aa r X i v : . [ m a t h . S G ] M a y K ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY
JUNGSOO KANG
Abstract.
Rabinowitz Floer homology has been investigated on submanifolds of contacttype. The contact condition, however, is quite restrictive. For example, a product of contacthypersurfaces is rarely of contact type. In this article, we study Rabinowitz Floer homologyfor product manifolds which are not necessarily of contact type. We show for a class ofproduct manifolds that there are infinitely many leafwise intersection points by proving theK¨unneth formula for Rabinowitz Floer homology. Introduction
Rabinowitz Floer homology has been extensively studied in recent years because of its in-terrelation with the leafwise intersection problem. However Rabinowitz Floer homology (tobe honest, the perturbed Rabinowitz action functional) has worked principally on a contactsubmanifold and little research has been conducted on a non-contact case. Our primaryobjective in this paper is to find leafwise intersection points and define Rabinowitz Floerhomology for this class of submanifolds which are not necessarily of contact type. In additionwe show for the class that there are infinitely many leafwise intersection points by provingthe K¨unneth formula for Rabinowitz Floer homology. For simplicity, throughout this paper,we use Z / Z -coefficient.We consider restricted contact hypersurfaces (Σ , λ ) resp. (Σ , λ ) in exact symplecticmanifolds ( M , ω = dλ ) resp. ( M , ω = dλ ). Moreover we assume that Σ resp. Σ bounds a compact region in M resp. M and that those M and M are convex at infinity;that is, they are symplectomorphic to the symplectization of a compact contact manifold atinfinity. Given F ∈ ( S × M ), F ∈ C ∞ ( S × M ), the operation (cid:0) F ⊕ F (cid:1) ( t, x, y ) = F ( t, x ) + F ( t, y ) , ( t, x, y ) ∈ S × M × M provides a time-dependent Hamiltonian function F ⊕ F ∈ C ∞ ( S × M × M ). We also intro-duce projection maps π : M × M → M and π : M × M → M ; then ( M × M , ω ⊕ ω )admits the symplectic structure ω ⊕ ω = π ∗ ω + π ∗ ω .On (Σ × Σ , M × M ), we define the perturbed Rabinowitz action functional A e H , e H F asin (2.1). Since Σ × Σ is a stable submanifold, we can define Floer homology of A e H , e H when F ≡ A e H , e H ) iscalled Rabinowitz Floer homology and denoted by RFH(Σ × Σ , M × M ), see Section 3. Bythe standard continuation method in Floer theory, HF( A e H , e H F ) and RFH(Σ × Σ , M × M )are isomorphic whenever HF( A e H , e H F ) is defined. Key words and phrases.
K¨unneth formula, Rabinowitz Floer homology, leafwise intersections.2000
Mathematics Subject Classification.
Theorem A.
The Floer homologies
RFH(Σ × Σ , M × M ) and HF( A e H , e H F ⊕ F ) are well-defined.Moreover, we have the following K¨unneth formula in Rabinowitz Floer homology: RFH n (Σ × Σ , M × M ) ∼ = n M p =0 RFH p (Σ , M ) ⊗ RFH n − p (Σ , M ) . Here, RFH p (Σ , M ) (resp. RFH n − p (Σ , M )) is the Rabinowitz Floer homology for therestricted contact hypersurface Σ in M (resp. Σ in M ), see [1] or Section 3. Remark 1.1.
In this paper, we unfortunately establish compactness of gradient flow linesof the Rabinowitz action functional only for perturbations of the form F = F ⊕ F . Thuswe cannot study the existence problem of leafwise intersection points for an arbitrary pertur-bation. However, if Σ × Σ has contact type in the sense of Bolle [10, 11] (see Section 4),the Floer homology HF( A e H , e H F ) is defined for all perturbations, see [24]. We note that, ingeneral, Σ × Σ is not of contact type in the sense of Bolle. For example, S × S is not acontact submanifold in R , see Remark 4.2. Question 1.2.
What perturbations have a leafwise intersection point on (Σ × Σ , M × M )? Remark 1.3.
Once one verifies compactness of gradient flow lines of the Rabinowitz actionfunctional for a given perturbation F , it guarantees the existence of leafwise intersectionpoints for that F by using the stretching the neck argument in [1]. In this paper, we are ableto compactify gradient flow lines of A e H , e H F ⊕ F , and thus it guarantees the existence of leafwiseintersection points of F ⊕ F ; but, this directly follows from the result in [1] that each F and F has a leafwise intersection point on Σ and Σ respectively. Definition 1.4.
The
Hamiltonian vector field X F on a symplectic manifold ( M, ω ) is definedexplicitly by i X F ω = dF for a Hamiltonian function F ∈ C ∞ ( S × M ), and we call itstime one flow φ F the Hamiltonian diffeomorphism . We denote by Ham c ( M, ω ) the group ofHamiltonian diffeomorphisms generated by compactly supported Hamiltonian function. Thisgroup has a well-known norm introduced by Hofer (see Definition 2.2).
Definition 1.5.
We denote by ℘ (Σ , λ ) > minimal period of closed Reeb orbits of(Σ , λ ) which are contractible in M . If there is no contractible closed Reeb orbit we set ℘ (Σ , λ ) = ∞ .In Theorem B we do not consider Σ , and M need to be closed. Theorem B.
Let ( M , ω ) be a closed and symplectically aspherical, i.e. ω | π ( M ) = 0 ,symplectic manifold. Then, although Σ × M is not a contact hypersurface, (B1) Σ × M has a leafwise intersection point for φ ∈ Ham c ( M × M , ω ⊕ ω ) withHofer-norm || φ || < ℘ (Σ , λ ) even if Σ does not bound a compact region in M . (B2) The Rabinowitz Floer homology
RFH(Σ × M , M × M ) can be defined when Σ bounds a compact region in M . Moreover, we have the K¨unneth formula: RFH n (Σ × M , M × M ) ∼ = n M p =0 RFH p (Σ , M ) ⊗ H n − p ( M ) . To prove Theorem B without any contact conditions, we need to show a special version ofisoperimetric inequality, see Lemma (3.2). ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 3
Remark 1.6.
It is worth emphasizing that Σ × M is not necessarily of restricted contacttype. For instance, if M is not exact, then Σ × M is never of restricted contact type. Never-theless, interestingly enough, we can achieve compactness of gradient flow lines of Rabinowitzaction functional for an arbitrary perturbation F ∈ Ham c ( M × M , ω ⊕ ω ); accordinglyFloer homology of the Rabinowitz action functional with any perturbations is well-defined.The K¨unneth formula enable us to compute the Rabinowitz Floer homology of a productmanifold in terms of Rabinowitz Floer homology of each manifolds. As applications, in Section4 we shall prove the following two corollaries. Corollary A.
Let N be a closed Riemannian manifold of dim N ≥ with dim H ∗ (Λ N ) = ∞ where Λ N is the free loop space of N . Then there exists infinitely many leafwise intersectionpoints for a generic φ ∈ Ham c ( T ∗ S × T ∗ N ) on ( S ∗ S × S ∗ N, T ∗ S × T ∗ N ) . Remark 1.7.
Since ( S ∗ S × S ∗ N, T ∗ S × T ∗ N ) is of restricted contact type in the sense ofBolle (Lemma 4.6), φ in Corollary A is not necessarily of product type. If φ has product type,then the above result is obvious by [1, 2]. Unlike Corollary A, the following Corollary B doesnot assume the contact condition since Theorem B does not need any contact conditions. Corollary B.
Let M be a closed and symplectically aspherical symplectic manifold and N be as above. Then a generic φ ∈ Ham c ( T ∗ N × M ) has infinitely many leafwise intersectionpoints on ( S ∗ N × M, T ∗ N × M ) . Remark 1.8. If π ( N ) is finite then dim H ∗ (Λ N ) = ∞ by [30]. If the number of conjugacyclasses of π ( N ) is infinite then dim H (Λ N ) = ∞ . Therefore, the only remaining case is if π ( N ) is infinite but the number of conjugacy classes of π ( N ) is finite.1.1. Leafwise intersections.
Let (
M, ω ) be a 2 n dimensional symplectic manifold and Σbe a coisotropic submanifold of codimension 0 ≤ k ≤ n . Then the symplectic structure ω determines a symplectic orthogonal bundle T Σ ω ⊂ T Σ as follows: T Σ ω := { ( x, ξ ) ∈ T Σ | ω x ( ξ, ζ ) = 0 for all ζ ∈ T x Σ } Since ω is closed, T Σ ω is integrable, thus Σ is foliated by the leaves of the characteristicfoliation and we denote by L x the isotropic leaf through x . We call x ∈ Σ a leafwise inter-section point of φ ∈ Ham(
M, ω ) if x ∈ L x ∩ φ ( L x ). In the extremal case k = n , Lagrangiansubmanifold consists of only one leaf. Thus a leafwise intersection point is nothing but aLagrangian intersection point in the case k = n . In the other extremal case that k = 0, aleafwise intersection corresponds to a periodic orbit of φ .The leafwise intersection problem was initiated by Moser [27] and pursued further in [9,22, 17, 20, 16, 21, 1, 2, 3, 4, 31, 23, 24, 25, 6, 7]. We refer to [1, 24] for the history ofthe problem. In particular, Albers-Frauenfelder approached the problem by means of theperturbed Rabinowitz action functional and much relevant research has been conducted in[1, 2, 3, 4, 13, 14, 15, 8, 23, 24, 25, 7]. We refer to [5] for a brief survey on Rabinowitz Floertheory. 2. Rabinowitz action functional on product manifolds
Since Σ and Σ are contact hypersurfaces, there exist associated Liouville vector fields Y resp. Y on M resp. M such that L Y i ω i = ω i and Y i ⋔ Σ i for i = 1 ,
2. We denote by φ tY i theflow of Y i and fix δ > φ tY i | Σ i is defined for | t | < δ . Since Σ resp. Σ bounds a JUNGSOO KANG compact region in M resp. M , we are able to define Hamiltonian functions G ∈ C ∞ ( M )and G ∈ C ∞ ( M ) so that(1) G − (0) = Σ and G − (0) = Σ are regular level sets;(2) dG and dG have compact supports;(3) G i ( φ tY i ( x i )) = t for all x i ∈ Σ i , i = 1 ,
2, and | t | < δ ;We extend G , G to be defined on the whole of M × M : e G i : M × M −→ R i = 1 , x , x ) G i ( x i ) . Definition 2.1.
Given time-dependent Hamiltonian functions e H , e H , F ∈ C ∞ ( S × M × M ), a triple ( e H , e H , F ) is called a Moser triple if it satisfies(1) their time supports are disjoint, i.e. e H ( t, · ) = e H ( t, · ) = 0 for ∀ t ∈ [0 ,
12 ] and F ( t, · ) = 0 for ∀ t ∈ [ 12 , . (2) F = F ⊕ F for some F ∈ C ∞ c ( S × M ), F ∈ C ∞ c ( S × M ).(3) e H and e H are weakly time-dependent Hamiltonian functions. That is, e H and e H areof the form (cid:0) e H ( t, x ) , e H ( t, x ) (cid:1) = χ ( t ) (cid:0) e G ( x ) , e G ( x ) (cid:1) for χ : S → S with R χdt = 1and Supp χ ⊂ ( , Definition 2.2.
Let F ∈ C ∞ c ( S × M, R ) be a compactly supported time-dependent Hamil-tonian function on a symplectic manifold ( M, ω ). We set || F || + := Z max x ∈ M F ( t, x ) dt || F || − := − Z min x ∈ M F ( t, x ) dt = || − F || + and || F || = || F || + + || F || − . For φ ∈ Ham c ( M, ω ) the Hofer norm is || φ || = inf {|| F || | φ = φ F , F ∈ C ∞ c ( S × M, R ) } . Lemma 2.3.
For all φ ∈ Ham c ( M, ω ) || φ || = ||| φ ||| := inf (cid:8) || F || | φ = φ F , F ( t, · ) = 0 ∀ t ∈ [ , (cid:9) . Proof . To prove || φ || ≥ ||| φ ||| , pick a smooth monotone increasing map r : [0 , → [0 , r (0) = 0 and r (1 /
2) = 1. For F with φ F = φ we set F r ( t, x ) := r ′ ( t ) F ( r ( t ) , x ). Thena direct computation shows φ F r = φ F , || F r || = || F || , and F r ( t, x ) = 0 for all t ∈ [ , (cid:3) We denote by L = L M × M ⊂ C ∞ ( S , M × M ) the component of contractible loopsin M × M . With a Moser triple ( H , H , F ), the perturbed Rabinowitz action functional A e H , e H F ( v, η , η ) : L × R −→ R is defined as follows: A e H , e H F ( v, η , η ) = − Z v ∗ λ ⊕ λ − η Z e H ( t, v ) dt − η Z e H ( t, v ) dt − Z F ( t, v ) dt (2.1) ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 5 where λ ⊕ λ = π ∗ λ + π ∗ λ . The real numbers η and η can be thought of as Lagrangemultipliers.Critical points ( v, η , η ) ∈ Crit A e H , e H F satisfy ∂ t v = η X e H ( t, v ) + η X e H ( t, v ) + X F ( t, v ) , Z e H ( t, v ) dt = 0 , Z e H ( t, v ) dt = 0 . (2.2)Albers-Frauenfelder [1] observed that a critical point of A e H , e H F gives rise to a leafwiseintersection point. (In fact, they proved the following proposition for the codimensional onecase, yet their proof continues to hold in our case, see [24] also.) Definition 2.4.
A leafwise coisotropic intersection point x ∈ Σ × Σ is called periodic if theleaf L x contains neither a closed Reeb orbit in Σ nor a closed Reeb orbit in Σ . Proposition 2.5. [1]
Let ( v, η , η ) ∈ Crit A e H , e H F . Then x = v (1 / satisfies φ F ( x ) ∈ L x .Thus, x is a leafwise intersection point. Moreover, the map Crit A e H , e H F −→ (cid:8) leafwise intersections (cid:9) is injective unless there exists a periodic leafwise intersection. We choose a compatible almost complex structure J on M and define the metric on( M , ω ) by g ( · , · ) = ω ( · , J · ). Analogously we also define the metric on ( M , ω ), g ( · , · ) = ω ( · , J · ). Then g = g ⊕ g which is the metric on ( M × M , ω ⊕ ω ) induces a metric m on the tangent space T ( v,η ,η ) ( L × R ) ∼ = T v L × R as follows: m ( v,η ,η ) (cid:0) (ˆ v , ˆ η , ˆ η ) , (ˆ v , ˆ η , ˆ η ) (cid:1) := Z g v (ˆ v , ˆ v ) dt + ˆ η ˆ η + ˆ η ˆ η . Definition 2.6.
A map w = ( v, η , η ) ∈ C ∞ ( R , L × R ) which solves ∂ s w ( s ) + ∇ m A e H , e H F ( w ( s )) = 0 (2.3)is called a gradient flow line of A e H , e H F with respect to the metric m .According to Floer’s interpretation, the gradient flow equation (2.3) can be interpreted asmaps v ( s, t ) : R × S → M × M and η ( s ), η ( s ) : R → R solving ∂ s v + J ( v ) (cid:0) ∂ t v − η X e H ( t, v ) − η X e H ( t, v ) − X F ( t, v ) (cid:1) = 0 ,∂ s η − Z e H ( t, v ) dt = 0 ,∂ s η − Z e H ( t, v ) dt = 0 . (2.4) JUNGSOO KANG
Definition 2.7.
The energy of a map w ∈ C ∞ ( R , L × R ) is defined by E ( w ) := Z ∞−∞ || ∂ s w || m ds . Lemma 2.8.
Let w be a gradient flow line of A e H , e H F . Then E ( w ) = A e H , e H F ( w − ) − A H ,H F ( w + ) . (2.5)where w ± = lim s →±∞ w ( s ). Proof . It follows from the gradient flow equation (2.3). E ( w ) = Z ∞−∞ m (cid:0) − ∇ m A e H , e H F ( w ( s )) , ∂ s w ( s ) (cid:1) ds = − Z ∞−∞ d A e H , e H F ( w ( s ))( ∂ s w ( s )) ds = − Z ∞−∞ dds (cid:16) A e H , e H F ( w ( s )) (cid:17) ds = A e H , e H F ( w − ) − A e H , e H F ( w + ) . (cid:3) Compactness of gradient flow lines.
In order to define Rabinowitz Floer homology,we need compactness of gradient flow lines of the Rabinowitz action functional with fixedasymptotic data. More specifically we show the following theorem. In the rest of this section,our perturbation F ∈ C ∞ c ( S × M × M ) is of the form F ⊕ F for some F ∈ C ∞ c ( S × M )and F ∈ C ∞ c ( S × M ). Theorem 2.9.
Let { w n } n ∈ N be a sequence of gradient flow lines of A e H , e H F for which thereexist a < b such that a ≤ A e H , e H F ( w n ( s )) ≤ b, for all s ∈ R . Then for every reparametization sequence σ n ∈ R , the sequence w n ( · + σ n ) has a subsequencewhich is converges in C ∞ loc ( R , L × R ) . Proof . In order to prove the theorem, we need to verify the following three ingredients.(1) a uniform L ∞ -bound on v n ,(2) a uniform L ∞ -bound on η n , η n ,(3) a uniform L ∞ -bound the derivatives of v n .for a sequence of gradient flow lines { ( v n , η n , η n ) } n ∈ N . Once we establish (2), the othersfollow by standard arguments in Floer theory. At the end of this section, we prove Theorem2.15 which proves (2) and thus completes the proof of Theorem 2.9. (cid:3) First of all, we introduce two auxiliary action functionals A , A : L M × M × R −→ R : A ( v, η , η ) := Z v ∗ π ∗ λ − η Z H ( t, v ) dt − Z F ( t, v ) dt, A ( v, η , η ) := Z v ∗ π ∗ λ − η Z H ( t, v ) dt − Z F ( t, v ) dt. ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 7 Lemma 2.10.
Let w = ( v, η , η ) ∈ C ∞ ( R , L × R ) be a gradient flow line of A e H , e H F withasymptotic ends w − = ( v − , η − , η − ) and w + = ( v + , η , η ). Then the action values of A and A are bounded along w in terms of the asymptotic data:(i) A ( w ( s )) ≤ |A ( w − ) | + |A ( w + ) | + 4 || F || + , ∀ s ∈ R ;(ii) A ( w ( s )) ≤ |A ( w − ) | + |A ( w + ) | + 4 || F || + , ∀ s ∈ R . Proof . We only show the first inequality, the later one is proved in a similar way. Since itholds that π ∗ X F = X F , π ∗ X F = X F , and i X e H π ∗ ω = 0, we compute dds A ( w ( s )) = d A ( w ( s ))[ ∂ s w ( s )]= Z π ∗ ω (cid:0) ∂ t v, ∂ s v ) − Z ω ⊕ ω (cid:0) η X H ( t, v ) + X F ( t, v ) , ∂ s v (cid:1) − (cid:16) Z e H ( t, v ) dt (cid:17) = Z π ∗ ω (cid:0) ∂ t v − η X e H ( t, v ) − X F ( t, v ) , ∂ s v (cid:1) dt − Z π ∗ ω ( X F ( t, v ) , ∂ s v ) dt − (cid:16) Z e H ( t, v ) dt (cid:17) = − Z π ∗ ω ( ∂ s v, J ∂ s v ) dt − Z dds F ( t, π ◦ v ) dt − (cid:16) Z e H ( t, v ) dt (cid:17) . Integrating the above equality from −∞ to any s ∈ R , we have A ( w ( s )) − A ( w − ) = Z s −∞ dds A ( w ( s )) ds = − Z s −∞ Z π ∗ ω ( ∂ s v, J ∂ s v ) dtds − Z s −∞ Z dds F ( t, π ◦ v ) dtds − Z s −∞ (cid:16) Z e H ( t, v ) dt (cid:17) ds = − Z s −∞ B ( s ) ds − Z F ( t, π ◦ v ( s )) − F ( t, π ◦ v − ) dt. (2.6)where B ( s ) is defined as B ( s ) := Z π ∗ ω ( ∂ s v, J ∂ s v ) dt + (cid:16) Z e H ( t, v ) dt (cid:17) . Therefore the following estimate can be derived for any s ∈ R |A ( w ( s )) | ≤ |A ( w + ) | + 2 || F || + + (cid:12)(cid:12)(cid:12) Z s −∞ B ( s ) ds (cid:12)(cid:12)(cid:12) , and it remains to find a bound for | R s −∞ B ( s ) ds | . Since B ( s ) is nonnegative, we are able toestimate as the following. By setting s = ∞ in formula (2.6), we have A ( w + ) − A ( w − ) = − Z ∞−∞ B ( s ) ds − Z F ( t, π ◦ v + ) − F ( t, π ◦ v − ) dt JUNGSOO KANG
Using the above formula, we obtain (cid:12)(cid:12)(cid:12) Z s −∞ B ( s ) ds (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Z ∞−∞ B ( s ) ds (cid:12)(cid:12)(cid:12) ≤ |A ( w + ) | + |A ( w − ) | + 2 || F || + . Thus we finally deduce |A ( w ( s )) | ≤ |A ( w + ) | + 2 |A ( w − ) | + 4 || F || + , ∀ s ∈ R . (cid:3) Once we have Lemma 2.10, the rest of the proof of Theorem 2.9 is quite similar as in [1].
Lemma 2.11.
Given a gradient flow line w ( s ) = ( v, η , η )( s ) ∈ C ∞ ( R , L × R ) of A e H , e H F ,assume that v ( t ) ∈ U δ := e G − ( − δ, δ ) ∩ e G − ( − δ, δ ) for all t ∈ (1 / ,
1) with 0 < δ < min { , δ } .Then there exists C i > | η i | ≤ C i (cid:16) |A i ( v, η ) | + ||∇ m A e H , e H F || m + 1 (cid:17) , i = 1 , . Proof . We estimate |A i ( v, η , η ) | = (cid:12)(cid:12)(cid:12) Z v ∗ π ∗ i λ i + η i Z e H i ( t, v ) dt + Z F ( t, v ) dt (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) η i Z π ∗ i λ i ( v ) (cid:0) X e H i ( t, v ) (cid:1) dt (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) Z π ∗ i λ i ( v ) (cid:0) X F ( t, v ) (cid:1) dt (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) η i Z e H i ( t, v ) dt (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) Z F ( t, v ) dt (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) Z π ∗ i λ i ( v ) (cid:0) ∂ t v − η X e H ( t, v ) − η X e H ( t, v ) − X F ( t, v ) (cid:1) dt (cid:12)(cid:12)(cid:12) ≥ | η i | − δ | η i | − C i,δ || ∂ t v − η X e H ( t, v ) − η X e H ( t, v ) − X F ( t, v ) || L − C i,δ,F ≥ | η i | − δ | η i | − C i,δ ||∇ m A e H , e H F || m − C i,F where C i,δ := || π ∗ i λ i | U δ || L ∞ and C i,δ,F := || F || L ∞ + C i || X F || L ∞ . The second inequality holdssince π ∗ i λ i ( X e H j ) = 0 if i = j . This estimate finishes the lemma with C i := max n − δ , C i,δ − δ , C i,δ,F − δ o , i = 1 , . (cid:3) Lemma 2.12.
Given a gradient flow line w ( s ) = ( v, η , η )( s ) ∈ C ∞ ( R , L × R ) of A H ,H F , ifthere exists t ∈ ( ,
1) such that v ( t ) / ∈ U δ then ||∇ m A e H , e H F ( v, η , η ) || m > ǫ for some ǫ = ǫ δ . Proof . Since v ( t ) / ∈ U δ for some t ∈ ( , v ( t ) / ∈ U δ := e G − ( − δ, δ ) or v ( t ) / ∈ U δ := e G − ( − δ, δ ) for that t ∈ ( , v ( t ) / ∈ U δ . If in addition v ( t ) / ∈ U δ/ for all t ∈ ( , (cid:12)(cid:12)(cid:12)(cid:12) ∇ m A e H , e H F ( v, η , η ) (cid:12)(cid:12)(cid:12)(cid:12) m ≥ (cid:12)(cid:12)(cid:12) Z e H ( t, v ( t )) dt (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z e H ( t, v ( t )) dt (cid:12)(cid:12)(cid:12) ≥ δ . Otherwise there is t ′ ∈ ( ,
1) such that v ( t ′ ) ∈ U δ/ . Thus there exist t , t ∈ ( ,
1) satisfyingone of the following two cases. v ( t ) ∈ ∂U δ/ , v ( t ) ∈ ∂U δ and v ( s ) ∈ U δ − U δ/ for all s ∈ [ t , t ] (2.7) ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 9 or v ( t ) ∈ ∂U δ , v ( t ) ∈ ∂U δ/ and v ( s ) ∈ U δ − U δ/ for all s ∈ [ t , t ] . We only treat the first case (2.7) and the second case follows analogously. With κ := max x ∈ U δ ||∇ g e G ( x ) || g we estimate κ ||∇ m A e H , e H F ( v, η , η ) || m ≥ κ || ∂ t v − η X e H ( t, v ) − η X e H ( t, v ) − X F ( t, v ) || L ≥ κ || ∂ t v − η X e H ( t, v ) − η X e H ( t, v ) − X F ( t, v ) || L ≥ Z t t || ∂ t v − η X e H ( t, v ) − η X e H ( t, v ) − X F ( t, v ) || g · ||∇ e G ( x ) || g dt ≥ (cid:12)(cid:12)(cid:12)(cid:12) Z t t (cid:10) ∇ e G ( v ( t )) , ∂ t v − η X e H ( t, v ) − η X e H ( t, v ) − X F ( t, v ) (cid:11) dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z t t d e G ( v ( t )) (cid:0) ∂ t v − η X e H ( t, v ) − η X e H ( t, v ) − X F ( t, v | {z } =0 (cid:1) dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z t t ddt e G ( v ( t )) dt − d e G ( v ( t )) (cid:0) η X e H ( t, v ) − η X e H ( t, v ) (cid:1)| {z } = η χω ( X e G ,X e G )+ η χω ( X e G ,X e G )=0 (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12) e G ( v ( t )) (cid:12)(cid:12) − (cid:12)(cid:12) e G ( v ( t )) (cid:12)(cid:12) = δ . Hence, the lemma follows with ǫ δ := min (cid:8) δ/ , δ/ (2 κ ) (cid:9) . (cid:3) Combining Lemma 2.11 and Lemma 2.12, we deduce the following fundamental lemma . Lemma 2.13.
For a gradient flow line w ( s ) = ( v, η , η )( s ) ∈ C ∞ ( R , L × R ) of A e H , e H F , thefollowing assertion holds for i = 1 , C, ǫ > | η i | ≤ C (cid:0) |A i ( w − ) | + |A i ( w + ) | + 1 (cid:1) if ||∇ m A e H , e H F ( v, η , η ) || m < ǫ. Proof . According to Lemma 2.12, v ( t ) lies in U δ for all t ∈ ( ,
1) under the assumptionthat ||∇ m A e H , e H F ( v, η , η ) || m < ǫ . Thus we are able to apply Lemma 2.11 and the followingcomputation concludes the proof of the lemma. | η i | ≤ C i ( |A i ( v, η , η ) | + ||∇ m A e H , e H F ( v, η , η ) || m + 1) ≤ C i (2 |A i ( w − ) | + |A i ( w + ) | + 4 || F || + + ||∇ m A e H , e H F ( v, η , η ) || m + 1) ≤ C i (2 |A i ( w − ) | + |A i ( w + ) | + 4 || F || + + 1 + ǫ ) . (cid:3) Lemma 2.14.
For given a gradient flow line w of A e H , e H F and σ ∈ R , we define τ ( σ ) := inf (cid:8) τ ≥ (cid:12)(cid:12) ||∇ m A e H , e H F ( w ( σ + τ )) || m ≤ ǫ (cid:9) , Then we obtain a bound on τ ( σ ) as follows: τ ( σ ) ≤ A e H , e H F ( w − ) − A e H , e H F ( w + ) ǫ . Proof . Using Lemma 2.8, we compute ǫ τ ( σ ) ≤ Z σ + τ ( σ ) σ (cid:12)(cid:12)(cid:12)(cid:12) ∇ m A e H , e H F ( w ) (cid:12)(cid:12)(cid:12)(cid:12) m ds ≤ E ( w ) ≤ A e H , e H F ( w − ) − A e H , e H F ( w + ) . Dividing both sides through by ǫ , the lemma follows. (cid:3) Theorem 2.15.
Given two critical points w − and w + , there exists a constant Θ > dependingonly on w − and w + such that every gradient flow line w ( s ) = ( v, η , η )( s ) of A e H , e H F withfixed asymptotic ends w ± satisfies || η i || L ∞ ≤ Θ for i = 1 , . Proof . Using Lemma 2.10 and Lemma 2.14, we estimate | η i ( σ ) | ≤ | η i ( σ + τ ( σ )) | + Z σ + τ ( σ ) σ | ∂ s η i ( s ) | ds ≤ C (cid:0) |A i ( w − ) | + |A i ( w + ) | + 1 (cid:1) + τ ( σ ) || e H i || L ∞ ≤ C (cid:0) |A i ( w − ) | + |A i ( w + ) | + 1 (cid:1) + A e H , e H F ( w − ) − A e H , e H F ( w + ) ǫ ! || H i || L ∞ . (cid:3) As we mentioned before, Theorem 2.15 completes the proof of Theorem 2.9.3.
K¨unneth formula in Rabinowitz Floer homology
Thanks to the previous section, we are now able to define Rabinowitz Floer homologyof (Σ × Σ , M × M ) for admissible perturbations of the form F ⊕ F (or unperturbed).Whilst A e H , e H F is generically Morse (Lemma 4.7), A H ,H is never Morse because there is a S -symmetry coming from time-shift on the critical point set. However A e H , e H is genericallyMorse-Bott, so we are able to compute its Floer homology by choosing an auxiliary Morsefunction on the critical manifold and counting gradient flow lines with cascades, see [18, 13].Using the continuation method in Floer theory, we know that the Floer homology of A e H , e H F is isomorphic to the Floer homology of A e H , e H = A e H , e H whenever these Floer homologiesare defined. Thus we only treat the unperturbed Rabinowitz action functional A e H , e H andits Floer homology. Furthermore, we derive the K¨unneth formula by making use of the factthat all critical points and gradient flow lines can be split. In the last subsection, we proveTheorem B using similar steps to those in the proof of Theorem A; but we need to prove a ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 11 special version of an isoperimetric inequality (Lemma 3.2) since unlike Theorem A, we havenot insisted on any restrictions on perturbations in Theorem B.3.1. Rabinowitz Floer homology.
Firstly, we define a chain complex and a boundaryoperator for the Rabinowitz action functional. In order to define a chain complex we choosean additional Morse function f on the critical manifold Crit A e H , e H . We define a Z / n ( A e H , e H , f ) := n ξ = X ( v,η ,η ) ξ ( v,η ,η ) ( v, η , η ) (cid:12)(cid:12)(cid:12) ( v, η , η ) ∈ Crit n f, ξ ( v,η ,η ) ∈ Z / o where ξ ( v,η ,η ) satisfy the finiteness condition: (cid:8) ( v, η , η ) ∈ Crit n f (cid:12)(cid:12) ξ ( v,η ,η ) = 0 , A H ,H ( v, η , η ) ≥ κ (cid:9) < ∞ , ∀ κ ∈ R . The grading for the chain complex, µ = µ RFH , is described in the appendix of this paper.To define the boundary operator, we roughly explain the notion of a gradient flow linewith cascades . For rigorous and explicit constructions, we refer to [18]. Consider a gradientflow line with cascades interchanging w − ⊂ C − and w + ⊂ C + where w ± ∈ Crit f and C ± ⊂ Crit A e H , e H ; it starts with a gradient flow line of f in C − with the negative asymptoticend w − and meets the negative asymptotic ends of a gradient flow line of A e H , e H (solving(2.4) with F ≡ cascade . Its positive asymptoticend encounters a gradient flow line of f in C + which converges to w + . Several cascades andno cascades are also allowed. Now, we define a moduli space c M{ w − , w + } := ( w ∈ C ∞ ( R , L × R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w is a gradient flow line with cascadeswith lim s →±∞ w ( s ) = w ± ∈ Crit f ) and divide out the R -action from shifting the gradient flow lines in the s -variable. Then weobtain the moduli space of unparametrized gradient flow lines, denoted by M := c M / R . The standard transversality theory shows that this moduli space is a smooth manifoldfor a generic choice of the almost complex structure and the metric, see [19, 12]. Fromthe calculation (5.1) in the appendix, we also know that the dimension of M is equal to µ RFH ( w − ) − µ RFH ( w + ) −
1. Therefore if µ RFH ( w − ) − µ RFH ( w + ) = 1, M ( w − , w + ) is a finiteset because of Theorem 2.9. We let M{ w − , w + } be the parity of this moduli space. Wedefine the boundary maps { ∂ e H , e H n } n ∈ Z as follows: ∂ e H , e H n +1 : CF n +1 ( A e H , e H ) −→ CF n ( A e H , e H ) w − X w + ∈ Crit n f M{ w − , w + } w + . Due to the Floer’s central theorem, we know that ∂ H ,H n ◦ ∂ H ,H n +1 = 0 so that (cid:0) CF ∗ ( A H ,H ) , ∂ H ,H ∗ (cid:1) is a chain complex. We define Rabinowitz Floer homology byRFH n (Σ × Σ , M × M ) := HF n (cid:0) A e H , e H (cid:1) = H n (cid:0) CF ∗ ( A e H , e H ) , ∂ e H , e H ∗ (cid:1) . Remark 3.1.
Since in the previous section, we achieved the compactness result for A e H , e H F ⊕ F ,the Floer homology HF n (cid:0) A e H , e H F ⊕ F (cid:1) can be defined; besides, it is isomorphic to RFH n (Σ × Σ , M × M ) by the continuation homomorphism which counts gradient flow lines of A e H , e H F s where F s is a homotopy between F ⊕ F and F ≡ Proof of Theorem A.
At first, we set H ( t, x ) = χ ( t ) G ( x ) ∈ C ∞ ( S × M ) , H ( t, x ) = χ ( t ) G ( x ) ∈ C ∞ ( S × M )where χ : S → [0 , ∞ ) with R χdt = 1 and Supp χ ⊂ ( , π i ) ∗ X e H i ( x , x ) = X H i ( x i ) , i = 1 , . We consider the Rabinowitz action functionals A H : L M × R → R and A H : L M × R → R : • A H ( v , η ) = − Z v ∗ λ − η Z H ( t, v ) dt, • A H ( v , η ) = − Z v ∗ λ − η Z H ( t, v ) dt. In fact, we can accomplish compactness of gradient flow lines of each action functional withminor modifications of our case, or see [1]. We observe that ( v , η ) ∈ Crit A H solves ∂ t v = η X H ( t, v ) & Z H ( t, v ) dt = 0 , (3.1)and ( v , η ) ∈ Crit A H solves ∂ t v = η X H ( t, v ) & Z H ( t, v ) dt = 0 . (3.2)Moreover a gradient flow line w ( s, t ) = (cid:0) v ( s, t ) , η ( s ) (cid:1) : R × S → M × R resp. w ( s, t ) = (cid:0) v ( s, t ) , η ( s ) (cid:1) : R × S → M × R is characterized by • ∂ s v + J ( v ) (cid:0) ∂ t v − η X H ( t, v ) (cid:1) = 0 , ∂ s η − Z H ( t, v ) dt = 0 , resp. • ∂ s v + J ( v ) (cid:0) ∂ t v − η X H ( t, v ) (cid:1) = 0 , ∂ s η − Z H ( t, v ) dt = 0 . (3.3)Then we define chain complexes CF( A H ), CF( A H ) and their boundary operators ∂ H , ∂ H analogously as before, or see [1] and denote their Floer homologies byRFH(Σ , M ) = H (cid:0) CF( A H ) , ∂ H (cid:1) , RFH(Σ , M ) = H (cid:0) CF( A H ) , ∂ H (cid:1) . Next, for the K¨unneth formula, we define the tensor product of chain complexes by (cid:0) CF ∗ ( A H ) ⊗ CF ∗ ( A H ) (cid:1) n = n M i =0 CF i ( A H ) ⊗ CF n − i ( A H ) . together with the boundary operator ∂ ⊗ n given by ∂ ⊗ n (cid:0) ( v , η ) i ⊗ ( v , η ) n − i (cid:1) = ∂ H i ( v , η ) i ⊗ ( v , η ) n − i + ( v , η ) i ⊗ ∂ H n − i ( v , η ) n − i . Analyzing the critical point equations (2.2) when F ≡
0, (3.1), and (3.2), we easily noticethat (cid:0) ( v , v ) , η , η (cid:1) = ( v, η , η ) ∈ Crit A H ,H if and only if ( v , η ) ∈ Crit A H and ( v , η ) ∈ ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 13 Crit A H where v = π ◦ v : S → M and v = π ◦ v : S → M for the projections π , π .Here, ( v , v ) ∈ C ∞ ( S , M × M ) is defined by( v , v ) : S −→ M × M ,t ( v ( t ) , v ( t )) . Moreover the index behaves additively (see (5.2)), thus we haveCrit n ( A e H , e H ) = [ i + j = n Crit i ( A H ) × Crit j ( A H ) , and we are able to define a chain homomorphism: P n : (cid:0) CF ∗ ( A H ) ⊗ CF ∗ ( A H ) (cid:1) n −→ CF n ( A e H , e H ) , ( v , η ) ⊗ ( v , η ) (cid:0) ( v , v ) , η , η (cid:1) . To verify that P n is a chain homomorphism, we need to show that ∂ H ,H n ◦ P n = P n − ◦ ∂ ⊗ n . For w − = ( v − , η − ) ∈ Crit A H and w − = ( v − , η − ) ∈ Crit A H , we compute ∂ e H , e H n ◦ P n ( w − ⊗ w − ) = ∂ e H , e H n (cid:0) ( v − , v − ) , η − , η − (cid:1)| {z } =: w − = X w + ∈ Crit A e H , e H ; µ ( w + )= µ ( w − ) − M{ w − , w + } w + = X ( v ,η ) ∈ Crit A H ; µ ( w )= µ ( w − ) − M (cid:8) w − , (( v , v − ) , η , η − ) (cid:9)(cid:0) ( v , v − ) , η , η − (cid:1) + X ( v ,η ) ∈ Crit A H ; µ ( w )= µ ( w − ) − M (cid:8) w − , (( v − , v ) , η − , η ) (cid:9)(cid:0) ( v − , v ) , η − , η (cid:1) = X ( v ,η ) ∈ Crit A H ; µ ( w )= µ ( w − ) − M (cid:8) w − , w (cid:9) P n − ( w ⊗ w − )+ X ( v ,η ) ∈ Crit A H ; µ ( w )= µ ( w − ) − M (cid:8) w − , w (cid:9) P n − ( w − ⊗ w )= P n − ( ∂ H i w − ⊗ w − ) + P n − ( w − ⊗ ∂ H n − i w − )= P n − ◦ ∂ ⊗ n ( w − ⊗ w − ) . where M (cid:8) w − , w (cid:9) resp. M (cid:8) w − , w (cid:9) is the moduli space which consists of gradient flowlines with cascades of A H resp. A H . The fourth equality follows by comparing (2.4) togetherwith (3.3). Therefore we have an isomorphism( P • ) ∗ : H • (cid:0) CF( A H ) ⊗ CF( A H ) (cid:1) ∼ = −→ H • (CF( A e H , e H )) = RFH • (Σ × Σ , M × M ) . Finally, the algebraic K¨unneth formula enable us to derive the desired (topological) K¨unnethformula in Rabinowitz Floer homology.RFH n (Σ × Σ , M × M ) ∼ = n M p =0 RFH p (Σ , M ) ⊗ RFH n − p (Σ , M ) . Proof of Theorem B.
In this subsection, we do not consider Σ and let ( M , ω ) beclosed and symplectically aspherical, i.e. ω | π ( M ) . To prove Statement (B1) in TheoremB, we need compactness of gradient flow lines of the perturbed Rabinowitz action functionalon (Σ × M , M × M ) for an arbitrary perturbation F ∈ C ∞ c ( S × M × M ). For thatreason, we analyze the Rabinowitz action functional again as in Section 4; once we obtain thefundamental lemma, then the remaining steps are exactly same as before. Moreover due tocompactness of gradient flow lines, we can find a leafwise intersection point for Hofer-smallHamiltonian diffeomorphisms using the stretching the neck argument in [1]. We assume thatΣ × M bounds a compact region in M × M for Statement (B2) in Theorem B throughoutthis subsection; but, when it comes to the existence of leafwise intersections, Σ × M neednot bound a compact region in M × M using the techniques in [23, 24]. As before, we choosea defining Hamiltonian function G ∈ C ∞ ( M ) so that(1) G − (0) = Σ is a regular level set and dG has a compact support.(2) G i ( φ tY ( x )) = t for all x ∈ Σ i , and | t | < δ ;where Y is the Liouville vector field for Σ ⊂ M . We define e G ∈ C ∞ ( M × M ) by e G ( x , x ) = G ( x ). Thus e G is a defining Hamiltonian function for Σ × M . We let e H ( t, x ) = χ ( t ) e G ( x ) ∈ C ∞ ( S × M × M ). With a perturbation F ∈ C ∞ c ( S × M × M ), the perturbed Rabinowitzaction functional A e HF : L × R −→ R is given by A e HF ( v, η ) = − Z D ¯ v ∗ ω ⊕ ω − η Z e H ( t, v ) dt − Z F ( t, v ) dt where L = L M × M ⊂ C ∞ ( S , M × M ) is the component of contractible loops in M × M and ¯ v : D → M × M is a filling disk of v . The symplectic asphericity condition impliesthat the value of the above action functional is independent of the choice of filling disc.Next, we prove the following lemma using a kind of isoperimetric inequality. Lemma 3.2.
Let w ( s, t ) = ( v ( s, t ) , η ( s )) ∈ C ∞ ( R × S , M × M ) × C ∞ ( R , R ) be a gradientflow line of A e HF . We set γ ( t ) = v ( s , t ) ∈ C ∞ ( S , M × M ) for some fixed s ∈ R . Then R D ¯ γ ∗ π ∗ ω is uniformly bounded provided ||∇ m A e HF ( v ( s , · ) , η ( s )) || m < ǫ for some ǫ > (cid:12)(cid:12)(cid:12) Z D ¯ γ ∗ π ∗ ω (cid:12)(cid:12)(cid:12) ≤ max x ∈ g M (cid:8) || λ f M ( x ) || ˜ g (cid:12)(cid:12) d ˜ g ( x, f M ⋆ ) < ǫ + || X F || L ∞ (cid:9)(cid:0) ǫ + || X F || L ∞ (cid:1) . (3.4)where f M is the universal covering of M ; ˜ g is the lifting of the metric g ( · , · ) = ω ( · , J · ) on M ; f M ⋆ is a fundamental domain in f M ; d ˜ g ( x, f M ⋆ ) is the distance between x and f M ⋆ ; thevalue on the right hand side of (3.4) is finite since f M ⋆ ∼ = M is compact. Proof . We write v ( s, t ) as v ( s, t ) = ( v , v )( s, t ) where v : R × S → M and v : R × S → M . Let γ ∈ C ∞ ( S , M × M ) be defined by γ ( t ) = v ( s , t ) for some s ∈ R . Since γ is contractible and M is symplectically aspherical, the value of R D ¯ γ ∗ π ∗ ω is well-defined.Let γ := π ◦ γ . We also consider ( f M , f ω ) the universal cover of M where f ω is the lift ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 15 of ω and we also lift the metric g on M which we write as ˜ g . Since we have assumedthe symplectically asphericity of ( M , ω ), there exists a primitive one form λ f M of f ω . Let f M ⋆ ( ∼ = M ) be one of the fundamental domains in f M and ˜ v ( s, t ) : R × S → M × f M be thelift of v such that ˜ v ( s , t ) = ˜ γ ( t ) intersects M × f M ⋆ . Now, we can show the following kind ofisoperimetric inequality. This inequality concludes the proof. (cid:12)(cid:12)(cid:12) Z D ¯ γ ∗ π ∗ ω (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z D (˜¯ γ ) ∗ f ω (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z ˜ γ ∗ λ f M (cid:12)(cid:12)(cid:12) ≤ || λ f M | γ ( S ) || L ∞ Z || ∂ t ˜ γ || ˜ g dt = || λ f M | γ ( S ) || L ∞ Z || ∂ t γ || g dt = || λ f M | γ ( S ) || L ∞ Z || J ∂ s γ + π ∗ X F ( t, γ ) || g dt ≤ λ Max (cid:0) ||∇ m A e HF ( v ( s , · ) , η ( s )) || m + || X F || L ∞ (cid:1) . where λ Max := max x ∈ g M n || λ f M ( x ) || ˜ g (cid:12)(cid:12)(cid:12) d ˜ g ( x, f M ⋆ ) < Z || ∂ t γ || g dt o ≤ max x ∈ g M (cid:8) || λ f M ( x ) || ˜ g (cid:12)(cid:12) d ˜ g ( x, f M ⋆ ) < ||∇ m A e HF ( v ( s , · ) , η ( s )) || m + || X F || L ∞ (cid:9) . (cid:3) The following two lemmas can be proved similarly to the corresponding lemmas in theprevious section.
Lemma 3.3.
We assume that for ( v, η ) ∈ C ∞ ( S , M × M ) × R , v ( t ) ∈ U δ := e G − ( − δ, δ )for all t ∈ ( ,
1) with 0 < δ < min { , δ } . Then there exists C > | η | ≤ C (cid:16) |A e HF ( v, η ) | + ||∇ m A e HF ( v, η ) || m + (cid:12)(cid:12)(cid:12) Z D ¯ v ∗ π ∗ ω (cid:12)(cid:12)(cid:12) + 1 (cid:17) . Lemma 3.4.
For ( v, η ) ∈ C ∞ ( S , M × M ) × R if there exists t ∈ [ ,
1] such that v ( t ) / ∈ U δ ,then ||∇ m A e HF ( v, η ) || m > ǫ for some ǫ = ǫ δ .Due to the three previous lemmata, we deduce the fundamental lemma in the situation ofTheorem B, and thus we obtain a uniform L ∞ -bound on the Lagrange multiplier η by thesame argument as in the previous section. Lemma 3.5.
For a gradient flow line w ( s ) = ( v, η )( s ) ∈ C ∞ ( R , L × R ), the followingassertions holds with some C, ǫ >
0. If ||∇ m A e HF ( v, η ) || m < ǫ , | η | ≤ C (cid:0) |A e HF ( w − ) | + |A e HF ( w + ) | + ǫ + Ξ ǫ + 1 (cid:1) provided that ||∇ m A e HF ( v, η ) || m < ǫ where Ξ ǫ = max (cid:8) || λ g M ( x ) || ˜ g | d ˜ g ( x, f M ⋆ ) < ǫ + || X F || L ∞ (cid:9)(cid:0) ǫ + || X F || L ∞ (cid:1) < ∞ . Proof . The proof is almost same as the proof of Lemma 2.13. Since ||∇ m A e HF ( v, η ) || m < ǫ , v ( t ) ⊂ U δ for t ∈ ( ,
1) by Lemma 3.4. Thus Lemma 3.2 and Lemma 3.3 prove the lemma. (cid:3)
This fundamental lemma proves compactness of gradient flow lines as before. Let φ ∈ Ham c ( M × M , ω ⊕ ω ) be a Hamiltonian diffeomorphism with the Hofer norm less than ℘ (Σ , λ ). We consider a moduli space of gradient flow lines of the Rabinowitz action func-tional perturbed by a special smooth family of Hamiltonian functions. Then in the boundaryof this moduli space, there is a broken gradient flow line of which one asymptotic end givesrise to either a leafwise intersection point of φ or a closed Reeb orbit with period less than || φ || . But since || φ || < ℘ (Σ , λ ), there is no such a closed Reeb orbit and hence we obtain aleafwise intersection point. This is so called the stretching the neck argument, see [1, 24]. Evenfurther, there exists a leafwise intersection point even if Σ × M does not bound a compactregion in M × M due to the arguments in [23, 24]. Next, we define the Rabinowitz Floerhomology for (Σ × M , M × M ) in the same way as before and derive the K¨unneth formulain this situation. First of all, we consider another two action functionals A H : L M × R → R and A : L M → R defined by A H ( v , η ) := − Z v ∗ λ − η Z H ( t, v ) dt, A ( v ) := − Z D ¯ v ∗ ω . where H ( t, x ) = χ ( t ) G ( x ) ∈ C ∞ ( S × M ). We note that A e H is defined on L M × M × R .As in the proof of Theorem A, we compare critical points of A e H and A H as follows.Crit n ( A e H ) = [ i + j = n Crit i ( A H ) × Crit j ( A ) . Since Crit A consists of one component M , any gradient flow line with cascades of A nec-essarily has zero cascades, and hence is simply a gradient flow line of an additional Morsefunction f ∈ C ∞ ( M ). Thus the chain group for the Morse-Bott homology of A is given byCF( A , f ) = CM( f ). Here CM stands for the Morse complex. The following map is a chainisomorphism, which can be verified using the methods of the previous subsection. P n : (cid:0) CF ∗ ( A H ) ⊗ CM ∗ ( f ) (cid:1) n −→ CF n ( A e H ) , ( v , η ) ⊗ v (cid:0) ( v , v ) , η (cid:1) . Therefore it induces an isomorphism on the homology level:( P • ) ∗ : H • (cid:0) CF( A H ) ⊗ CM( f ) (cid:1) ∼ = −→ H • (cid:0) CF( A e H ) (cid:1) = RFH • (Σ × M , M × M ) . Finally, the K¨unneth formula for (Σ × M , M × M ) directly follows:RFH n (Σ × M , M × M ) ∼ = n M p =0 RFH p (Σ , M ) ⊗ H n − p ( M ) . Applications
As we have mentioned in the introduction, we cannot achieve compactness of gradient flowlines of A e H , e H F for an arbitrary perturbation F . For that reason, the existence problem ofleafwise intersection points for a product submanifold which is not of contact type is stillopen. On the other hand, the existence of leafwise intersection points for contact coisotropicsubmanifolds was already proved in [21, 24]. Furthermore, due to the K¨unneth formula,we can deduce the existence of infinitely many leafwise intersection points for some kind ofproduct submanifolds of contact type. First, we recall the notion of contact condition oncoisotropic submanifolds introduced by Bolle [10, 11]. ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 17 Definition 4.1.
A coisotropic submanifold Σ of codimension k in a symplectic manifold( M, ω ) is called of restricted contact type if there exist global one forms λ , . . . , λ k ∈ Ω ( M )which satisfy(1) dλ i = ω for i = 1 , . . . , k ;(2) λ ∧ · · · ∧ λ k ∧ ω n − k | Σ = 0. Remark 4.2. [11, 20] Let Σ be closed and have contact type in M . Then a one form λ = a λ + · · · + a k λ k with a + · · · + a k = 0 is closed and hence defines an element of H (Σ).In addition, λ = 0 is not exact; otherwise λ = df for some f ∈ C ∞ (Σ), and hence λ ( x ) = 0at a critical point x of f , but condition (ii) yields that λ , . . . , λ k are linearly independent onΣ; thus λ ( x ) = · · · λ k ( x ) = 0. As a result, dim H (Σ) ≥ k −
1. It imposes restrictions onthe contact condition; for instance, S × S is not of contact type in R .We note that if the codimension of Σ is bigger than one, Σ never bounds a compact regionin M . In spite of such a dimension problem, the condition that global coordinates exist(roughly speaking, Poisson-commuting Hamiltonian functions whose common zero locus isonly Σ) enable us to unfold the generalized Rabinowitz Floer homology theory [24]. It turnsout that a product of contact hypersurfaces bounding respective ambient symplectic manifoldshas global coordinates. Theorem 4.3. [24] If Σ is a contact coisotropic submanifold of M which admits global coor-dinates, then the Floer homology of the perturbed Rabinowitz action functional is well-defined. Since the Rabinowitz action functional can be defined for each homotopy classes of loops,we can define the Rabinowitz Floer homology RFH(Σ , M, γ ) for γ ∈ [ S , M ]. We note that theRFH(Σ , M ) considered so far, is equal to RFH(Σ , M, pt). Moreover we also define RabinowitzFloer homology on the full loop space and denote it by RFH (Σ , M ). Then we have RFH ∗ (Σ , M ) = M γ ∈ [ S ,M ] RFH ∗ (Σ , M, γ ) . We recall the computation of Rabinowitz Floer homology on the (unit) cotangent bundle( S ∗ N, T ∗ N ) for a closed Riemannian manifold N . Theorem 4.4. [14, 8, 25]
RFH ∗ ( S ∗ N, T ∗ N ) ∼ = ( H ∗ (Λ N ) , ∗ > , H −∗ +1 (Λ N ) , ∗ < . Here Λ N stands for the free loop space of N . Since the K¨unneth formula obviously holds for
RFH also, the following corollary directlyfollows.
Corollary 4.5. If RFH ∗ (Σ , M ) = 0 , and dim H ∗ (Λ N ) = ∞ then dim RFH ∗ (Σ × S ∗ N, M × T ∗ N ) = ∞ . Accordingly, if Σ × S ∗ N has contact type again, Σ × S ∗ N has infinitely many leafwiseintersection points or a periodic leafwise intersections for a generic perturbation. Proof of Corollary A and B.
From now on, we investigate leafwise intersections on( S ∗ S × S ∗ N, T ∗ S × T ∗ N ). Lemma 4.6. S ∗ S × S ∗ N is a contact submanifold of codimension two in T ∗ S × T ∗ N . Proof . ( T ∗ S , ω S , can ) ∼ = ( S × R , dθ ∧ dr ) where θ is the angular coordinate on S and r is the coordinate on R . Then dθ ∧ dr has two global primitives − rdθ and − rdθ + dθ .We can easily check that S ∗ S × S ∗ N carries a contact structure with − rdθ ⊕ λ N , can and( − rdθ + dθ ) ⊕ λ N , can where λ N , can is the canonical one form on T ∗ N . (cid:3) To exclude periodic leafwise intersections, we consider the loop space Ω defined byΩ := (cid:8) v = ( v , v ) ∈ C ∞ ( S , T ∗ S × T ∗ N ) (cid:12)(cid:12) v is contractible in T ∗ S (cid:9) . Then we define the Rabinowitz action functional on this loop space, A H ,H F : Ω × R −→ R ,and construct the respective Rabinowitz Floer homology RFH( S ∗ S × S ∗ N, T ∗ S × T ∗ N, Ω)as before. Moreover the following type of the K¨unneth formula holds.RFH n ( S ∗ S × S ∗ N, T ∗ S × T ∗ N, Ω) ∼ = n M p =0 RFH p ( S ∗ S , T ∗ S ) ⊗ RFH n − p ( S ∗ N, T ∗ N ) . Therefore RFH( S ∗ S × S ∗ N, T ∗ S × T ∗ N, Ω) is of infinite dimensional if dim H ∗ (Λ N ) = ∞ and Lemma 4.7 below yields that there are infinitely many leafwise intersection points for ageneric perturbation whenever dim N ≥
2. This proves Corollary A.In order to prove that there is generically no periodic leafwise intersections, we review theargument in [2]. We denote by R the set of closed Reeb orbits in T ∗ N which has dimensionone. It is convenient to introduce the following sets: F j = (cid:8) F ∈ C jc ( S × T ∗ S × T ∗ N ) (cid:12)(cid:12) F ( t, · ) = 0 , ∀ t ∈ (cid:2) , (cid:3)(cid:9) , F = ∞ \ j =1 F j . Lemma 4.7.
If dim N ≥
2, then the set F S ∗ S × S ∗ N := ( F ∈ F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A H ,H F is Morse & v (0) ∩ ( S ∗ S × R ) = ∅ for all ∀ ( v, η , η ) ∈ A H ,H F , R ∈ R ) is dense in the set F . Proof . In this proof, we denote byΩ , := (cid:8) v = ( v , v ) ∈ W , ( S , T ∗ S × T ∗ N ) (cid:12)(cid:12) v is contractible in T ∗ S (cid:9) . the loop space which is indeed a Hilbert manifold. Let E be the L -bundle over Ω , with E v = L ( S , v ∗ T ( S ∗ S × S ∗ N )). We consider the section S : Ω , × R × F j −→ E ∨ × R defined by S ( v, η , η , F ) := d A H ,H F ( v, η , η ) . Here the symbol ∨ represents the dual space. At ( v, η , η , F ) ∈ S − (0), the vertical differen-tial DS : T ( v,η ,η ,F ) Ω , × R × F j −→ E ∨ v × R is given by the pairing (cid:10) DS ( v,η ,η ,F ) [ˆ v , ˆ η , ˆ η , ˆ F ] , [ˆ v , ˆ η , ˆ η ] (cid:11) = H A H ,H F [(ˆ v , ˆ η , ˆ η ) , (ˆ v , ˆ η , ˆ η )] + Z ˆ F ( t, v ) dt. ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 19 where H A H ,H F is the Hessian of A H ,H F . Due to the arguments in [1] (in fact they provedthe surjectivity for A HF , but their proof obviously can be extended to our situation, see also[24]), we know that for ( v, η , η , F ) ∈ S − (0), DS ( v,η ,η ,F ) is surjective on the space V := (cid:8) (ˆ v, ˆ η , ˆ η , ˆ F ) ∈ T ( v,η ,η ,F ) (Ω , × R × F j ) (cid:12)(cid:12) ˆ v (0) = 0 (cid:9) . Next, we consider the evaluation mapev :
M −→ S ∗ S × S ∗ N, ( v, η , η , F ) v (0) . Since DS ( v,η ,η ,F ) | V is surjective, Lemma 4.8 below implies that ev is a submersion. Then M R := ev − ( S ∗ S × R ) is a submanifold in M ofcodim( M R / M ) = codim( S ∗ S × R /S ∗ S × S ∗ N ) . We consider the projections Π :
M −→ F j and Π R := Π |M R . Then A H ,H F is Morse if andonly if F is a regular value of Π, which is a generic property by Sard-Smale theorem (for j large enough). The set Π − ( F ) of leafwise intersection points for F is manifold of requireddimension zero since it is a critical set of A H ,H F . On the other hand, Π − R ( F ) is a manifoldof dimension 0 + dim M R − dim M = − codim( M R / M ) < N ≥
2. Therefore ev does not intersect S ∗ S × R , so the set F jS ∗ S × S ∗ N := F S ∗ S × S ∗ N ∩ F j is dense in F for all j ∈ N . Since F S ∗ S × S ∗ N is the countable intersection of F jS ∗ S × S ∗ N for j ∈ N , it is dense again in F and the lemma is proved. (cid:3) Lemma 4.8. (Salamon)
Let
E −→ B be a Banach bundle and s : B −→ E a smooth section.Moreover, let φ : B −→ N be a smooth map into the Banach manifold N . We fix a point x ∈ s − (0) ⊂ B and set K := ker dφ ( x ) ⊂ T x B and assume the following two conditions.(1) The vertical differential Ds | K : K −→ E x is surjective.(2) dφ ( x ) : T x B −→ T φ ( x ) N is surjective.Then dφ ( x ) | ker Ds ( x ) : ker Ds ( x ) −→ T φ ( x ) N is surjective. Proof . Given ξ ∈ T φ ( x ) N , condition (ii) implies that there exists η ∈ T x B satisfying dφ ( x ) η = ξ . In addition, by condition (i), there exists ζ ∈ K ⊂ T x B satisfying Ds ( x ) ζ = Ds ( x ) η . Weset τ := η − ζ and compute Ds ( x ) τ = Ds ( x ) η − Ds ( x ) ζ = 0thus, τ ∈ ker Ds ( x ). Moreover, dφ ( x ) τ = dφ ( x ) η − dφ ( x ) ζ | {z } =0 = dφ ( x ) η = ξ proves the lemma. (cid:3) In the case of Theorem B, we also redefine the Rabinowitz action functional A HF : Ω M × R → R by A HF ( v, η ) = − Z v ∗ λ − Z D ¯ v ∗ ω − η Z H ( t, v ) dt − Z F ( t, v ) dt where Ω M : (cid:8) v = ( v , v ) ∈ C ∞ ( S , M × M ) (cid:12)(cid:12) v is contractible in M (cid:9) . We can also define the respective Rabinowitz Floer homology and derive an appropriateK¨unneth formula as before.
Corollary 4.9.
Let ( M , ω ) be a closed, symplectically aspherical symplectic manifold. If aclosed manifold N has dim H ∗ (Λ N ) = ∞ , then dim RFH ∗ ( S ∗ N × M , T ∗ N × M , Ω M ) = ∞ . Therefore, if dim N ≥ , S ∗ N × M has infinitely many leafwise intersection points for ageneric perturbation. The previous corollary proves Corollary B.
Remark 4.10.
The corollaries still holds when we deal with a generic fiber-wise star shapedhypersurface Σ ⊂ T ∗ N instead of S ∗ N , see [2].5. Appendix : Index for Rabinowitz Floer homology
In fact, we are able to derive the K¨unneth formula and obtain applications without definingindices. Nevertheless, for the sake of completeness, we briefly recall the index for generatorsof the Rabinowitz Floer chain complex in this appendix (see [13] for the detailed arguments).Let Σ be a contact hypersurface in M . Under the following assumption the Rabinowitz Floerhomology has Z -grading,(H1) Closed Reeb orbits on (Σ , λ ) is of Morse-Bott type [13].(H2) The first chern class c vanishes on T M . Remark 5.1.
Without any hypothesis on the first chern class, the Rabinowitz Floer homologyhas Z / M be the moduli space of all finite energy gradient flow lines of A H and w = ( v, η ) ∈ C ∞ ( R × S , M ) × C ∞ ( R , R ) be a gradient flow line of A H with lim s →±∞ w ( s ) = w ± =( v ± , η ± ) ∈ Crit f and v ± ⊂ C ± where C ± ⊂ Crit A H are connected components of the criticalmanifold and f is an additional Morse function on a critical manifold Crit A H . The lineariza-tion of the gradient flow equation along ( v, η ) gives rise to an operator D A H ( v,η ) . For suitableweighted Sobolev spaces, D A H ( v,η ) is a Fredholm operator. Then the local virtual dimension of M at ( v, η ) is defined to bevirdim ( v,η ) M := ind D A H ( v,η ) + dim C − + dim C + . Here ind D A H ( v,η ) stands for the Fredholm index of D A H ( v,η ) . Cieliebak-Frauenfelder [13] investi-gated the spectral flow of the Hessian Hess A H and consequently proved the following indexformula: virdim ( v,η ) M = µ CZ ( v + ) − µ CZ ( v − ) + dim C − + dim C + . Here µ CZ is the Conley-Zehnder index defined below. Since a closed Reeb orbit v + is con-tractible in M , we have a filling disk ¯ v + : D → M such that ¯ v + | ∂D = v + . The filling disk ¯ v + determines homotopy class of trivialization of the symplectic vector bundle (¯ v + ) ∗ T M . Thelinearized flow of the Reeb vector field along v + defines a path in Sp (2 n, R ) the group of ¨UNNETH FORMULA IN RABINOWITZ FLOER HOMOLOGY 21 symplectic matrices. The Conley Zehnder index of v + is defined by the Maslov index of [26]this path. This index is independent of choice of filling disk due to (H2) because the ConleyZehnder indices of different filling disks are differ by c . In the same way µ CZ ( v − ) is alsodefined. Now, we are in a position to define a grading µ RFH on CF( A H ) by µ RFH ( v ± , η ± ) := µ CZ ( v ± ) + µ fσ ( v ± ) . where µ fσ is the signature index defined by µ σf ( v ± ) = −
12 sign(Hess f ( v ± ))= − (cid:26) positive eigenvalues ofthe Hessian of f at v ± (cid:27) − (cid:26) negative eigenvalues ofthe Hessian of f at v ± (cid:27) ! . We note that by definition, µ σf ( v ± ) = µ Morse f ( v ± ) −
12 dim C ± . Then we notice that the dimension of gradient flow lines of A HF interchanging w − and w + equals the index difference of the two critical points by the following computation.dim c M{ w − , w + } = virdim ( v,η ) M − dim C + − dim C − + dim W uf ( v − ) + dim W sf ( v + )= µ CZ ( v − ) − µ CZ ( v + ) − dim C − + dim C + µ Morse f ( w − ) + dim C + − µ Morse f ( w + )= µ CZ ( v − ) − µ CZ ( v + ) − dim C − + dim C + µ σf ( v − ) + 12 dim C − + dim C + − ( µ fσ ( v + ) + 12 dim C + )= µ CZ ( v − ) − µ CZ ( v + ) + µ fσ ( v − ) − µ σ ( v + )= µ RFH ( v − ) − µ RFH ( v + ) (5.1)where W sf ( v + )( W uf ( v − )) is the (un)stable manifold with respect to ( f, v ± ).Furthermore, the RFH-index of (cid:0) ( v , v ) , η , η (cid:1) ∈ Crit A e H , e H (as used in Theorem A)splits into the indices of ( v , η ) and ( v , η ). µ RFH (cid:0) ( v , v ) , η , η (cid:1) = µ RFH ( v , η ) + µ RFH ( v , η ) (5.2)since the Conley-Zehnder index (in fact, the Maslov index) and the Morse index behaveadditively under the direct sum operation. Acknowledgments.
I am deeply indebted to Urs Frauenfelder for numerous fruitful discus-sions. I also thanks the anonymous referee for careful reading and comments.