. Let tφ t u 0ďtď1 be a smooth family of symplectic diffeomorphisms on (M, ω) such that φ 0 = id and all the fixed points of φ 1 are non-degenerate. Then we have As in Remark 1.5, the conclusion of Theorem 3.1 also holds for any field instead of Q, if (M, ω) is weakly monotone.
Finally we derive Theorem 1.4 from Theorem 3.1 using a known relation between the rank of the Floer-Novikov cohomology and with the number of fixed points of φ. §3.1. The ranks of (Floer-)Novikov cohomologies. Recall that the rank of a module L over an integral domain R is defined to be the dimension of the vector space F (R) b R L over the field of fractions F (R) of R, cf. [12, page 560]. Now assume that R is an integral domain and C is a (co)chain complex with coefficients in R such that C is a finitely generated free module over R. In what follows, C is a cochain complex over R. Clearly the cohomology H˚(C) is a module over R. Then we define the rank of H˚(C) as follows Proof. (i) Since R Ă F we have F (R(Γ, ϕ)) Ă F (F(Γ, ϕ)). Next we note that F (R(Γ, ϕ)) Ą F (F(Γ, ϕ)) since F (R(Γ, ϕ)) Ą F. This proves the first assertion of Proposition 3.2.
(ii) The second assertion of Proposition 3.2 follows from the universal coefficient theorem. Since the field of fractions F (R) is a flat R-module, we obtain rank H˚(C) = dim F H˚(C b R F) ď rankC.  Let η t be a 1-periodic family of closed 1-forms in the same cohomology class. Denote by ψ tη t u t the symplectic flow on M that is generated by the time-depending symplectic vector field X t defined by i(X t ) = η t with ψ tηu 0 = id. The definition of a smaller variant Floer-Novikov cochain complex in [12] extends to any connected component of LM as follows. .
The finiteness condition for [ζ] P U is that the set is finite for any c P R.
In [12], we proved that, if [ζ] is sufficiently small, we can arrange a representative ζ such that A a tψ tζu t˝φ t u increases along solutions of (2.7). Namely, Proposotion 3.3. (cf. [12, page 558]) Let δ be the boundary operator defined in (2.9). Then, for a sufficiently small U , the boundary homomorphism δ preserves the finiteness condition for each [ζ] P U . In particular,

Remark 3.4.
In [12] the second author stated the proposition in the case a = 0 P π 1 (LM ) and R = Q, but the same argument works for any connected component of LM . In [8] the first author considered another subring of Λ 0 ω,η containing Λ 0 ω,η,U for U containing the segment [τ 1´1 , τ 2´1 ] with coefficients in R = Q as follows Then she defined a subcomplex CFN (tφ t u, R), which is nothing but . She called this chain complex reduced Floer-Novikov chain complex.
In what follows we shall compare smaller variant of Floer-Novikov chain complexes associated to symplectic isotopies of different fluxes. Firstly, we prepare the following lemma. Pick and fix a norm on the finite dimensional vector space H 1 (M ; R). Let tγ 1 , . . . , γ N u be the set of all 1-periodic orbits of tX t u and c P H 1 (M ; R). For each t, pick contractible neighborhoods, say a smooth family of embeddings of geodesic balls U ρ (γ j (t)) around γ j (t) with a sufficiently small radius ρ for some Riemannian metric on M . Then we have the following 10 Lemma 3.5. For any ϵ ą 0, there exists δ ą 0 with the following property 11 . Let }¨} H 1 be a norm on the real vector space H 1 (M ; R). If }c} H 1 ă δ, there exists a smooth 1-periodic family tθ t u of closed 1-forms on M such that [θ t ] = c, θ t vanishes on U ρ (γ j (t)), j = 1, . . . , N , for each t and the C 1norm of θ t is less than ϵ.
Let X θ t be the vector field defined by i(X θ t )ω = θ t and ψ tθ t u the symplectic flow generated by tX θ t u.
Then, in the same way as in Section 3.2 in [12], φ t and ψ tθ t u˝φ t have the same 1-periodic orbits and all of them are non-degenerate.
Take tθ t u representing c as in Lemma 3.5 and compare CFN˚(tφ t u; Ă M ) and CFN˚(tψ tθ t u˝φ t u; Ă M ) as in [12,Proposition 4.7]. Here we do not assume c is proportional to [η]. The following lemma in [9,12] was proved in the case of a = 0 (contractible loops), but the same argument works for any a. Roughly speaking, we proved that the functional A tφ t u increases along solutions of (2.12). The proof of [9, Lemma 5.4] yields the slightly stronger statement as follows.
Note that, for any ϵ ą 0, if we can take U , a sufficiently small neighborhood of the origin in V , the following condition (˚ϵ) holds: (˚ϵ) For any c P U , there exists tθ t u as in Lemma 3.5 and }θ t } C 1 ď ϵ.
Using Lemma 3.7 we obtained the following . For c P U , pick tθ t u as in (˚ϵ). Then z HFNŮ (tφ t u) is isomorphic to z HFNŮ1(tψ tθ t u˝φ t u) as Λ ω,η,U = Λ ω,η+c,U 1 -modules, where The ranks of HFN˚(tφ t u, Ă M ) a and HFN˚(tψ tθ t u˝φ t u; Ă M ) a are compared using the isomorphism in Proposition 3.9, see [12, page 560]. Namely, we have Remark 3.14. When Ă M is the covering space of M associated with ker η Ă π 1 (M ), HFN˚(tφ t u; Ă M ) is equal to Ć HFN˚(tφ t u).