S 1 -equivariant Rabinowitz-Floer homology
aa r X i v : . [ m a t h . S G ] M a y S -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY URS FRAUENFELDER AND FELIX SCHLENK
Abstract.
We define the S -equivariant Rabinowitz–Floer homology of a bounding con-tact hypersurface Σ in an exact symplectic manifold, and show by a geometric argumentthat it vanishes if Σ is displaceable.In the appendix we describe an approach to transversality for Floer homologies forwhich the moduli space c M J of all gradient flow lines is compact for some almost complexstructure J . This approach uses a large set of perturbations, namely vector fields on theloop space, and selects from the possibly non-compact perturbed moduli spaces a partnear c M J that turns out to be compact for small enough perturbations. Contents
1. Introduction 22. Recollections on Rabinowitz–Floer homology 42.1. The Rabinowitz action functional 52.2. Rabinowitz–Floer homology 63. Construction of equivariant Rabinowitz–Floer homology 73.1. The equivariant Rabinowitz action functional 73.2. Equivariant Rabinowitz–Floer homology 84. Proof of Theorem A 104.1. The perturbed Rabinowitz action functional, and leafwise intersections 114.2. The perturbed equivariant Rabinowitz action functional 114.3. Proof of Theorem A 145. Invariance 146. Other approaches 22Appendix on transversality 25Preface 25Appendix A. Transversality for Morse homology on finite-dimensional manifolds 26A.1. The set-up 26A.2. Perturbations 27A.3. Selection of compact perturbed moduli spaces 28A.4. The Hilbert manifold of lines 29A.5. The Hilbert bundle set-up 29
Date : October 19, 2018.
Key words and phrases. equivariant Rabinowitz–Floer homology, displaceable hypersurface.FS partially supported by SNF grant 200020-144432/1.2000
Mathematics Subject Classification.
Primary 53D40, Secondary 37J45, 53D35.
A.6. Finding smooth perturbed moduli spaces 30Appendix B. Transversality for Floer homology with one compact moduli space 34B.1. The Conley-type argument in the Floer case 34B.2. A compactness theorem for non-local perturbations of the Cauchy–Riemannoperator 36References 421.
Introduction
Consider a bounding contact hypersurface Σ in an exact convex symplectic manifold(
M, λ ). (Definitions are recalled in Section 2.) In this situation, Kai Cieliebak and thefirst author defined in [13] a homology group RFH(Σ , M ), the Rabinowitz–Floer homologyof Σ, as the Floer homology associated to the Rabinowitz action functional A F : L × R → R , ( v, η )
7→ − Z S v ∗ λ − η Z S F (cid:0) v ( t ) (cid:1) dt. Here, F : M → R is a suitable function with F − (0) = Σ, and S = R / Z denotes the circleand L = C ∞ ( S , M ) the free loop space of M . Note that the Rabinowitz action functionalis invariant under the circle action τ v ( · ) v ( · − τ ) obtained by rotating the loop v . Thismakes it possible to construct the equivariant Rabinowitz–Floer homology RFH S (Σ , M )as well.Recall that Σ is said to be Hamiltonian displaceable if there exists a compactly supportedHamiltonian diffeomorphism that disjoins Σ from itself. One of the most useful propertiesof the Rabinowitz–Floer homology of Σ is that it vanishes if Σ is displaceable. The mainresult of this note is that this fact continues to hold in the equivariant case. Theorem A.
Assume that Σ is Hamiltonian displaceable. Then RFH S (Σ , M ) = { } . We shall prove this result by a leafwise intersection argument, following [2]. A morealgebraic proof of Theorem A was given in [10] in the framework of symplectic homology,and their proof should also apply to Rabinowitz–Floer homology, cf. Section 6.The main body of this note is organized as follows. In Section 2 we recall the con-struction of non-equivariant Rabinowitz–Floer homology RFH(Σ , M ), and in Section 3 weconstruct S -equivariant Rabinowitz–Floer homology RFH S (Σ , M ). The core of this partis Section 4 in which we prove Theorem A. In Section 5 we give an alternative and some-what easier approach to the invariance of RFH S (Σ , M ). In Section 6 we briefly discussother approaches to proving RFH S (Σ , M ) = 0 for displaceable hypersurfaces. Transversality.
Consider a Floer theory on a symplectic manifold (
M, ω ) with actionfunctional A , fix an ω -compatible almost complex structure J on M , and let ∇ A be the L -gradient of A defined by J . Fix a < b and let Crit A ba be the critical points of A with action in [ a, b ]. Denote by G ba the space of all flow lines of −∇ A between elements -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 3 in Crit A ba , and by G ( c − , c + ) its subset of flow lines between two critical points c − , c + . Fora perturbation v of −∇ A that vanishes at Crit A ba , denote by G ba ( v ) the space of all flowlines of −∇ A + v between elements in Crit A ba , and by G ( c − , c + , v ) its subset of flow linesbetween c − , c + . The transversality problem in this situation is to show that for a genericchoice of perturbations v , the spaces G ( c − , c + , v ) are cut out transversally from “the spaceof all lines” from c − to c + , and are thus smooth manifolds. Geometric perturbations.
The traditional approach to achieve transversality is to choosea very small space of perturbations, namely the space of ω -compatible almost complexstructures on M , that may also depend on t ∈ S (and for RFH also on η ∈ R ), see [22, 8].The advantage of this approach is that if compactness of G ba holds, then it holds for allspaces G ba ( v ) for the same reason; its disadvantage is that one has few perturbations athand, which makes it sometimes very hard to achieve transversality. This approach hasbeen carried out for Rabinowitz–Floer homology in [1, § S N +1 -factor of the S -equivariant theory causes no additional problems. Polyfolds.
A more conceptual approach to transversality, that applies in very general sit-uations, is the theory of polyfolds [28, 29]. The group R acts on the spaces G ( c − , c + ) bytime shift, and the quotient G ( c − , c + ) / R of unparametrized gradient flow lines can be com-pactified to G ( c − , c + ) / R by adding the unparametrized broken gradient flow lines from c − to c + . While the space G ( c − , c + ) can be interpreted as the zero-set of a section from aHilbert manifold to a Hilbert bundle, this is not possible anymore for G ( c − , c + ) / R for tworeasons: First of all the R -action on G ( c − , c + ) is not smooth in the usual sense. How-ever, it is smooth in a new sense discovered by Hofer, Wysocki and Zehnder, namely scalesmooth. Scale smoothness does not require just one Hilbert manifold, but a whole scaleof Hilbert manifolds, and therefore leads to exciting interactions between Floer homologyand interpolation theory [45]. The second issue is the presence of broken gradient flowlines, which is an analytical limit phenomenon. However, the space of all unparametrizedbroken or unbroken lines from c − to c + can be interpreted as the fixed point set of a scalesmooth retraction. (This is in sharp contrast to the Hilbert set-up, where by the last the-orem of Cartan the fixed point set of a smooth retraction is itself a Hilbert manifold [11].)These two facts allow to interpret the moduli space G ( c − , c + ) / R as the zero-set of a Fred-holm section from an M-polyfold to an M-polyfold bundle. Here, M stands for “manifoldflavoured”, indicating that no orbifold technology is required. A detailed account of thisstory is currently written up by Albers and Wysocki [4].One can now directly perturb the Fredholm section from the M-polyfold to the M-polyfold bundle as in [29] to make it transverse to zero. Alternatively, arguing as in [17], byfinite-dimensional approximation one can write a compact zero-set of a Fredholm sectionfrom an M-polyfold to an M-polyfold bundle as the zero-set of a section from a finite-dimensional manifold to a finite-rank vector bundle over the manifold, and it is well-knownthat such a section can be made transverse to zero by a small perturbation. The differenceof the rank of the vector bundle and the dimension of the underlying manifold correspondsto the Fredholm index. In the language of Cieliebak–Mundet–Salamon one can think of URS FRAUENFELDER AND FELIX SCHLENK such a section as a finite-dimensional G -moduli problem for the trivial Lie group G ; in thelanguage of Fukaya–Ono [25] and Fukaya–Oh–Ohta–Ono [26], such a section correspondsto a global M-Kuranishi structure, namely a Kuranishi structure consisting of one chartwith no orbifold flavour.When applying this approach to RFH, it is sufficient to work with one J , that one is freeto choose independent of t and η . In this approach the elements of the abstractly perturbedmoduli spaces do not correspond to gradient flow lines anymore. That the gradient flowlines with cascades of RFH fit into the M-polyfolds set-up has not yet been worked out indetail. A Conley-type argument.
In the appendix we outline an intermediate approach to transver-sality, that is inspired by Conley index theory. It allows for more general perturbationsthan the traditional approach, namely non-local vector fields on the loop space L , but stillstays in the framework of gradient flow lines. Assume that one knows that G ba is compactfor some J . The larger class of non-local perturbations then makes it easy to achievetransversality. The danger is now that even for arbitrarily small perturbations the spaces G ba ( v ) become non-compact. We shall show, however, that for sufficiently small perturba-tions v one can select a compact part of G ba ( v ) near G ba : For a flow line x of −∇ A + v let ev( x ) = x (0) ∈ L be the evaluation at 0. Since G ba is compact, K := ev( G ba ) ⊂ L iscompact. Choose a bounded open neighbourhood N of K in L . Then it turns out thatfor sufficiently small perturbations v the part G ba ( v, N ) of G ba ( v ) that lies in N is compactand completely contained in N , that is, N is an isolating neighbourhood of the “flow”of −∇ A + v for every small enough v . One can thus use the isolated invariant sets G ba ( v, N )with regular v to define Floer homology. In Appendix A we describe this approach totransversality in detail in the framework of Morse homology on a finite-dimensional butpossibly non-compact manifold. The arguments are chosen in such a way that they trans-late to Floer homology (see § B.1), with the key difference that now the compactness ofthe selected components G ba ( v, N ) does not just follow from the Arzel`a–Ascoli theorem,because L is not locally compact. To remedy for this, we prove in § B.2 a compactnessresult for non-local perturbations of the Cauchy–Riemann operator.
Acknowledgments.
We thank Alberto Abbondandolo for pointing out the reference [34].We are grateful to the referees for valuable comments and suggestions.2.
Recollections on Rabinowitz–Floer homology
In this section we recall the construction of the (non-equivariant) Rabinowitz–Floerhomology of a hypersurface Σ of restricted contact type, following [13] and [2]. Our con-struction of equivariant Rabinowitz–Floer homology in the next section will be based onthis construction.Consider an exact convex symplectic manifold (
M, λ ). This means that λ is a one-formon the connected manifold M such that dλ is a symplectic form, and that ( M, dλ ) isconvex at infinity, i.e., there exists an exhaustion M = S k M k of M by compact subsets M k ⊂ M k +1 with smooth boundaries ∂M k such that λ | ∂M k is a contact form. We further -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 5 fix a closed connected smooth hypersurface Σ in M that is bounding and of contact type.The former means that M \ Σ has two components, one compact and one non-compact,and the latter means that λ | Σ is a contact form, or equivalently that the vector field Y λ implicitly defined by ι Y λ dλ = λ is transverse to Σ.For a smooth function F on M , the Hamiltonian vector field X F is defined by ι X F dλ = dF , and ϕ tF denotes the flow of X F . The Reeb flow ϕ tR on Σ is the flow of the vector field R defined by dλ ( R, · ) = 0 and λ ( R ) = 1.2.1. The Rabinowitz action functional. A defining Hamiltonian for Σ is a smoothfunction F : M → R such that Σ = F − (0), such that dF has compact support, and suchthat ϕ tF restricts on Σ to the Reeb flow ϕ tR of (Σ , λ | Σ ). The set of defining Hamiltonians isnon-empty and convex. Given a defining Hamiltonian F , the Rabinowitz action functional A F : L × R → R is defined by A F ( v, η ) = − Z S v ∗ λ − η Z S F (cid:0) v ( t ) (cid:1) dt. (1)Its critical points ( v, η ) are the solutions of the problem˙ v ( t ) = η X F ( v ( t )) , Z S F ( v ( t )) dt, i.e., pairs ( v, η ) with η ∈ R and v a closed curve on Σ of the form v ( t ) = ϕ ηtF , t ∈ R .The critical points therefore correspond to closed orbits of X F on the fixed energy surfaceΣ = F − (0) of arbitrary period | η | > Since v ⊂ Σ and ϕ tF = ϕ tR along Σ, A F ( v, η ) = − Z S v ∗ λ = − η, that is, the critical values of A F are zero and minus the periods of the closed Reeb orbitson Σ.The action functional A F is invariant under the S -action on L × R given by τ · (cid:0) v ( · ) , η (cid:1) (cid:0) v ( · − τ ) , η (cid:1) . (2)Therefore, the functional A F is never Morse. The component { ( p, | p ∈ Σ } ∼ = Σ of thecritical set is always Morse–Bott for A F , see [2, Lemma 2.12]. The following assumptionon Σ is sufficient for A F to be Morse–Bott: Every periodic orbit of the Reeb flow ϕ tR is non-degenerate. (3)In other words, for a T -periodic orbit γ of the Reeb flow, 1 is not in the spectrum of thelinearization T p ϕ TR : ξ p → ξ p at p = γ (0), where ξ = ker λ denotes the contact structure Despite J. Moser’s explicit statement that the action functional (1) is useless for finding periodic orbits,[32, p. 731], P. Rabinowitz in [35, p. 161 and (2.7)] used precisely this functional to prove his celebratedexistence theorem for periodic orbits on starshaped hypersurfaces in R n , thus pioneering the use of globalcritical point methods in Hamiltonian mechanics. In [13] and subsequent papers, the functional (1) wastherefore called Rabinowitz action functional. Other good names for this functional may be “fixed energyaction functional” or “Hamiltonian free period action functional”, since it selects solutions on the prescribedenergy level { H = 0 } , allowing for arbitrary period | η | . URS FRAUENFELDER AND FELIX SCHLENK of Σ. This holds if and only if for any defining Hamiltonian F of Σ, for every periodic orbitof ϕ tF on Σ the Floquet multiplier 1 has multiplicity 2.2.2. Rabinowitz–Floer homology.
Rabinowitz–Floer homology RFH(Σ , M ) is the Floerhomology of the functional A F , where F is any defining Hamiltonian for Σ. We assumethe reader to be familiar with the construction in [13], and also refer to [2] and to thesurvey [3]. Here, we only point out a few aspects in the construction of RFH(Σ , M ) thatdo not arise in the construction of usual Hamiltonian Floer homology.
1. The chain groups.
The functional A F is not Morse, but Morse–Bott. One thereforechooses an auxiliary Morse function h : Crit A F → R , and generates the chain groups by thecritical points of h . However, even though the symplectic form dλ is exact, the generatorsof the Rabinowitz–Floer chain groups FC( A F , h ) are not finite sums P ξ c c with ξ c ∈ Z and c ∈ Crit h , but possibly infinite sums P ξ c c that for every κ ∈ R satisfy the finitenesscondition (cid:8) c ∈ Crit h | ξ c = 0 , A F ( c ) > κ (cid:9) < ∞ . This must be done so for the following reason: Assume that c lies on the critical point( v, η ) of A F , with η = 0. Then A F ( v, η ) = − η . Since with ( v, η ) also ( v, kη ) belongs toCrit A F for each k ∈ Z , we see that A F is not bounded from below on Crit A F . Hencethere may be infinitely many critical points that appear in the image ∂c of the boundaryoperator.
2. The almost complex structures.
Let J con be the set of almost complex structureson M that are dλ -compatible and convex at infinity. The choice of the set of almost complexstructures used to define RFH(Σ , M ) depends on the method that one uses to establishtransversality (cf. the introduction). If one works with polyfolds or with the Conley-typeapproach explained in the appendix, one can take a fixed J ∈ J con . In the next paragraphwe describe the boundary operator in the traditional way. For this we fix J ∗ ∈ J con andfollowing [1] consider the set J of smooth S × R -families J = { J t ( · , η ) } ⊂ J con such thatsup t,η k J t ( · , η ) k C ℓ < ∞ for all ℓ ∈ N (4)and such that there exists a constant c > c k J ∗ ( x ) k k J t ( x, η ) k c k J ∗ ( x ) k for all x ∈ M and ( t, η ) ∈ S × R . (5)Here, k · k is the norm taken with respect to some background Riemannian metric on M .
3. The boundary operator.
The boundary operator ∂ on FC( A F , h ) is defined bycounting gradient flow lines with cascades (see [23, Appendix A]). These flow lines consistof (partial) negative gradient flow lines of h and finite energy Floer gradient flow linesof A F . Given a family J ∈ J and two critical points ( v − , η − ) and ( v + , η + ) of A F , a Floergradient flow line is a solution ( v, η ) ∈ C ∞ ( R × S , M × R ) of the problem ∂ s v ( s, t ) + J t (cid:0) v ( s, t ) , η ( s ) (cid:1)(cid:0) ∂ t v ( s, t ) − η ( s ) X F (cid:0) v ( t ) (cid:1) = 0 , ˙ η ( s ) + R S F (cid:0) v ( s, t ) (cid:1) dt = 0 , ) (6) -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 7 with asymptotic boundary conditions ( v − , η − ) and ( v + , η + ). The main analytical issue indefining the boundary operator ∂ is to prove a uniform L ∞ -bound on the η -component ofthe solutions of (6) with given boundary conditions. This is done in [13, Corollary 3.3] for η -independent J , and the proof goes through thanks to (5). Assumption (4) is imposedto avoid bubbling, so that the space of all solutions of (6) is C ∞ loc -compact. Transversalityfor the space of solutions of (6) between two critical points for a generic set of J ∈ J isproven in [1, § S -equivariant Rabinowitz–Floer homology described in the next section.The foundational work coming closest to the holomorphic curve set-up considered in thispaper is in [6, 7] and [40, § S is by verifying that the flow lines with cascades fit into the M -polyfold set-up (cf. theintroduction).
4. Invariance.
The resulting homology group FH( A F ) := ker ∂/ im ∂ does not dependon the choice of a defining function F for Σ. One can therefore define RFH(Σ , M ) :=FH( A F ) for any choice of F . Moreover, given two bounding contact hypersurfaces Σ and Σ that are isotopic through a family { Σ s } s of contact hypersurfaces,RFH(Σ , M ) ∼ = RFH(Σ , M ) . (7)For the proof, one chooses a smooth family F s : M → R of defining Hamiltonians for Σ s such that F s = F for s F s = F for s >
1, and uses solutions of (6) with F replacedby F s to construct a chain homotopy equivalence between FC( A F , h ) and FC( A F , h ).The main analytical issue is again proving a bound on the η -components, which can bedone as in [13, Corollary 3.4] thanks to (5).Recall that we have worked for now under the assumption (3). This assumption on Σ isgeneric in the C ∞ -topology. In view of (7) we can define the Rabinowitz–Floer homologyRFH(Σ , M ) of any bounding contact hypersurface as RFH(Σ ′ , M ) where Σ ′ is a close-byhypersurface meeting assumption (3).3. Construction of equivariant Rabinowitz–Floer homology
In this section we give a Borel-type construction of S -equivariant Rabinowitz–Floerhomology, closely following the construction of S -equivariant symplectic homology givenby Viterbo in [46, § The equivariant Rabinowitz action functional.
For each integer N > S N +1 the odd-dimensional unit sphere in C N +1 . The circle S acts on S N +1 by τ · ( z , . . . , z N +1 ) = ( τ z , . . . , τ z N +1 ) . The quotient of this action is complex projective space C P N = S N +1 /S . Recall theaction (2) of S on the loop space L , and let S act on L × R × S N +1 by the diagonal URS FRAUENFELDER AND FELIX SCHLENK action τ · (cid:0) v ( · ) , η, z (cid:1) = (cid:0) v ( · − τ ) , η, τ · z (cid:1) . (8)We shall denote the circle S with this action on L × R × S N +1 by T . Denote the quotientof this action by L × R × T S N +1 . The functional e A F,N ; T : L × R × S N +1 → R defined by e A F,N ; T ( v, η, z ) = − Z S v ∗ λ − η Z S F (cid:0) v ( t ) (cid:1) dt (9)is Morse–Bott if and only if the functional A F defined in (1) is Morse–Bott. Indeed, thecritical set of e A F,N ; T is the critical set of A F times S N +1 . Since the functional (9) isinvariant under the action (8), we can define the equivariant Rabinowitz action functional A F,N ; T : L × R × T S N +1 → R by A F,N ; T ([ v, η, z ]) = − Z S v ∗ λ − η Z S F (cid:0) v ( t ) (cid:1) dt, (10)and since the action (8) is free, this functional is Morse–Bott under the assumption (3)on Σ.3.2. Equivariant Rabinowitz–Floer homology. T -equivariant Rabinowitz–Floer ho-mology RFH T (Σ , M ) is the direct limit in N of the Floer homology of the functional A F,N ; T ,where F is any defining Hamiltonian for Σ.
1. The chain groups.
Fix a defining Hamiltonian F for Σ meeting assumption (3), andfix N ∈ N . Then e A F,N ; T is Morse–Bott, with critical manifolds the union of Σ ×{ }× S N +1 and C i × { k η i } × S N +1 , k ∈ Z \ { } , where each C i × { η i } is a circle of simple Reeb orbitsof period η i . Since the action of T on L × R × S N +1 is free,Crit A F,N ; T = Crit e A F,N ; T / T = Crit A F × T S N +1 is a closed manifold. Denote by g S N +1 the round Riemannian metric on S N +1 , andchoose a Riemannian metric g Σ on Σ and S -invariant Riemannian metrics g C i on C i .Then the Riemannian metric g N on Crit e A F,N ; T defined by g N | Σ ×{ }× S N +1 = g Σ ⊕ g S N +1 and g N | C i ×{ k η i }× S N +1 = g C i ⊕ g S N +1 is T -invariant, and hence descends to the Riemannianmetric g T N on Crit A F,N ; T . Choose a Morse function h N : Crit A F,N ; T → R such that the pair( h N , g T N ) is Morse–Smale (that is, the stable and unstable manifolds of the negative gradientflow of h N with respect to g T N intersect transversally). The chain group FC( A F,N ; T , h N )consists of Novikov sums P ξ c c with c ∈ Crit h N , as in Section 2.2.
2. The almost complex structures.
If one works with polyfolds or with the Conley-type approach, one can, again, just take a fixed J ∈ J con . Here, we again fix J ∗ ∈ J con and look at smooth S × S N +1 × R -families J = { J t,z ( · , η ) } ⊂ J con such thatsup t,z,η k J t,z ( · , η ) k C ℓ < ∞ for all ℓ ∈ N (11)and such that there exists a constant c > c k J ∗ ( x ) k k J t,z ( x, η ) k c k J ∗ ( x ) k for all x ∈ M and ( t, z, η ) ∈ S × S N +1 × R . (12) -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 9 Furthermore, we impose that the family J is S -invariant: J t + τ,τz ( · , η ) = J t,z ( · , η ) for all ( t, z, η ) ∈ S × S N +1 × R and τ ∈ S . (13)The space J S of all families J in J con satisfying (11), (12) and (13) is non-empty (sinceproperty (13) is obtained by averaging over S ) and contractible.
3. The boundary operator.
Let e h N : Crit e A F,N ; T → R be the lift of h N . Then e h N is Morse–Bott, with T -orbits as critical manifolds. Given two critical points c + , c − of h N ,denote by C + , C − the corresponding critical circles of e h N . Given J ∈ J S consider allgradient flow lines with cascades c M ( c + , c − ) from a point in C + to a point in C − . Here,the (partial) Morse flow lines are (partial) negative gradient flow lines of e h N on Crit e A F,N ; T with respect to g N , and the cascades (i.e., the Floer gradient flow lines) are finite energysolutions ( v, η, z ) ∈ C ∞ ( R × S , M × R × S N +1 ) of the problem ∂ s v ( s, t ) + J t,z ( s ) (cid:0) v ( s, t ) , η ( s ) (cid:1)(cid:0) ∂ t v ( s, t ) − η ( s ) X F (cid:0) v ( t ) (cid:1) = 0 , ˙ η ( s ) + R S F (cid:0) v ( s, t ) (cid:1) dt = 0 , ˙ z ( s ) + ∇ g S N +1 e h N ( z ( s )) = 0 . (14)Here, ∇ g S N +1 e h N ( z ) denotes the component of ∇ g N e h N ( z ) along T z S N +1 . Since g N and J are T -invariant, T freely acts on c M ( c + , c − ). The space c M ( c + , c − ) therefore decomposesas c M ( c + , c − ) = a c ∈ C + c M ( c, c − )where c M ( c, c − ) is the space of gradient flow lines with cascades from c ∈ C + with thelast gradient flow line of e h N converging to an arbitrary point in C − , and c M ( c + , c − ) / T ∼ = c M ( c, c − ) for any c ∈ C + . One shows as in [1, § J ∈ J S the spaces c M ( c, c − ) are smooth manifolds.The real numbers s ∈ R freely act by shift on each Floer gradient flow line in a gradientflow line with cascades in c M ( c + , c − ). The space M ( c + , c − ) ∼ = ` c ∈ C + M ( c, c − ) obtained bymodding out these R -actions is compact. The main point in the proof is, again, a uniform L ∞ -bound on the η -component of the solutions of (14) with given boundary conditions.Such a bound is obtained exactly as in [13, Corollary 3.3], thanks to (12).Now the boundary operator on FC( A F,N ; T , h N ) is defined by ∂ ( c + ) = X c − ν ( c + , c − ) c − where the sum runs over those c − for which M ( c + , c − ) / T ∼ = M ( c, c − ) is 0-dimensional andwhere ν ( c + , c − ) is the number mod 2 of elements in this space.
4. Invariance.
Let FH( A F,N ; T , h N , J ) := ker ∂/ im ∂ be the resulting homology groups.The inclusion S N +1 → S N +3 is T -equivariant. In particular, Crit A F,N ; T ⊂ Crit A F,N +1; T .Since g S N +3 restricts to g S N +1 on S N +1 , the Riemannian metric g N +1 restricts to g N on Crit e A F,N ; T . Given a Morse function h N on Crit A F,N ; T as above, we choose a Morse function h N +1 on Crit A F,N +1; T such that h N +1 extends h N , such that Crit h N ⊂ Crit h N +1 ,and such that the pair ( h N +1 , g T N +1 ) is Morse–Smale. Further, we choose the family J N +1 = J t,z ( · , η ) with z ∈ S N +3 such that it extends the family J N = J t,z ( · , η ) with z ∈ S N +1 .The chain complex FC( A F,N ; T , h N , J N ) is thus a subcomplex of FC( A F,N +1; T , h N +1 , J N +1 ).We thus obtain a homomorphism ι N : FH( A F,N ; T , h N , J N ) → FH( A F,N +1; T , h N +1 , J N +1 ) . (15)The groups FH( A F,N ; T , h N , J N ) do not depend on the choice of h N and J N , nor on the choiceof g Σ in the definition of g N , nor on the defining Hamiltonian F for Σ. This is proven byFloer continuation as in [13] (see also Section 5). These continuation isomorphisms com-mute with the inclusion homomorphisms in (15): Given another defining Hamiltonian F ′ and other choices h ′ N and J ′ N , there is a commutative diagramFH( A F,N ; T , h N , J N ) ∼ = (cid:15) (cid:15) ι N / / FH( A F,N +1; T , h N +1 , J N +1 ) ∼ = (cid:15) (cid:15) FH( A F ′ ,N ; T , h ′ N , J ′ N ) ι ′ N / / FH( A F ′ ,N +1; T , h ′ N +1 , J ′ N +1 ) . The direct limit RFH T (Σ , M ) := lim −→ FH( A F,N ; T , h N , J N ) (16)therefore only depends on Σ. In fact, RFH T (Σ , M ) is invariant under isotopies of boundingcontact hypersurfaces (cf. Section 2.2). Remarks 3.1. 1.
Our homology groups RFH T (Σ , M ) are not graded. We therefore donot need to assume that the first Chern class of ( M, dλ ) vanishes on π ( M ). Under thisassumption, the groups RFH T (Σ , M ) carry a Z -grading (with values in + Z ), cf. [13,Section 4]. The above construction of S -equivariant Rabinowitz–Floer homology should givethe same result as the construction in [9] which uses parametrized symplectic homology,when applied to the parameter space R × S N +1 : The difference in the construction is thatour parameter space R × S N +1 is not compact, and that we work with cascades instead ofsuitable perturbations of the Hamiltonian F . We expect that combining the constructionin [9] with the L ∞ -estimates on the η -component from [13, Section 3] leads to the samegroups RFH T (Σ , M ) in view of a version of the Correspondence Theorem 3.7 in [7].A construction of an S -equivariant Rabinowitz–Floer homology that stays within thesetting of S -equivariant symplectic homology was given recently in [18]. We expect thatalso this homology is isomorphic to RFH T (Σ , M ).4. Proof of Theorem A
In this section we prove our main result: RFH T (Σ , M ) = 0 if Σ is displaceable. For theproof, we first recall how the analogous result is proven in the non-equivariant case. Weshall apply the same method in the non-equivariant case. -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 11 The perturbed Rabinowitz action functional, and leafwise intersections.
Ithas been shown in [13] that RFH(Σ , M ) vanishes if Σ is displaceable. This result has beenreproved in [2] by a more geometric argument, in which the functional A F is perturbedto a functional whose critical points are leafwise intersections. While the argument in [13]can be useful in problems where the leafwise intersection argument does not help (such asproving the existence of a closed characteristic on a displaceable stable hypersurface [16]),we here apply the leafwise intersection argument from [2].A perturbation pair for the Rabinowitz action functional is a tuple( χ, H ) ∈ C ∞ (cid:0) S , [0 , ∞ ) (cid:1) × C ∞ (cid:0) M × S , R (cid:1) such that R S χ ( t ) dt = 1. For a perturbation pair, the perturbed Rabinowitz action func-tional A Fχ,H : L × R → R is defined by A Fχ,H ( v, η ) = − Z S v ∗ λ − η Z S χ ( t ) F (cid:0) v ( t ) (cid:1) dt − Z S H (cid:0) v ( t ) , t (cid:1) dt. (17)The critical points ( v, η ) of this perturbed action functional are the solutions of the system˙ v ( t ) = η χ ( t ) X F (cid:0) v ( t ) (cid:1) + X H (cid:0) v ( t ) , t (cid:1) , R S χ ( t ) F (cid:0) v ( t ) (cid:1) dt. ) (18)As noticed in [2], it is useful to look at special perturbation pairs: Definition 4.1.
A perturbation pair ( χ, H ) is called of
Moser type if there exists t ∈ S such that the time support of H lies in [ t , t + 1 /
2] and the support of χ lies in [ t − / , t ].The energy hypersurface Σ = F − (0) is foliated by its leaves L x = { ϕ tF ( x ) | t ∈ R } .Given a perturbation H as above, a point x ∈ Σ is called a leafwise intersection pointfor H if ϕ H ( x ) ∈ L x . The following lemma was observed in [2]. Lemma 4.2.
If a perturbation pair is of Moser type and ( v, η ) is a solution of (18) , then v ( t ) is a leafwise intersection point for H on Σ = F − (0) . The perturbed equivariant Rabinowitz action functional.
In order to showthat RFH T (Σ , M ) vanishes for displaceable Σ, we wish to apply the same method as in thenon-equivariant case.In the following S acts diagonally on S × S N +1 by τ ( · , z ) = ( ·− τ, τ z ), and S × S S N +1 is the quotient of S × S N +1 under this action. A perturbation triple is a triple( ψ, G, k ) ∈ C ∞ (cid:0) S × S S N +1 , [0 , ∞ ) (cid:1) × C ∞ (cid:0) M × S × S S N +1 , R (cid:1) × C ∞ (cid:0) C P N , R (cid:1) such that for every z ∈ S N +1 , Z S ψ (cid:0) [ t, z ] (cid:1) dt = 1 , (19)and such that k is a Morse function on C P N . For a perturbation triple we define theperturbed equivariant Rabinowitz action functional A ψ,G,k := A F,N ; T ψ,G,k : L × R × T S N +1 → R (20) by A ψ,G,k (cid:0) [ v, η, z ] (cid:1) = − Z S v ∗ λ − η Z S ψ (cid:0) [ t, z ] (cid:1) F (cid:0) v ( t ) (cid:1) dt − Z S G (cid:0) v ( t ) , [ t, z ] (cid:1) dt − k (cid:0) [ z ] (cid:1) . Denote by e ψ ∈ C ∞ (cid:0) S × S N +1 , [0 , ∞ ) (cid:1) , e G ∈ C ∞ ( M × S × S N +1 , R ) , e k ∈ C ∞ ( S N +1 , R )the lifts of ψ , G and k . We can then write the lift of A ψ,G,k to L × R × S N +1 as A e ψ, e G, e k ( v, η, z ) = − Z S v ∗ λ − η Z S e ψ ( t, z ) F (cid:0) v ( t ) (cid:1) dt − Z S e G (cid:0) v ( t ) , t, z (cid:1) dt − e k ( z ) . (21)The critical points ( v, η, z ) of A e ψ, e G, e k are the solutions of the system˙ v ( t ) = η e ψ ( t, z ) X F (cid:0) v ( t ) (cid:1) + X e G (cid:0) v ( t ) , t (cid:1) , R S e ψ ( t, z ) F (cid:0) v ( t ) (cid:1) dt, η R S F (cid:0) v ( t ) (cid:1) ∂ z e ψ ( t, z ) dt − R S ∂ z e G (cid:0) v ( t ) , t, z ) dt − d e k ( z ) . (22) Definition 4.3.
A perturbation triple ( ψ, G, k ) is called admissible if the following twoconditions hold.(i) For each z ∈ S N +1 and each solution ( v, η ) of equation (18) with respect to theperturbation ( e ψ z , e G z ) the identity F (cid:0) v ( t ) (cid:1) d e ψ t ( z ) = 0 holds for all t ∈ S .(ii) | d e G x,t ( z ) ˆ z | < | d e k ( z ) ˆ z | for all z / ∈ Crit e k , ˆ z = 0 ∈ T z S N +1 , ( x, t ) ∈ M × S . Lemma 4.4.
Assume that ( ψ, G, k ) is an admissible perturbation triple. Then criticalpoints [ v, η, z ] of A ψ,G,k have the property that [ z ] is a critical point of k , and for each z ∈ S N +1 over [ z ] , the pair ( v, η ) is a solution to equation (18) for the perturbation ( e ψ z , e G z ) .Proof. In view of the first two equations in (22), we see that ( v, η ) is a solution of (18) forthe perturbation ( e ψ z , e G z ). It remains to show that [ z ] is a critical point of k . In view ofthe last equation in (22), for every ˆ z ∈ T z S N +1 the equation η Z S F (cid:0) v ( t ) (cid:1) d e ψ t ( z ) ˆ z dt + Z S d e G v,t ( z ) ˆ z dt + d e k ( z ) ˆ z = 0has to be met. By assertion (i) of Definition 4.3, the first term vanishes. Now assertion (ii)implies d e k ( z ) ˆ z = 0, hence [ z ] is a critical point of k . (cid:3) Definition 4.5.
Given a perturbation pair of Moser type ( χ, H ), we call a perturbationtriple ( ψ, G, k ) an equivariant extension of ( χ, H ) if the following conditions hold.(I) The perturbation triple ( ψ, G, k ) is admissible. -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 13 (II) For every z ∈ Crit e k there exists t z ∈ S such that for every t ∈ S and every x ∈ M the identities e G ( x, t, z ) = H ( x, t + t z ) and e ψ ( t, z ) = χ ( t + t z ) hold true. Lemma 4.6.
For any perturbation pair ( χ, H ) of Moser type, there exists an equivariantextension.Proof. Choose a Morse function k on C P N . For every y ∈ Crit k choose open neighborhoods y ∈ U y ⊂ U y ⊂ V y ⊂ V y ⊂ W y with the property that W y is contractible, and for different critical points y and y ′ of k theneighborhoods W y and W y ′ are disjoint. Since W y is contractible, the principal S -bundle π : S N +1 → C P N can be trivialized over W y . We abbreviate X = [ y ∈ Crit k π − ( W y )and choose a trivialization Φ : X → π ( X ) × S . We further choose smooth cutoff functions β , β ∈ C ∞ ( C P N , [0 , y ∈ Crit k , β | U y = 1 , β | V y = 1and supp ( β ) ⊂ [ y ∈ Crit k V y , supp ( β ) ⊂ [ y ∈ Crit k W y . We further abbreviate by p : X × S → S the projection to the second factor. We now set e G ( x, t, z ) = (cid:26) β ([ z ]) H (cid:0) x, t + p (Φ( z )) (cid:1) , z ∈ X, , z / ∈ X. and e ψ ( t, z ) = (cid:26) β ([ z ]) χ (cid:0) t + p (Φ( z )) (cid:1) + 1 − β ([ z ]) , z ∈ X, , z / ∈ X. Define G and ψ by G ( x, [ t, z ]) = e G ( x, t, z ) and ψ ([ t, z ]) = e ψ ( t, z ). Then the perturbationtriple ( ψ, G, k ) satisfies condition (II) of an equivariant extension. Moreover, since theperturbation pair ( χ, H ) is of Moser type, the triple ( ψ, G, k ) also meets condition (i) ofadmissibility. It does not necessarily satisfy condition (ii) of admissibility. However, wecan remedy this by replacing k by Ck for a large enough positive constant C . This finishesthe proof of the lemma. (cid:3) Proof of Theorem A.
Assume that Σ is displaceable in M , and choose a definingHamiltonian F : M → R for Σ meeting assumption (3). In view of the definition (16) ofRFH T (Σ , M ), it suffices to show that FH( A F,N ; T ) = 0 for each N . So fix N ∈ N .Choose χ : S → [0 , ∞ ) with supp ( χ ) ⊂ (0 , ) and R S χ ( t ) dt = 1, and choose a Hamil-tonian function H : M × S → R with H ( · , t ) = 0 for all t ∈ [0 , ] whose time 1-flow ϕ H displaces Σ. By Lemma 4.6, the pair ( χ, H ) has an equivariant extension ( ψ, G, k ). Let[ v, η, z ] be a critical point of A ψ,G,k . Choose z ∈ S N +1 over [ z ]. By Lemma 4.4 and by (II)of Definition 4.5, ˙ v ( t ) = η χ ( t + t z ) X F (cid:0) v ( t ) (cid:1) + X H ( · ,t + t z ) (cid:0) v ( t ) , t (cid:1) , R S χ ( t + t z ) F (cid:0) v ( t ) (cid:1) dt. ) By Lemma 4.2, v ( t z ) is a leafwise intersection point for H ( · , t + t z ). This is impossiblebecause ϕ H displaces Σ. It follows that the functional A ψ,G,k = A F,N ; T ψ,G,k has no criticalpoints. The Floer homology FH( A F,N ; T ψ,G,k ) is defined along the lines of Section 3.2, see Sec-tion 5. Since A F,N ; T ψ,G,k has no critical points, the Floer complex of A F,N ; T ψ,G,k is trivial, and henceFH( A F,N ; T ψ,G,k ) = 0. Theorem A thus follows from the invariance FH( A F,N ; T ψ,G,k ) ∼ = FH( A F,N ; T ),which is proven in the next section. 5. Invariance
The goal of this section is to prove
Proposition 5.1.
FH( A F,N ; T ψ,G,k ) ∼ = FH( A F,N ; T ) . This isomorphism can be proven along the lines of the proof of Corollary 3.4 in [13]. Inthis section we give a different proof.We start with reviewing two continuation methods for showing invariance of a Floer-typehomology. For simplicity, we describe these methods in the setting of Morse homology andMorse–Bott homology on a non-compact manifold M . For i = 0 , f i : M → R besmooth Morse functions with compact critical sets Crit f i . Method 1.
Assume that there is a smooth family { f s } s of Morse functions f s : M → R such that the critical sets Crit f s are all isotopic. More precisely, assume that there is adiffeomorphismΨ : Crit f × [0 , → a s Crit f s × { s } , (cid:0) x, s (cid:1) (cid:0) x s , s (cid:1) . For a Riemannian metric j s on M and for x s , y s ∈ Crit f s denote by c M f s ,j s ( x s , y s ) the setof negative gradient flow lines from x s to y s , and by M f s ,j s ( x s , y s ) := c M f s ,j s ( x s , y s ) / R thespace of unparametrized gradient flow lines.For s = 0 , j s on M such that the pair ( f s , j s ) is Morse–Smale. Then one can define the Morse homology of f s by counting elements of M f s ,j s ( x s , y s )for x s , y s ∈ Crit f s with ind( x s ) = ind( y s ) + 1, s = 0 ,
1. For a generic smooth path of -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 15 Riemannian metrics { j s } from j to j and for x, y ∈ Crit f with ind( x ) = ind( y ) + 1, theunion of moduli spaces M { f,j } ( { x } , { y } ) = [ s M f s ,j s (cid:0) x s , y s (cid:1) × { s } is then a 1-dimensional smooth manifold with boundary that is “transverse at 0 and 1”, i.e.,for s = 0 , M f s ,j s (cid:0) x s , y s (cid:1) × { s } belong to the boundary of M { f,j } ( { x } , { y } ),see Figure 1. If one can show that the sets c M f s ,j s (cid:0) x s , y s (cid:1) , 0 s
1, are uniformlybounded, it follows that the Morse homologies of f and of f are isomorphic.PSfrag replacements0 1 s M f s ,j s (cid:0) x s , y s (cid:1) Figure 1.
The union of moduli spaces S s M f s ,j s (cid:0) x s , y s (cid:1) × { s } .Indeed, M { f,j } ( { x } , { y } ) is the union M ` M of two types of components: Thecomponents of M are compact intervals with boundary over 0 and 1, and the componentsof M are half-open intervals (with boundary over 0 or 1) or open intervals. If M is empty,then the coefficients ν ( x i , y ki ) = M f i ,j i ( x i , y ki ) mod 2 in the boundary operator ∂ i x i = X k ν ( x i , y ki ) y ki are the same for i = 0 ,
1. The components of M may change the coefficients ν ( x i , y ki ), butthey do not alter the Morse homology. Indeed, the contribution of the components of M to the boundary operator can be computed explicitely, and from this one can write downan explicit chain homotopy equivalence between the Morse chain complexes of ( f , j ) and( f , j ), see [20, Lemmata 3.5 and 3.6]. We illustrate this by an example:Suppose Crit f s has three critical points, a s , b s of index 1 and c s of index 0. Supposethat at s = 0 there is exactly one gradient flow line γ , from b to c . Then the Morsehomology is generated by a :MH( f , j ) = MH ( f , j ) = Z h a i . Assume now that at some time s ∗ ∈ (0 ,
1) a gradient flow line γ ab from a s ∗ to b s ∗ appears.This flow line is not generic, and immediately disappears. The flow line γ ab affects the twofamilies of moduli spaces M f s ,j s ( b s , c s ) and M f s ,j s ( a s , c s ) as follows: The moduli spaces M f s ,j s ( b s , c s ) are not affected: Before time s ∗ this space contains exactly one gradient flowline γ s , which persists beyond time s ∗ . PSfrag replacements a b c a b c a s ∗ b s ∗ c s ∗ γ γ γ s ∗ γ ab Figure 2.
The gradient flow lines at s = 0, s = s ∗ , s = 1.The moduli spaces M f s ,j s ( a s , c s ) were empty for s < s ∗ . At time s ∗ there is a brokengradient flow line from a s ∗ to c s ∗ , namely γ ab followed by the gradient flow line γ s ∗ from b s ∗ to c s ∗ . These two flow lines can be glued together to a unique gradient flow line from a s to c s . Hence ν ( a s , c s ) changes at s ∗ from 0 to 1. For s > s ∗ we now have one gradientflow line from a s to c s and one from b s to c s . But this change does not affect the Morsehomology: c s is still in the image of the boundary operator ∂ s , and while now neither a s nor b s are in the kernel, a s − b s is in the kernel of ∂ s . Hence we still haveMH( f , j ) = MH ( f , j ) = Z h a − b i . PSfrag replacements M f s ,j s ( b s , c s ) M f s ,j s ( a s , c s )0 0 11 s ∗ s ∗ γ s ∗ A bifurcation as above, that creates a component in M , is called a slide bifurcation,or a handle slide, since such a bifurcation acts on the corresponding handle decompositionof M by sliding one handle over another. The other type of bifurcation that appears ina generic isotopy between Morse functions are birth bifurcations and death bifurcations,namely the birth of two critical points or the cancellation of two critical points. Suchbifurcations do not arise in the situation at hand.Below we shall apply this method in a Morse–Bott set-up: Assume there is a smoothfamily { f s } s of Morse–Bott functions f s : M → R with compact critical sets Crit f s -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 17 and a diffeomorphismΨ : Crit f × [0 , → a s Crit f s × { s } , (cid:0) x, s (cid:1) (cid:0) x s , s (cid:1) . Choose a Morse function h on Crit f . Then the functions h s ( x s ) := h ( x )are Morse functions on Crit f s , and the sets Crit h s are isotopic. For a Riemannian met-ric g s on Crit f s , for a Riemannian metric j s on M and for x s , y s ∈ Crit h s denote by c M f s ,j s ,h s ,g s ( x s , y s ) the set of negative gradient flow lines with cascades from x s to y s , andby M f s ,j s ,h s ,g s ( x s , y s ) the space of unparametrized gradient flow lines with cascades.For s = 0 , g s on Crit f s such that the pair ( h s , g s ) isMorse–Smale. For generic Riemannian metrics j s on M one can then define the Morse–Bott homology of the quadruples ( f s , j s , h s , g s ), s = 0 ,
1, by counting elements of the0-dimensional components M f s ,j s ,h s ,g s ( x s , y s ), see [23, Appendix A]. For a generic smoothpath of Riemannian metrics { g s } on Crit f s and for a generic smooth path of Riemannianmetrics { j s } on M from ( j , g ) to ( j , g ), we have that for each pair x, y ∈ Crit h forwhich M f ,j ,h ,g ( x, y ) is 0-dimensional, the union of moduli spaces M { f,j,h,g } ( { x } , { y } ) = (cid:8) ( u, s ) | u ∈ M f s ,j s ,h s ,g s (cid:0) x s , y s (cid:1) ; 0 s (cid:9) , is a 1-dimensional smooth manifold with boundary that is “transverse at 0 and 1”. If onecan show that the sets c M f s ,j s ,g s ,h s (cid:0) x s , y s (cid:1) , 0 s
1, are uniformly bounded, it followsthat the Morse homologies of f and of f are isomorphic. Method 2 (Floer continuation).
Choose a smooth monotone function β : R → [0 , β ( s ) = 0 for s β ( s ) = 1 for s >
1. For s ∈ R define the function f s = (1 − β ( s )) f + β ( s ) f . For x ∈ Crit f and y ∈ Crit f and for a smooth family of Riemannian metrics { g s } with g s = g for s g s = g for s > ( ˙ u ( s ) = −∇ g s f s ( u ( s )) , s ∈ R ;lim s →−∞ u ( s ) = x, lim s →∞ u ( s ) = y. (23)For a generic choice of the path { g s } and for x ∈ Crit f and y ∈ Crit f with ind( x ) =ind( y ), the space of solutions to (23) is a smooth 0-dimensional manifold. If one canshow that this space is bounded, then it is finite. Counting these solutions then definesa chain homomorphism between the Morse chain complexes of f and f , that induces anisomorphism between the Morse homologies of f and f .Similarly, given triples ( j s , h s , g s ) for s = 0 , h s , g s ) Morse–Smale pairs and j s generic, Floer continuation can be used to show that the Morse homologies of ( f , j , h , g )and ( f , j , h , g ) are isomorphic, see [23, Theorem A.17]. Historical Remark.
Floer used Method 1 in [20] to prove invariance of his homologyfor Lagrangian intersections. (He also dealt with isolated bifurcations of the critical sets, namely birth and death bifurcations, by first putting them into normal form and thenconstructing a chain map between the complex before and after the bifurcation that inducesan isomorphism in homology.) Such a bifurcation analysis was later also used in [19, 30,44].) The powerful and flexible Method 2 was invented by Floer only later in [21]. ♦ Proposition 5.1 can be proven by Method 2, by adapting the proof of Corollary 3.4in [13]. We leave the minor modifications to the interested reader. Here we give a differentargument, that takes into account the structure of the functional A F,N ; T ψ,G,k , and uses Method 1once and Method 2 twice.Consider the four functionals on L × R × T S N +1 , A ([ v, η, z ]) = − Z S v ∗ λ − η Z S F (cid:0) v ( t ) (cid:1) dt, A ([ v, η, z ]) = − Z S v ∗ λ − η Z S F (cid:0) v ( t ) (cid:1) dt − e k ( z ) , A ([ v, η, z ]) = − Z S v ∗ λ − η Z S e ψ ( t, z ) F (cid:0) v ( t ) (cid:1) dt − e k ( z ) , A ([ v, η, z ]) = − Z S v ∗ λ − η Z S e ψ ( t, z ) F (cid:0) v ( t ) (cid:1) dt − Z S e G (cid:0) v ( t ) , t, z (cid:1) dt − e k ( z ) . The functionals A and A are Morse–Bott by our assumption (3) on Σ and since k isMorse, while A is Morse–Bott by Lemma 5.2 below. The functional A is Morse–Bottbecause it has no critical points. Hence the four lifted functionals e A i : L × R × S N +1 → R are also Morse–Bott.The Floer homology FH( A ) = FH( A , h , J ) was defined in Section 3.2, and the Floerhomology FH( A i ) for i = 1 , , h i and a Riemannian metric g i on Crit A i such that ( h i , g i ) is a Morse–Smale pair, liftsthem to the Morse–Bott function e h i and the T -invariant metric e g i on Crit e A i , and definesthe boundary of a critical point c + of h i by counting rigid T -families of unparametrized neg-ative gradient flow lines with cascades in M ( c + , c − ) between critical T -orbits C + and C − of e h i , with respect to the T -invariant Riemannian metric e g i on Crit e A i and a generic fam-ily J t,z ( · , η ) in J S .It follows from Method 2 that FH( A ) ∼ = FH( A ) and that FH( A ) ∼ = FH( A ). This iseasy for the passage A A : The summand e k ( z ) is bounded with all its derivatives. Theclaim thus follows from the L ∞ -bound on each space c M ( c + , c − ) of gradient flow lines withcascades between a pair of critical circles of e h given in the proof of Corollary 3.3 in [13].For the passage A A , invariance follows as in [2, Section 2], by either choosing e G sufficiently small in L ∞ (which we are free to do) or by decomposing the isotopy A A into many small isotopies.The isomorphism FH( A ) ∼ = FH( A ) can also be shown by applying Method 2 to theparts of a sufficiently fine decomposition of the isotopy A A (see the proof of Corol-lary 3.4 in [13]). This argument is somewhat harder, since η appears in front of the -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 19 summand that is altered. We circumvent this difficulty by applying Method 1. Choose asmooth monotone function β : [1 , → [0 ,
1] with β ( s ) = 0 for s near 1 and β ( s ) = 1 for s near 2. For s ∈ [1 ,
2] set e ψ s ( t, z ) = (cid:0) − β ( s ) (cid:1) · β ( s ) · e ψ ( t, z ) = 1 + β ( s ) (cid:0) e ψ ( t, z ) − (cid:1) . Then R S e ψ s ( t, z ) dt = 1 for all s . Consider the family of functionals e A s ( v, η, z ) := − Z S v ∗ λ − η Z S e ψ s ( t, z ) F ( v ( t )) dt + e k ( z ) , s . Then e A s = e A for s near 1 and e A s = e A for s near 2.The critical manifolds Crit e A s are in canonical bijection with Crit e A . Indeed, lookingat (22) with e G = 0 and e ψ replaced by e ψ s , we see that they all contain Σ × { } × Crit e k .Moreover, given z ∈ Crit e k , and with s z ( t ) := Z t e ψ s ( τ, z ) dτ, the periodic orbit ( v ( t ) , η, z ) of X F with period | η | corresponds to the reparametrized orbit( v ( s z ( t )) , η, z ) of e ψ s ( t, z ) X F with period | η | . (The orbit v ( s z ( t )) also has period | η | because s z (1) = 1.) More formally, the reparametrization map e Ψ : Crit e A × [1 , → a s Crit e A s × { s } , (cid:0) ( v ( · ) , η, z ) , s (cid:1) (cid:0) ( v ( s z ( · )) , η, z ) , s (cid:1) is a diffeomorphism. Lemma 5.2.
For each s ∈ [1 , the critical set Crit e A s is a Morse–Bott submanifold of e A s . Before giving the proof, we use the lemma to prove Proposition 5.1. All the function-als e A s and all the sets Crit e A s are T -invariant. Choose a Morse function h on Crit A .Then the functions h s ([ v ( s z ( · )) , η, z ]) := h ([ v ( · ) , η, z ])are Morse functions on Crit A s , and the sets Crit h s are isotopic.For a Riemannian metric g s on Crit A s , for a family J s := ( J t,z ( · , η )) s in J S and for c + s , c − s ∈ Crit h s , denote by c M A s , J s ,h s ,g s ( c + s , c − s ) the set of negative gradient flow lines withcascades from c + s to c − s , and by M A s , J s ,h s ,g s ( c + s , c − s ) the space of unparametrized T -familiesof gradient flow lines with cascades, as constructed in Section 3.2.3.For s = 1 , g s and J s as in the definition of the Floer homologies FH( A s ): g s is aRiemannian metric on Crit A s such that ( h s , g s ) is a Morse–Smale pair, and J s is a genericfamily in J S . Then for a generic smooth path of Riemannian metrics { g s } on Crit A s and for a generic smooth path of families { J s } in J S from ( g , J ) to ( g , J ), we havethat for each pair c + , c − ∈ Crit h for which M A , J ,h ,g ( c + , c − ) is 0-dimensional, the unionof moduli spaces M A , J ,h,g ( { c + } , { c − } ) = [ s M A s , J s ,h s ,g s (cid:0) c + s , c − s (cid:1) × { s } is a 1-dimensional smooth manifold that is “transverse at 0 and 1”. Notice that the map e Ψ is action-preserving: e A s ( x s ) = e A ( x ). The space c M A , J ,h ,g ( c + , c − ) is L ∞ -bounded,and in fact there is a uniform L ∞ -bound on the spaces c M A s , J s ,h s ,g s (cid:0) c + s , c − s (cid:1) , 1 s
2, seethe proof of Corollary 3.3 in [13]. It follows that FH( A ) ∼ = FH( A ). Proof of Lemma 5.2.
We use the method in Appendix A.1 of [2]. Fix s , and fix a criticalpoint ( v , η , z ) ∈ L × R × S N +1 . We decompose e A s as e A s ( v, η, z ) = A ( v ) + η F ∆ ψ ( v, z ) + ( η − η ) F ψ ( v, z ) + e k ( z )where A ( v ) := − Z S v ∗ λ − η Z S e ψ s ( t, z ) F ( v ( t )) dt, F ∆ ψ ( v, z ) := Z S (cid:16) e ψ s ( t, z ) − e ψ s ( t, z ) (cid:17) F ( v ( t )) dt, F ψ ( v, z ) := Z S e ψ s ( t, z ) F ( v ( t )) dt. In order to compute the Hessian of e A s at ( v , η , z ), we apply “a change of coordinates”:Consider the twisted loop space L η F := (cid:8) w ∈ C ∞ ([0 , , M ) | w (0) = φ η F ( w (1)) (cid:9) and the diffeomorphism Φ η F : L η F → L = C ∞ ( S , M ) given byΦ η F ( w )( t ) = φ tη F s ( w ( t ))where we abbreviated F s ( · ) := e ψ s ( t, z ) F ( · ). Then the path w = Φ − η F ◦ v = v (0) ∈ Σ isconstant. Hence tangent vectors ˆ w ( t ) at w are curves in the linear space T w M withˆ w (1) = dφ − η F s ( w ) ˆ w (0) . (24)We are going to compute the kernel of the Hessian of the pulled-back functional A Φ s := (Φ η F × id R × id S N +1 ) ∗ e A s : L η F × R × S N +1 → R at the critical point ( w , η , z ). First notice that Φ ∗ η F d A ( w )[ ˆ w ] = R ω ( ddt w, ˆ w ) dt for any w ∈ L η F and ˆ w ∈ T w L η F , and that(Φ η F × id S N +1 ) ∗ F ∆ ψ ( w, z ) = Z (cid:16) e ψ s ( t, z ) − e ψ s ( t, z ) (cid:17) F ( w ( t )) dt, (Φ η F × id S N +1 ) ∗ F ψ ( w, z ) = Z e ψ s ( t, z ) F ( w ( t )) dt -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 21 (since F is preserved under φ tη F s ). The differential of A Φ s therefore is d A Φ s ( w, η, z )[ ˆ w, ˆ η, ˆ z ] = Z ω ( ddt w, ˆ w ) dt + η Z n(cid:16) e ψ s ( t, z ) − e ψ s ( t, z ) (cid:17) dF ( w ( t )) ˆ w ( t ) − ∂ z e ψ s ( t, z ) ˆ z F ( w ( t )) o dt − ˆ η Z e ψ s ( t, z ) F ( w ( t )) dt +( η − η ) Z e ψ s ( t, z ) dF ( w ( t )) ˆ w ( t ) + ∂ z e ψ s ( t, z ) ˆ z F ( w ( t )) dt + d e k ( z )ˆ z. At the critical point x := ( w , η , z ) the Hessian of A Φ s applied to ξ i := ( ˆ w i , ˆ η i , ˆ z i ) thereforeisHess A Φ s ( x )[ ξ , ξ ] = Z ω ( ddt ˆ w , ˆ w ) dt − η Z n ∂ z e ψ s ( t, z ) ˆ z dF ( w ) ˆ w ( t ) + ∂ z e ψ s ( t, z ) ˆ z dF ( w ) ˆ w ( t ) o dt − ˆ η Z e ψ s ( t, z ) dF ( w ) ˆ w ( t ) dt − ˆ η Z e ψ s ( t, z ) dF ( w ) ˆ w ( t ) dt +Hess e k ( z )(ˆ z , ˆ z )where we have used that F ( w ) = 0. A tangent vector ( ˆ w, ˆ η, ˆ z ) therefore belongs to thekernel of Hess A Φ s ( w , η , z ) if and only if0 = ddt ˆ w ( t ) − ˆ η e ψ s ( t, z ) X F ( w ) − η ∂ z e ψ s ( t, z ) ˆ z X F ( w ) , (25)0 = Z e ψ s ( t, z ) dF ( w ) ˆ w ( t ) dt, (26)0 = − η Z dF ( w ) ˆ w ( t ) ∂ z e ψ s ( t, z )( · ) dt + Hess e k ( z )(ˆ z, · ) . (27)Denote by H z = { τ z | τ ∈ S } the Hopf circle in S N +1 through z .Assume first that η = 0. Then ( v , η , z ) belongs to the critical component Σ ×{ }× H z of “constant in Σ loops”. Since η = 0, (27) yields ˆ z ∈ T z H z , and integrating (25) yieldsˆ w (1) = ˆ w (0) + ˆ η X F ( w )(since s z (1) = 1). Since in this case Φ η F : L → L is the identity mapping, ˆ w (1) = ˆ w (0),and so ˆ η = 0. By now, (25) reads ddt ˆ w ( t ) = 0, that is, ˆ w ( t ) ≡ ˆ w (0) ∈ T w M is constant. Finally, (26) shows that ˆ w (0) ∈ T w Σ. The kernel of the Hessian of e A s at ( v , η , z ) =( w , , z ) is thus identified with T w Σ × T z H z .Assume now that η = 0. Then S v := { v ( · − τ ) | τ ∈ S } is an embedded circle in L .Hence the critical component of ( v , η , z ) is the torus S v × { η } × H z . It is clear thatthe kernel of the Hessian of e A s at ( v , η , z ) has dimension at least two, and we mustshow that the dimension is two. By assumption (3), 1 has multiplicity 2 in the spectrumof dφ − η F ( w ). Since φ η F s = φ η F , the same holds true for L s := dφ − η F s ( w ). Recall that s z ( t ) = R t e ψ s ( τ, z ) dτ . Integrating (25) we getˆ w ( t ) = ˆ w (0) + ˆ η s z ( t ) X F ( w ) + η ∂ z (cid:12)(cid:12) z s z ( t ) ˆ z X F ( w ) . (28)In particular (since s z (1) = 1 for all z ), and by (24),ˆ w (1) = ˆ w (0) + ˆ η X F ( w ) = L s ˆ w (0) . (29)Consider the sub-vector space V of T w M spanned by ˆ w (0) and X F ( w ). Assume that V is 2-dimensional. Then (29) and the fact that 1 has multiplicity 2 in the spectrum of L s show that V is the whole 1-eigenspace of L s . In particular, V is symplectic. On the otherhand, since dF ( X F ) = 0, equations (28) and (26) show that dF ( w ) ˆ w (0) = Z e ψ s ( t, z ) dF ( w ) ˆ w (0) dt = Z e ψ s ( t, z ) dF ( w ) ˆ w ( t ) dt = 0and hence ˆ w (0) ∈ T w Σ. Since X F ( w ) generates the kernel of ω | T w Σ , this contradicts V being symplectic.It follows that ˆ w (0) = r X F ( w ) for some r ∈ R . In particular, L s ˆ w (0) = ˆ w (0). Thesecond equation in (29) thus shows that ˆ η = 0. Since ˆ w (0) ∈ V = span ( X F ( w )), equa-tion (28) shows that ˆ w ( t ) ∈ V for all t . Therefore (27) gives ˆ z ∈ ker Hess e k ( z ) = T z H z .We conclude with (28) that the kernel of Hess A Φ s ( w , η , z ) is { ( ˆ w ( t ) , , ˆ z ) | ˆ z ∈ T z H z } = n(cid:0) r + η ∂ z (cid:12)(cid:12) z s z ( t ) ˆ z (cid:1) X F ( w ) , , ˆ z (cid:1) | r ∈ R , ˆ z ∈ T z H z o . Hence dim ker Hess A Φ s ( w , η , z ) = dim ker Hess e A s ( v , η , z ) = 2. (cid:3) Other approaches
In this note we have defined T -equivariant Rabinowitz–Floer homology RFH T (Σ , M )via the Borel construction and Floer homology with cascades, and we have proven thevanishing of RFH T (Σ , M ) for displaceable Σ by a leave-wise intersection argument. Thereare several other approaches to construct a T -equivariant Rabinowitz–Floer homology (twoare mentioned in Remark 3.1, and one more is outlined in 3. below), all of which areexpected to give the same result. And there are different ways to prove the vanishing ofRFH T (Σ , M ) or of these other versions for displaceable Σ. In particular, the arguments in1. and 2. below imply the vanishing of the version defined in [18], see items (4) and (3) onpage 70 of [18]. Let V be the bounded component of M \ Σ, and denote by SH ∗ ( V ) itssymplectic homology and by SH T ∗ ( V ) its equivariant symplectic homology. -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 23
1. Vanishing of
RFH T (Σ , M ) via vanishing of SH T ( V ) . There should be a T -equivariant version of the long exact sequence · · · −→ SH −∗ ( V ) −→ SH ∗ ( V ) −→ RFH ∗ (Σ , M ) −→ SH −∗ +1 ( V ) → · · · from [15]. The vanishing of RFH T (Σ , M ) for displaceable Σ would then follow from thevanishing of SH T ( V ) proven in [10].
2. Vanishing of
RFH T (Σ , M ) via vanishing of RFH(Σ , M ) . It is shown in [10, Theo-rem 1.2] that SH( V ) = 0 ⇐⇒ SH T ( V ) = 0 . While the implication ⇐ = follows from the Gysin exact sequence in [9], the implication = ⇒ follows from the fact that SH T ( V ) is the limit of a spectral sequence whose second page isthe tensor product of the homology of the classifying space BS and of SH( V ), [10, § , M ) = 0 ⇐⇒ RFH T (Σ , M ) = 0 . In particular, the vanishing of RFH T (Σ , M ) for displaceable Σ would then follow from thevanishing of RFH(Σ , M ) proven in [13]. Together with the equivalence from [36, Theo-rem 13.3] we could then conclude the equivalencesRFH T (Σ , M ) = 0 ⇐⇒ RFH(Σ , M ) = 0 ⇐⇒ SH( V ) = 0 ⇐⇒ SH T ( V ) = 0 .
3. Chekanov’s construction of S -equivariant Floer homology. In the Borel-construction, approximations S N +1 of the classifying space S ∞ = ES are somewhat clum-sily added to the loop space as direct summands. In Chekanov’s version of S -equivariantFloer homology, S N +1 does not appear as a space, but is incorporated into the boundaryoperator: In the setting of Morse theory for a function f : M → R on a compact S -manifold M , with action S × M → M , ( s, x ) sx , one proceeds as follows. Given times t < · · · < t N ∈ R and angles s , . . . , s N ∈ S one considers the functions f t ( x ) = f ( x ) if t < t ,f ( s x ) if t t < t ,f (( s + s ) x ) if t t < t , ... f (( s N + · · · + s ) x ) if t N t, (30)and counts gradient “ N -jump flow lines” of the vector field −∇ f t . A neat way to seethat a point ( t , . . . , t N , s , . . . , s N ) corresponds to a point in S N − is through the joinconstruction, [10, § § construction and the isomorphism in [10] can be adapted to Rabinowitz–Floer homol-ogy, yielding a homology RFH T jump (Σ , M ) isomorphic to RFH T (Σ , M ). The vanishing ofRFH T (Σ , M ) for displaceable Σ then follows from the vanishing of RFH T jump (Σ , M ), whichin turn follows as for the non-equivariant RFH(Σ , M ) by a leafwise intersection argument,because the chain groups of RFH T jump (Σ , M ) are exactly the chain groups of RFH(Σ , M ). -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 25 Appendix on transversality
Preface.
Consider a smooth Morse function f on a closed Riemannian manifold ( X, g ). Inorder to define the Morse homology of X , one needs that the spaces of gradient flow linesbetween critical points are manifolds. This can be achieved by perturbing the Riemannianmetric g , see [42, § X .In Floer theory on a symplectic manifold ( M, ω ), the role of f and X is played bythe action functional A H and the loop space L = C ∞ ( S , M ), and one works with L -Riemannian metrics on L given by ω -compatible almost complex structures J on M : For v ∈ L and vector fields ξ , ξ along v , h ξ , ξ i v := Z ω ( v ( t )) (cid:0) ξ ( t ) , J ( v ( t )) ξ ( t ) (cid:1) dt. Floer’s equation (for H = 0, say) is hence local in M : ∂ s v ( s, t ) + J (cid:0) v ( s, t ) , t (cid:1) ∂ t v ( s, t ) = 0 . (31)Denote by c M J ( c − , c + ) the space of solutions of (31) asymptotic to critical points c − , c + of A H , and by c M J their union, with the C ∞ loc -topology. The traditional way to achievetransversality for the spaces c M J ( c − , c + ) is by perturbing J in the space J M of ω -compatiblealmost complex structures on M , or in the slightly larger space J M × S of 1-periodic loopsin J M .These are very small perturbation spaces: The almost complex structures in J M or J M × S are only “ M -dependent” or “ M × S -dependent”. A much larger space of perturbations,which is the precise analogue of the space of Riemannian metrics playing the role of pertur-bations in Morse homology, is the space J L of ω -compatible “ L -dependent” almost complexstructures: An element J ∈ J L associates with each v ∈ L a loop J v ( t ) of ω -compatiblealmost complex structures on T v ( t ) M . The L -Riemannian metric on L is now h ξ , ξ i v := Z ω ( v ( t )) (cid:0) ξ ( t ) , J v ( t ) ξ ( t ) (cid:1) dt. Hence, Floer’s equation for J ∈ J L is non-local in M : ∂ s v ( s, t ) + J v ( s ) ( t ) ∂ t v ( s, t ) = 0 . The two main features one requires for the moduli spaces c M J are that they are com-pact and that c M J ( c − , c + ) are cut out transversally. Both, compactness and transversality,which implies that c M J ( c − , c + ) are manifolds, are needed for defining the boundary oper-ator d and proving d = 0, as well as for proving that the resulting homology does notdepend on the choice of J .The advantage of working with a small space P small of perturbations, such as J M or J M × S , is that if one can prove compactness of one moduli space c M J , then for thesame reason one has compactness of c M J ′ for all J ′ ∈ P small . The disadvantage of these small perturbation spaces is that it is sometimes hard, or even impossible, to achievetransversality for a generic set of J ′ ∈ P small .Conversely, the advantage of working with a large space P large of perturbations, such as J L or the perturbation space used for transversality in the theory of M-polyfolds [29, § c M J for one J , compactnessof c M J ′ for nearby J ′ ∈ P large can fail!Assume now that one knows that c M J is compact for one J ∈ J M . The goal of thisappendix is to explain that in this situation one can simultaneously achieve compactnessand transversality for generic J ′ ∈ P large sufficiently close to J , not for the whole spaces c M J ′ , but for a compact part of c M J ′ near c M J .The natural framework for carrying out this argument is the scale calculus, which formsa central building block of M-polyfold theory, [29]. In this setting, the (completion ofthe) loop space L , the Hilbert-bundle over L , and also the space of perturbations P large ,are replaced by decreasing sequences of spaces and bundles, and the gradient of the ac-tion functional and its Hessian at critical points become a scale vector field and a scalesymmetric bilinear form. Since the scale calculus has not yet become common knowledge,and since setting up the scale structures in our situation would obscure the main idea, wewill stay in the framework of classical analysis and follow [37, 38], where “finite step scalestructures” are already implicit.To bring out the argument clearly, we first explain it in Appendix A in detail in the frame-work of Morse homology on a finite-dimensional but possibly non-compact manifold X .The arguments are chosen such that they extend to the infinite-dimensional situation ofFloer homology in § B.1, up to the compactness of the nearby parts of the spaces c M J ′ ,that follows from a compactness result for non-local perturbations of the Cauchy–Riemannoperator proven in § B.2. Other parts of the construction of this Floer homology (such asthe Fredholm theory and unique continuation) will be worked out in [24].
Appendix A. Transversality for Morse homology on finite-dimensionalmanifolds
A.1.
The set-up.
Let X be a C ∞ -smooth finite-dimensional manifold. We do not assumethat X is compact or complete, but for convenience we assume that X is connected. Choosea Riemannian metric g on X . Let f : X → R be a C ∞ -smooth Morse function. Denote by ∇ f = ∇ g f the gradient vector field of f with respect to g . We do not assume that thisvector field is complete. However, we make a compactness assumption on the “gradient flowlines” of ∇ f : A gradient flow line x ∈ C ∞ ( R , X ) is a solution of the ordinary differential This is the case for Rabinowitz–Floer homology on an exact symplectic manifold and any J ∈ J con if one looks only at the part of c M J between critical points with action in a fixed interval [ a, b ], see[13,Corollary 3.3], and similarly holds for these partial moduli spaces for S -equivariant Rabinowitz–Floerhomology and any J ∈ J con , cf. § § -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 27 equation ˙ x ( s ) = −∇ f ( x ( s )) , s ∈ R . For a, b ∈ R let G ba be the space of gradient flow lines x with a f ( x ( s )) b for all s ∈ R .Endow C ∞ ( R , X ) with the C ∞ loc -topology. (A) For all a b the space G ba is compact in C ∞ ( R , X ).Let X ba = { x ∈ X | a f ( x ) b } and Crit f ba = Crit f ∩ X ba . Assumption (A) impliesthat G ba is the disjoint union of the sets G ( x − , x + ) of gradient flow lines between points x − , x + ∈ Crit f ba , see [14, Lemma 2.1]. Note that critical points of f are gradient flow lines.Assumption (A) also implies that Crit f ba is a finite set. The set G ba is therefore the finiteunion of the sets G ( x − , x + ) with x ± ∈ Crit f ba .The spaces G ( x − , x + ) are not manifolds in general, but this holds true for a close-byRiemannian metric, [42, § g ′ the negative gradientvector field −∇ g ′ f is a pseudo-gradient vector field for f . Anticipating the perturbationvector fields in the M-polyfold theory, we work with pseudo-gradient vector fields −∇ f + v that agree with −∇ f near Crit f ba , see the next paragraph. Note that such pseudo-gradientvector fields bijectively correspond to Riemannian metrics g ′ that agree with g near Crit f ba .The key argument for this approach to transversality is given in A.3, where we showthat for every sufficiently small perturbation v one can select a compact part G ba ( v, N ) = S x − ,x + G ( x − , x + , v, N ) of G ba ( v ) = S x − ,x + G ( x − , x + , v ) near G ba . In A.4 and A.5 we describea Hilbert bundle H ←− E over the Hilbert manifold H of lines from x − to x + near G ba , andshow that the spaces G ( x − , x + , v, N ) can be viewed as the zero-set of suitable sections ofthis bundle. In A.6 we use (a minor variation of) the Sard–Smale theorem to show thatfor a generic set of perturbations, G ( x − , x + , v, N ) is smooth.To make Appendix A useful for Appendix B, we use the local compactness of X onlywhen invoking the Arzel`a–Ascoli theorem, a tool that is replaced in § B.1 by the compact-ness theorem in Floer homology proven in § B.2.A.2.
Perturbations.
Fix a < b . Let c , . . . , c N be the elements of Crit f ba . For each i choose an open neighbourhood U i of c i such that for i = j the closures of U i and U j aredisjoint. Set U = S U i .The evaluation G ba → X , x x (0), is continuous. By assumption (A) the image K := (cid:8) x (0) | x ∈ G ba (cid:9) ⊂ X is thus compact. Since ∇ f is a continuous vector field on X , wefind an open and bounded neighbourhood N of K in X and δ > k∇ f ( x ) k > δ for all x ∈ N \ U . Choose k ∈ N such that k > k > ind( c i ) − ind( c j ) for all c i , c j ∈ Crit f ba . The vector space V k of C k -vector fields on X that vanish on U and havebounded derivatives up to order k , endowed with the C k -norm, is a Banach space. Itssubset V kf = (cid:8) v ∈ V k | df ( −∇ f + v ) < N \ U (cid:9) contains the open ball B δ := (cid:8) v ∈ V k | k v k C k < δ (cid:9) , since df ( −∇ f + v ) = h−∇ f + v, ∇ f i . PSfrag replacements K N U i U j Figure 3.
The isolating neighbourhood N of K .A.3. Selection of compact perturbed moduli spaces.
For v ∈ V kf let G ba ( v, N ) bethe set of solutions x : R → X of˙ x ( s ) = −∇ f ( x ( s )) + v ( x ( s )) , such that a f ( x ( s )) b and x ( s ) ∈ N for all s ∈ R . The elements of G ba ( v, N ) are C k +1 -smooth, and the set of j th derivatives of elements of G ba ( v, N ) is uniformly boundedand equicontinuous for every j k , because N is bounded and because ∇ f and v arebounded on N with all their derivatives of order k . The Arzel`a–Ascoli theorem thusimplies that G ba ( v, N ) is C k loc -compact.The elements of G ba ( v, N ) may touch the boundary ∂ N , and even for “regular” v (asdefined in A.6) this may lead to new ends of G ba ( v, N ) that do not come from breaking offlow lines and thus may obstruct the property d = 0 necessary to define Morse homology.As the next lemma shows, this cannot happen if v is small enough. Recall that B δ ⊂ V kf . Lemma A.1.
There exists ε ∈ (0 , δ ) such that for all v ∈ B ε , x ( R ) ⊂ N for all x ∈ G ba ( v, N ) . In other words, N is an isolating neighbourhood for the flow of −∇ f + v simultaneouslyfor all v ∈ B ε , and for each such v the set G ba ( v, N ) is an isolated invariant set of this flow. Proof.
If not, there exists a sequence of perturbations v ν ∈ V kf with lim ν →∞ v ν → C k and a sequence of flow lines x ν ∈ G ba ( v ν , N ) such that x ν ( s ν ) ∈ ∂ N for some time s ν ∈ R .Since G ba ( v ν , N ) is invariant under time-shift, we can assume that s ν = 0 for all ν . Recallthat x ν solves ˙ x ν ( s ) = −∇ f ( x ν ( s )) + v ν ( x ν ( s )) , s ∈ R . Since x ν ( R ) ⊂ N and since the sequence v ν is uniformly bounded in C k , we can invokeagain the Arzel`a–Ascoli theorem to find a subsequence that converges in C k loc to an element x ∈ G ba . Hence x (0) ∈ K . On the other hand, x (0) = lim x ν j (0) ∈ ∂ N . But K ∩ ∂ N = ∅ ,a contradiction. (cid:3) -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 29 A.4.
The Hilbert manifold of lines.
Fix two points x − , x + ∈ N . Consider the “set oflines” from x − to x + in N , H := H x − ,x + ( N ) := “ (cid:8) x ∈ W , ( R , N ) | lim s →±∞ x ( s ) = x ± (cid:9) ” . We describe this set by constructing a Hilbert manifold atlas: Let C ∞ c ( R , N , x − , x + ) bethe set of x ∈ C ∞ ( R , N ) such that there exists T > x ( s ) = x − for s − T, x ( s ) = x + for s > T. Fix x ∈ C ∞ c ( R , N , x − , x + ). The pull-back bundle x ∗ T N → R is a trivializable vectorbundle over R of dimension n = dim N . Fix a trivialization Φ : x ∗ T N → R × R n . Choosean open neighbourhood V x ⊂ x ∗ T N of the zero-section such that the exponential mapexp g : T x ( r ) N ∩ V x → N is injective for all r ∈ R . The set U x := (cid:8) ξ ∈ W , ( R , R n ) | ( r, ξ ( r )) ∈ Φ( V x ) for all r ∈ R (cid:9) is an open subset of W , ( R , R n ). Define the injective map φ x : U x → C ( R , N , x − , x + ) , ξ exp g (cid:0) Φ − ( ξ ) (cid:1) , where C ( R , N , x − , x + ) is the set of continuous maps y : R → N with lim s →±∞ y ( s ) = x ± .Using elementary Sobolev theory one shows that the transition maps ( φ x ′ ) − ◦ φ x : U x ′ ∩ U x → U x ′ ∩ U x are smooth in W , ( R , R n ). Now define H := [ x ∈ C ∞ c ( R , N ,x − ,x + ) φ x ( U x ) . The space H is thus a Hilbert manifold modeled on W , ( R , R n ).A.5. The Hilbert bundle set-up.
The tangent space at x ∈ H is T x H = W , ( R , x ∗ T N ) . A larger bundle over H is the L -bundle with fibre E x = L ( R , x ∗ T N ), T H (cid:31) (cid:127) / / (cid:15) (cid:15) E | | ③③③③③③③③ H Fix now x − , x + ∈ Crit f . The vector fields −∇ f + v with v ∈ V kf are C k -smooth pseudo-gradient vector fields for f that agree with −∇ f on U ⊃ Crit f ba . For v ∈ V kf denote by G ( x − , x + , v, N ) the set of solutions x : R → N of˙ x ( s ) = ( −∇ f + v )( x ( s )) . (32)We can now interpret the set G ( x − , x + , v, N ) as the zero-locus of a section: Recall thata section of the bundle H p ←− E is a map H s −→ E such that p ◦ s = id H . To v ∈ V kf associate the C k -section s v : H → E defined by s v ( x ) := ˙ x + ( ∇ f − v )( x ) . Then x ∈ s − v (0) if and only if x is a W , -flow line of −∇ f + v in N from x − to x + . Usingequation (32) we see that such an x is actually C k +1 -smooth.For x in the zero-section H of the bundle E p −→ H we have the canonical splitting T x E = E x ⊕ T x H . (Defining such a splitting at a point x off the zero-section would require the choice of aconnection.) The differential ds v ( x ) is a map T x H → T s v ( x ) E . For x ∈ s − v (0) define the vertical differential by Ds v ( x ) : T x H → E x , Ds v ( x ) = π ◦ ds v ( x )where π : T x E = E x ⊕ T x H → E x is the projection along T x H .A C -section s : H → E is said to be transverse to the zero-section , s ⋔
0, if Ds ( x ) issurjective for all x ∈ s − (0). Furthermore, a C -section s : H → E is called Fredholm if Ds ( x ) : T x H → E x is a Fredholm operator for all x ∈ s − (0). Proposition A.2.
Let s : H → E be a C k -smooth Fredholm section such that s ⋔ . Then s − (0) is a C k -smooth manifold, and for x ∈ s − (0) we have dim x s − (0) = dim ker Ds ( x ) = ind Ds ( x ) . Here dim x s − (0) denotes the dimension of the connected component containing x , and ind Ds ( x ) := dim ker Ds ( x ) − dim coker Ds ( x ) is the Fredholm index of Ds ( x ) . The first identity follows from the implicit function theorem for C k -maps between Banachmanifolds, and the second identity holds because Ds ( x ) is surjective, i.e., coker Ds ( x ) = 0.Recall that the support of every perturbation v ∈ V kf of −∇ f is disjoint from Crit f ba .The sections s v are thus Fredholm, see [37, Theorem 2.1] or [42, § Corollary A.3. If x − , x + have Morse index ind( x ± ) and if s v ⋔ , then dim x s − v (0) = ind( x − ) − ind( x + ) . A.6.
Finding smooth perturbed moduli spaces.
Let ε ∈ (0 , δ ) be as in Lemma A.1.The open subset B ε of the Banach space V k is homeomorphic to V k and hence metrizableby a complete metric. In particular, B ε is a Baire space, i.e., every subset of B ε of thesecond category is dense in B ε .Fix x − , x + ∈ Crit f ba and write again H = H x − ,x + ( N ). Consider the C k -map S : B ε × H → E , S ( v, x ) = s v ( x ) . (33)By Lemma A.1 the set G ( x − , x + , v, N ) is the set of flow lines in G ba ( v, N ) from x − to x + .Write G ( x − , x + , N ) = [ v ∈ B ε { v } × G ( x − , x + , v, N ) . Then G ( x − , x + , N ) = S − (0). Theorem A.4.
There exists a subset V reg ( x − , x + ) ⊂ B ε of the second category such that s v : { v } × H → E is tranverse to for all v ∈ V reg ( x − , x + ) . -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 31 Before giving the proof, we draw the desired conclusions. Recall that the set B ε dependsonly on f and a, b , while the set V reg ( x − , x + ) ⊂ B ε in Theorem A.4 depends also on x − , x + . Proposition A.2, Corollary A.3 and Theorem A.4 show that for all v in the subset V reg ( x − , x + ) ⊂ B ε of the second category, the set s − v (0) = { v } × G ( x − , x + , v, N ) is a C k -smooth manifold of dimension ind( x − ) − ind( x + ). Recall that there are only finitelymany critical points in Crit f ba . With V reg := T x − ,x + V reg ( x − , x + ) we find Corollary A.5.
Assume that (A) holds. Then there exists a subset V reg ⊂ B ε of thesecond category such that for every v ∈ V reg all the sets G ( x − , x + , v, N ) with x ± ∈ Crit f ba are C k -manifolds of dimension ind( x − ) − ind( x + ) . Remark A.6.
An argument due to Taubes, for which we refer to [31, § k replaces by ∞ . We do not need this in the sequel.Recall that the sets G ba ( v, N ) are C k loc -compact for all v ∈ V kf . As in the proof ofLemma 2.1 in [14] if follows that for these v , G ba ( v, N ) = a x − ,x + G ( x − , x + , v, N ) , x ± ∈ Crit f ba . (34)By Lemma A.1 we have G ( x − , x + , v, N ) = G ( x − , x + , v, N ) for v ∈ B ε , and by Corollary A.5these sets are C k -manifolds in N of dimension ind( x − ) − ind( x + ) for every v ∈ V reg . Morsehomology can now be defined by using the spaces (34) with v ∈ V reg . Proof of Theorem A.4.
We invoke an abstract result, that is a minor variation of theclassical Sard–Smale theorem, proved by Sard and Quinn [34, Theorem 1] and put in theform below by Henry [27, Theorem 5.4], see also [41]. In the formulation we use the lettersfrom the previous text, but change the fonts of the spaces to stress the abstract setting.A map f : X → Y between topological spaces is called σ -proper if X can be written as acountable union X = S i X i such that the restrictions f : X i → Y are proper. Proposition A.7.
Let V and H be Banach manifolds and let E → H be a Banachbundle. Let S : V × H → E be a C k -map such that s v = S ( v, · ) : { v } × H → E is asection for each v ∈ V . Assume that (i) For every ( v, x ) ∈ S − (0) the vertical differential DS ( v, x ) is surjective. (ii) For every v ∈ V the section s v : { v } × H → E is a Fredholm map of index < k . (iii) The projection S − (0) → V , ( v, x ) v , is σ -proper.Then there exists a subset V reg ⊂ V of the second category such that s v ⋔ for all v ∈ V reg . In this version, the separability assumption on V and H in the classical Sard–Smaletheorem is replaced by the σ -properness assumption (iii). While this modification is irrele-vant in the case of finite-dimensional Morse homology, in which B ε and H are separable, itwill be relevant in the infinite-dimensional situation of Floer homology addressed in § B.1,where neither V nor H is separable anymore.We apply Proposition A.7 with V = B ε , H = H , E = E , and S as in (33), so that S − (0) = G ( x − , x + , N ). We first verify assumption (iii) of Proposition A.7. Let U be the neighbourhood of Crit f ba defined in § A.2. After choosing U smaller, if necessary, we can assume that the MorseLemma holds for f on U . For T ∈ N define G T ( x − , x + , N ) = (cid:8) ( v, x ) ∈ S − (0) | x ( s ) ∈ U for s / ∈ [ − T, T ] (cid:9) . In other words, ( v, x ) ∈ S − (0) = G ( x − , x + , N ) belongs to G T ( x − , x + , N ) if x ( s ) ∈ U ( x − )for s < − T and x ( s ) ∈ U ( x + ) for s > T . Then S − (0) = S T ∈ N G T ( x − , x + , N ). We thusneed to prove Lemma A.8.
For every T ∈ N the projection p : G T ( x − , x + , N ) → B ε , ( v, x ) v , isproper.Proof. The reason for this is that the only possible source of non-compactness of a sequence( v ν , x ν ) ⊂ G T ( x − , x + , N ) with ( v ν ) in a compact set of B ε is breaking, which does not occurbecause x ν ( s ) ⊂ U ( x − ) for s < − T and x ν ( s ) ⊂ U ( x + ) for s > T and because f is Morse.Given a compact subset K ⊂ B ε we wish to show that p − ( K ) ⊂ G T ( x − , x + , N ) iscompact. Let ( v ν , x ν ) be a sequence in p − ( K ). Since K is compact and by the Arzel`a–Ascoli theorem we can pass to a subsequence such that v ν → v in B ε and x ν | [ − T,T ] → x [ − T,T ] in C k ([ − T, T ] , N ). Recall that on U we have v ν = v = 0, and that the MorseLemma holds for f on U . Since x ν ( ± T ) → x [ − T,T ] ( ± T ), we also have x ν | ( −∞ , − T ] → x ( −∞ , − T ] in C k (( −∞ , − T ] , N ) and x ν | [ T, ∞ ) → x [ T, ∞ ) in C k ([ T, ∞ ) , N ), with exponen-tial decay of all derivatives of order k , by the Morse Lemma. For the concatena-tion x = x ( −∞ , − T ] x [ − T,T ] x [ T, ∞ ) we thus have x ν → x in C k ( R , N ), with exponentialdecay of all derivatives of order k . By the Arzel`a–Ascoli theorem and by Lemma A.1,( v, x ) ∈ G T ( x − , x + , N ). Furthermore, the convergence x ν → x in C ( R , N ) with exponen-tial decay of all derivatives of order v ν , x ν ) → ( v, x ) in B ε × H . (cid:3) That assertions (i) and (ii) of Proposition A.7 hold in our setting is a standard result(that in fact holds for the unrestricted map S : V kf × H → E ). We have already seen thisfor assertion (ii). It thus remains to prove Proposition A.9.
For every ( v, x ) ∈ S − (0) the vertical differential DS ( v, x ) is surjective.Proof. We distinguish two cases.
Case 1. x − = x + . Fix ( v, x ) ∈ S − (0). Then ˙ x + ( ∇ f − v )( x ) = 0. The tangent space T v B ε is the Banach space V k . The vertical differential DS ( v, x ) : V k ⊕ T x H → E x is givenby DS ( v, x )(ˆ v, ˆ x ) = Ds v ( x )ˆ x + ˆ v. (35)In particular, the image of Ds v ( x ) is contained in the image of DS ( v, x ). Since Ds v ( x )is Fredholm, its image is closed in E x and of finite codimension, and hence the imageof DS ( v, x ) is also closed in E x .The fiber E x is equipped with the L -inner product h η , η i g := Z ∞−∞ g x ( s ) ( η ( s ) , η ( s )) ds. -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 33 Assume that η ∈ coker DS ( v, x ) ⊂ E x , that is, η is h , i g -orthogonal to the image of DS ( v, x ).We must show that η = 0. By assumption, h DS ( v, x )(ˆ v, ˆ x ) , η i g = 0 for all (ˆ v, ˆ x ) ∈ V k ⊕ T x H . In particular h Ds v ( x )ˆ x, η i g = 0 for all ˆ x ∈ T x H , (36) h ˆ v, η i g = 0 for all ˆ v ∈ V k . (37)By (36), η ∈ coker Ds v ( x ) = ker Ds v ( x ) ∗ , where Ds v ( x ) ∗ denotes the adjoint operator.Choose a chart R n → N covering the whole flow line x . In this chart, Ds v ( x ) ˆ x = ˙ˆ x + Hess f ( x ) ˆ x − dv ( x ) ˆ x. Integrating by parts, we find that the adjoint operator is given by Ds v ( x ) ∗ η = − ˙ η + (cid:2) Hess f ( x ) − dv ( x ) (cid:3) T η. An element η ∈ coker Ds v ( x ) = ker Ds v ( x ) ∗ is thus a solution of the ordinary differentialequation with C k − -coefficients˙ η ( s ) = (cid:2) Hess f ( x ( s )) − dh T ( x ( s )) (cid:3) η ( s ) , s ∈ R . (38)Hence η ∈ C k ( R , x ∗ T N ). We must show that η ( s ) = 0 for all s ∈ R .Recall that the sets U i are mutually disjoint. Since x − = x + , the flow line x thereforeintersects the set W := N \ U . Assume that there exists s ∗ ∈ R such that x ∗ := x ( s ∗ ) ∈ W and η ( s ∗ ) = 0. Since η is continuous, there exists ε > η ( s ) = 0 for all s ∈ [ s ∗ − ε, s ∗ + ε ]. Since v ∈ B ε ⊂ V kf , x is a non-constant flow line of −∇ f + v , andtherefore the function f ◦ x is strictly decreasing. Hence the set W ε := { x ∈ N | f ( x ( s ∗ + ε )) < f ( x ) < f ( x ( s ∗ − ε )) } ∩ W is an open neighbourhood of x ∗ , and x ( s ) / ∈ W ε for s / ∈ ( s ∗ − ε, s ∗ + ε ). (39)Choose ˆ v ∈ V k with support in W ε such that h ˆ v ( x ∗ ) , η ( x ∗ ) i > h ˆ v ( x ( s )) , η ( x ( s )) i > s ∈ ( s ∗ − ε, s ∗ − ε ). By (39), h ˆ v ( x ( s )) , η ( x ( s )) i = 0 for s / ∈ ( s ∗ − ε, s ∗ − ε ). Therefore, h ˆ v, η i g := Z ∞−∞ (cid:10) ˆ v ( x ( s )) , η ( s ) (cid:11) ds = Z s ∗ + εs ∗ − ε (cid:10) ˆ v ( x ( s )) , η ( s ) (cid:11) ds > , in contradiction to (37).We are left with showing that η ( s ) = 0 also for x ( s ) ∈ U . Assume that x ( s ∗ ) ∈ U i . Sincethe flow line x is not entirely contained in U i , there exists s ′ with x ( s ′ ) ∈ W . We havealready seen that η ( s ) = 0 for s near s ′ . Since η is a solution of the ordinary differentialequation (38) with C k − -coefficients, and since k >
2, it follows that η ( s ′ ) = 0. We haveshown that η ( s ) = 0 for all s ∈ R . Case 2. x − = x + . In this case, S ( v, x ) = s v ( x ) = ˙ x + ∇ f ( x ) = 0 if and only if x ≡ x := x − = x + is the constant flow line. Recall from Case 1 that the image of Ds v ( x )is contained in the image of DS ( v, x ). It thus suffices to show that for every v ∈ B ε the operator Ds v ( x ) : T x H → E x is surjective, or, equivalently, that the adjoint operator Ds v ( x ) ∗ is injective. As in (38) an element η ∈ ker Ds v ( x ) ∗ is a solution of˙ η ( s ) = A η ( s ) , s ∈ R , (40)where A = Hess f ( x ). In a suitable basis, A = diag( a , . . . , a n ) with a i = 0 because f isMorse. The solutions of (40) are of the form η ( s ) = ( η ( s ) , . . . , η n ( s )) with η i ( s ) = η i (0) e a i s .They lie in L only if η i (0) = 0, i.e., if η = 0. (cid:3) Appendix B. Transversality for Floer homology with one compactmoduli space
B.1.
The Conley-type argument in the Floer case.
In this paragraph we outline howthe transversality scheme of Appendix A, and in particular the Conley-type argument inSection A.3, works in the setting of a Floer homology for which one has compactness of themoduli space c M J for one almost complex structure J . This is, for instance, the case forexact Liouville domains or for symplectically aspherical closed symplectic manifolds. Thekey difference to the Morse homology on a finite-dimensional manifold X is that now therole of X is taken by an infinite-dimensional manifold, that is not locally compact. TheArzel`a–Ascoli theorem is thus not available anymore; it is substituted by a compactnesstheorem for solutions of the Cauchy–Riemann equation with a non-local perturbation, thatwe address in the next paragraph. Dictionary.
Let (
M, ω ) be a symplectic manifold, choose an S -family J t of almost com-plex structures compatible with ω , and let H : M × S → R be a smooth function such thatall 1-periodic orbits of its Hamiltonian field X H t are non-degenerate. To avoid technical andnotational complications, we assume that M = C n . For ℓ > H ℓ = W ℓ, ( S , C n ). By assumption (A) below the set Crit A ba = { c , . . . , c N } ofcritical points of the action functional A := A H with action in [ a, b ] is finite. Choose k ∈ N such that k > k > ind( c i ) − ind( c j ) for all c i , c j ∈ Crit A ba .The role of the manifold X and of the Morse function f : X → R is now taken by H k andthe action functional A : H k → R , and the gradient flow line equation ˙ x ( s ) = −∇ f ( x ( s ))becomes ˙ x ( s ) = −∇ A ( x ( s )). If one takes ∇ to be the H -gradient with respect to theRiemannian metric on H k induced by ω and J t , this is Floer’s equation: A gradient flowline x : R → H k is a solution of the ordinary differential equation on H k ˙ x ( s ) = − J t ( x ( s )( t )) (cid:0) ∂ t x ( s ) + X H t ( x ( s )( t )) (cid:1) , s ∈ R . In the sequel, we omit the lower order term − J t X H t from the notation, and for the situationof ( S -equivariant) Rabinowitz–Floer homology we also neglect the R -factor of η and thespheres S N +1 . For a, b ∈ R let G ba be the space of gradient flow lines x with a A ( x ( s )) b for all s ∈ R . Endow C ∞ ( R , H k ) with the C ∞ loc -topology. We again assume that (A) For all a b the space G ba is compact in C ∞ ( R , H k ). -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 35 This assumption is satisfied if M is compact and [ ω ] vanishes on π ( M ), such as exactLiouville domains, and also for the gradient flow lines with cascades used to define ( S -equivariant) Rabinowitz–Floer homology.Choose open neighbourhoods U i of c i in H with disjoint closures. The evaluation G ba → H k at 0 is continuous and has compact image K ⊂ H k , and ∇ A : H k → H k − is smoothand the inclusion H k − → H is continuous. We can thus find an open and boundedneighbourhood N of K in H k and δ > k∇ A k H > δ on N \ U .Recall that given two Banach spaces ( Y, k k Y ), ( Z, k k Z ) that are continuously embeddedin the same Banach space, the space Y ∩ Z with norm k k Y + k k Z is a Banach space. For j > C k ( H j , H j ) be the (non-separable) Banach space of C k -vector fields H j → H j that have bounded derivatives up to order k , endowed with the C k -norm. For every j ∈ { , . . . , k } the restriction from H j to H k defines a continuous map C k ( H j , H j ) → C ( H k , H ), which is an embedding since the inclusion H k ⊂ H j is dense. With respect tothese embeddings define the Banach space C k ( H, H ) = C k ( H k , H k ) ∩ C k ( H k − , H k − ) ∩ · · · ∩ C k ( H , H ) . (41)For instance, C ( H, H ) = C ( H , H ) ∩ C ( H , H ) consists of all those C -maps H → H whose restriction to H takes values in H , and is still C as a map H → H .Now define V k to be the closed subspace of C k ( H, H ) formed by those vector fields thatvanish on U . The elements ˆ v ∈ V k used to extend the proof of Proposition A.9 are “bumpvector fields”, that in a chart around a point in H \ U are of the form x β ε ( k x k ) e i where ε > β ε : R → [0 , ε ] is a smooth function that has support in [ − ε, ε ] and thatis equal to ε near 0, and where e i is a vector of the Fourier basis of H . The subset V k A = (cid:8) v ∈ V k | d A ( −∇ A + v ) < N \ U (cid:9) of V k contains the open δ -ball in V k , since d A ( −∇ A + v ) = h−∇ A + v, ∇ A i H . TheHilbert manifold H x − ,x + ( N ) is constructed as in A.4, but is now modeled on W , ( R , H k )and is therefore not separable. The Hilbert bundle E → H has fibre E x = L ( R , H k − ).For v ∈ V k A let G ba ( v, N ) be the set of solutions x : R → H k of˙ x ( s ) = −∇ A ( x ( s )) + v ( x ( s ))such that a A ( x ( s )) b and x ( s ) ∈ N for all s ∈ R . The main difference to the case ofMorse homology on a finite-dimensional manifold studied in Appendix A is that now H k is not locally compact, whence we cannot appeal to the Arzel`a–Ascoli theorem anymore.It is now Theorem B.1 below that shows that for v ∈ B δ the spaces G ba ( v, N ) are C -compact in C ( R , H k ). Since v ∈ V k we can then use bootstrapping to see that G ba ( v, N )are C k loc -compact in C k ( R , H k ). We can now follow Appendix A to obtain transversality forthe perturbed moduli spaces G ba ( v, N ) for a generic set of perturbations V reg ⊂ B ε ⊂ B δ .Some of the further tools needed are: • The relevant Fredholm theory can now be established as in [37, § • The exponential decay of solutions for s → ±∞ used in Lemma A.8, that followedfrom the Morse Lemma, now follows as in [38], see also [39, Lemma 2.11]. • At the end of Case 1 in the proof of Proposition A.9 we used unique continuationfor ordinary differential equations. To establish the unique continuation resultneeded now one can use a technique of Agmon–Nirenberg as in [38, Lemma 3.3].B.2.
A compactness theorem for non-local perturbations of the Cauchy–Riemannoperator.
In this section we prove the compactness result used in the previous paragraph.As before we assume that M = C n . Suppose that J t ( z ) for z ∈ C n and t ∈ S = R / Z isa smooth family of almost complex structures on C n which is allowed to depend as wellsmoothly on the variable t ∈ S . We again abbreviate the Sobolev spaces H ℓ := W ℓ, ( S , C n )and suppose that we are given a vector field V in the space C k ( H, H ) defined in (41), with k >
2. For example if V t is a smooth vector field on C n which may also depend smoothlyon t ∈ S we can define such a vector field by V ( z )( t ) = V t ( z ( t )) , z ∈ H , t ∈ S . However, we do not require that V be of this form. In particular, V can be non-local inthe sense that for a loop z ∈ H and a time t ∈ S the value V ( z )( t ) ∈ C n depends onthe whole loop z and not just on the point z ( t ) and the time t . In fact, that V is allowedto be non-local is the main novelty of the discussion in this section. For T > I T = [ − T, T ] and C k ( I T , H ) = C ( I T , H k ) ∩ C ( I T , H k − ) ∩ · · · ∩ C k − ( I T , H ) . We are interested in solutions w ∈ C k ( I T , H ) of the following non-locally perturbedCauchy–Riemann equation on the finite cylinder I T × S : ∂ s w + J t ( w ) ∂ t w = V ( w ) . (42)The main result of this section is the following compactness statement. Theorem B.1.
Suppose that ( V ν ) is a sequence in C k ( H, H ) converging to V , and that ( w ν ) ⊂ C k ( I T , H ) is a sequence of solutions of ∂ s w ν + J t ( w ν ) ∂ t w ν = V ν ( w ν ) (43) for which there exists a constant C > such that k w ν ( s ) k H k C for all ν ∈ N , s ∈ I T . Then a subsequence of ( w ν ) converges in C k ( I T − , H ) to a solution w of (42) .Proof. For notational convenience we assume that k = 2. We are given a sequence( w ν ) ν > ⊂ C ( I T , H ) ∩ C ( I T , H ) of solutions of (43), and by assumption there existsa constant C > k w ν ( s ) k H C, k V ν ( w ν ( s )) k H C, k D V ν ( w ν ( s )) k L ( H ,H ) C, ν ∈ N , s ∈ I T . -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 37 We need to find a subsequence of ( w ν ) that converges in C ( I T − , H ) ∩ C ( I T − , H ) to asolution of (42). The proof requires the following regularity result. Proposition B.2.
Suppose that w ∈ C ( I T , H ) ∩ C ( I T , H ) is a solution of (42) . Then w ∈ L ( I T − , H ) ∩ W , ( I T − , H ) ∩ W , ( I T − , H ) .Proof. We first explain the proof of the proposition under the simplifying assumption thatthe almost complex structure is constant and given by multiplication with i , so that w isa solution of the equation ∂ s w + i∂ t w = V ( w ) . (44)The proof of this case is inspired by the proof of [37, Theorem 3.8]. For c ∈ R let A c : H k +1 → H k , ξ i ddt ξ + c ξ. Note that if c / ∈ π Z , then A c is an isomorphism simultaneously for all k ∈ N . We furtherabbreviate V c ∈ C ( H , H ) ∩ C ( H , H ) , V c ( z ) = V ( z ) + cz, z ∈ H . We can now rewrite (44) as ∂ s w + A c w = V c ( w ) . For ξ := ∂ s w ∈ C ( I T , H ) ∩ C ( I T , H )we have ∂ s ξ + A c ξ = D V c ( w ) ξ (45)where D V c ( z ) = D V ( z ) + c · id | H , z ∈ H . Choose a smooth cutoff function β ∈ C ∞ ( I T , [0 , β ( s ) = (cid:26) s ∈ [ − T + 1 , T − , s ∈ [ − T, − T + ] ∪ [ T − , T ] . Then ξ β := β ξ ∈ C ( I T , H ) ∩ C ( I T , H )has compact support in ( − T, T ). Pick further ρ : R → [0 , ∞ ) such that ρ ( s ) = 0 for | s | > R R ρ = 1. For δ > ρ δ ( s ) = δ ρ (cid:0) sδ (cid:1) . For 0 < δ < we use the notation ξ βδ := ρ δ ∗ ξ β ∈ C ( I T , H ) ∩ C ( I T , H ) . Because δ < it holds that ξ βδ still has compact support in ( − T, T ). For c / ∈ π Z we have ξ = − A − c ∂ s ξ + A − c D V c ( w ) ξ. Hence we get ξ βδ = ρ δ ∗ (cid:0) − A − c β ∂ s ξ + A − c D V c ( w ) ξ β (cid:1) = − ( ∂ s ρ δ ) ∗ ( A − c ξ β ) + ρ δ ∗ A − c (cid:16) ( ∂ s β ) ξ + D c V ( w ) ξ β (cid:17) . (46)From this formula we deduce that ξ βδ ∈ C ( I T , H ) ∩ C ( I T , H ) . We introduce the Hilbert spaces H := L ( I T , H ) , W := L ( I T , H ) ∩ W , ( I T , H ) . The inner product on W is given by h ζ , ζ i W = h ζ , ζ i L ( I T ,H ) + h ζ , ζ i W , ( I T ,H ) where h ζ , ζ i W , ( I T ,H ) = Z T − T h ζ ( s ) , ζ ( s ) i H ds + Z T − T (cid:10) ddt ζ ( s ) , ddt ζ ( s ) (cid:11) H ds. Between these spaces we have the operator D c : W → H , ζ ∂ s ζ + A c ζ . Using (46) we compute D c ξ βδ = ( ∂ s ρ δ ) ∗ ξ β − A c (cid:16) ( ∂ s ρ δ ) ∗ ( A − c ξ β ) (cid:17) + A c (cid:16) ρ δ ∗ A − c (cid:16) ( ∂ s β ) ξ + D c V ( w ) ξ β (cid:17)(cid:17) = ( ∂ s ρ δ ) ∗ ξ β − ( ∂ s ρ δ ) ∗ ξ β + ρ δ ∗ (cid:16) ( ∂ s β ) ξ + D c V ( w ) ξ β (cid:17) = ρ δ ∗ (cid:16) ( ∂ s β ) ξ + D c V ( w ) ξ β (cid:17) . In particular, with the constant κ := (cid:13)(cid:13) ( ∂ s β ) ξ + D c V ( w ) ξ β (cid:13)(cid:13) H independent of δ we have k D c ξ βδ k H κ . (47)For ζ ∈ W we compute k D c ζ k H = Z T − T k ∂ s ζ k H ds + 2 Z T − T h ∂ s ζ , A c ζ i H ds + Z T − T k A c ζ k H ds. If ζ has compact support in ( − T, T ) we obtain, using integration by parts, Z T − T h ∂ s ζ , A c ζ i H ds = − Z T − T h ζ , ∂ s ( A c ζ ) i H ds = − Z T − T h ζ , A c ∂ s ζ i H ds = − Z T − T h A c ζ , ∂ s ζ i H ds, and so Z T − T h ∂ s ζ , A c ζ i H ds = 0 , whence k D c ζ k H = Z T − T k ∂ s ζ k H ds + Z T − T k A c ζ k H ds > k ζ k W κ (48)for some constant κ > c . Putting (47) and (48) together we obtain k ξ βδ k W κ κ . -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 39 Since the bound κ κ is independent of δ , there exists a sequence δ ν → ν → ∞ suchthat ξ βδ ν converges weakly in W to some ξ β ∈ W . Since ξ βδ ν converges strongly in H to ξ β ,we conclude that ξ β = ξ β ∈ W . Hence ξ ∈ L ( I T − , H ) ∩ W , ( I T − , H ) , implying that w ∈ W , ( I T − , H ) ∩ W , ( I T − , H ) . Taking advantage of w = − A − c ∂ s w + A − c V ( w )we deduce that w ∈ L ( I T − , H ) ∩ W , ( I T − , H ) ∩ W , ( I T − , H ) . This proves the proposition in the case where the almost complex structure is constantand given by multiplication with i .The general case, where the almost complex structure is not constant, follows similarlyby a change of the framing. Changing the framing leads to an error term which is of lowerorder and does not affect the argument. This was for example used in [38]. Here is howthis works. Suppose that ∂ s w + J t ( w ) ∂ t w = V ( w ) . Choose a smooth family of matrices such thatΦ t ( z ) J t ( z ) = i Φ t ( z ) , z ∈ C n , t ∈ S . Set ξ ( s )( t ) := Φ t (cid:0) w ( s )( t ) (cid:1) ∂ s w ( s )( t )= Φ t (cid:0) w ( s )( t ) (cid:1)(cid:16) − J t (cid:0) w ( s )( t ) (cid:1) ∂ t w ( s )( t ) + V (cid:0) w ( s )( t ) (cid:1)(cid:17) = − i Φ t (cid:0) w ( s )( t ) (cid:1) ∂ t w ( s )( t ) + Φ t (cid:0) w ( s )( t ) (cid:1) V (cid:0) w ( s )( t ) (cid:1) . We introduce the map φ : C ( H , H ) ∩ C ( H , H ) → C ( H , H ) ∩ C ( H , H )which for a vector field V ∈ C ( H , H ) ∩ C ( H , H ) is given by φ ( V )( z )( t ) = Φ t ( z ( t )) ◦ V ( z )( t ) , z ∈ H , t ∈ S . Using thatΦ t (cid:0) w ( s )( t ) (cid:1) ∂ s ∂ t w ( s )( t ) = Φ t (cid:0) w ( s )( t ) (cid:1) ∂ t (cid:16) Φ − t (cid:0) w ( s )( t ) (cid:1) ξ ( s )( t ) (cid:17) = − (cid:16) ˙Φ t (cid:0) w ( s )( t ) (cid:1) + D Φ t (cid:0) w ( s )( t ) (cid:1) ∂ t w ( s )( t ) (cid:17) ∂ s w ( s )( t )+ ∂ t ξ ( s )( t ) we compute ∂ s ξ ( s )( t ) = − i (cid:16) D Φ t (cid:0) w ( s )( t ) (cid:1) ∂ s w ( s )( t ) (cid:17) ∂ t w ( s )( t ) − i Φ t (cid:0) w ( s )( t ) (cid:1) ∂ s ∂ t w ( s )( t )+ D ( φ ( V )) ∂ s w ( s )( t )= − i (cid:16) D Φ t (cid:0) w ( s )( t ) (cid:1) ∂ s w ( s )( t ) (cid:17) ∂ t w ( s )( t )+ i (cid:16) ˙Φ t (cid:0) w ( s )( t ) (cid:1) + D Φ t (cid:0) w ( s )( t ) (cid:1) ∂ t w ( s )( t ) (cid:17) ∂ s w ( s )( t ) − i∂ t ξ ( s )( t )+ D ( φ ( V )) ∂ s w ( s )( t ) . Writing Ψ c ( w ) ∈ C ( I T , H ) forΨ c ( w )( s )( t ) = − i (cid:16) D Φ t (cid:0) w ( s )( t ) (cid:1) ∂ s w ( s )( t ) (cid:17) ∂ t w ( s )( t ) (cid:16) i ˙Φ t + iD Φ t ∂ t w ( s )( t ) + D ( φ ( V )) (cid:17)(cid:0) w ( s )( t ) (cid:1) ∂ s w ( s )( t ) + c ξ ( s )( t )we then get the compact expression ∂ s ξ + A c ξ = Ψ c ( w ) , similar to (45). Using this, the proof for the general case now proceeds like the one in thespecial case where J was constant. (cid:3) The following lemma is an elaboration of [37, Lemma 3.6].
Lemma B.3.
For
T > the inclusion ι : L ( I T , H ) ∩ W , ( I T , H ) ∩ W , ( I T , H ) → C ( I T , H ) ∩ C ( I T , H ) is a compact operator.Proof. For N ∈ N abbreviate by V N ⊂ H = L ( S , C n ) the subspace of finite Fourierseries of the form z = N X k = − N z k e πik , z k ∈ C n . Note that V N is finite-dimensional. Indeed, its real dimension is 2 n (2 N + 1). Let π N : H → V N be the orthogonal projection. Note that the Fourier basis is a common orthogonal basis ofthe Sobolev spaces H k = W k, ( S , C n ) for every k ∈ N . In particular, the restriction π N | H k : H k → V N coincides with the orthogonal projection of H k to V N . LetΠ N : L ( I T , H ) ∩ W , ( I T , H ) ∩ W , ( I T , H ) → W , ( I T , V N ) , w π N ◦ w. Because V N is finite-dimensional, the inclusion I N : W , ( I T , V N ) → C ( I T , V N ) -EQUIVARIANT RABINOWITZ–FLOER HOMOLOGY 41 is a compact operator. We finally abbreviate by J N : C ( I T , V N ) → C ( I T , H ) ∩ C ( I T , H )the inclusion. Denote by ι N : L ( I T , H ) ∩ W , ( I T , H ) ∩ W , ( I T , H ) → C ( I T , H ) ∩ C ( I T , H )the composition of these three maps, ι N := J N ◦ I N ◦ Π N . Because I N is compact and the other two maps are continuous, ι N is a compact operator.We show that ι N converges to ι in the norm topology as N → ∞ . To see this we argueby contradiction and assume that there exists a constant c > N ∈ N there exists w N ∈ L ( I T , H ) ∩ W , ( I T , H ) ∩ W , ( I T , H ) with the property thatmax n k ( ι − ι N ) w N k C ( I T ,H ) , k ( ι − ι N ) w N k C ( I T ,H ) o = 1 (49)but max n k w N k L ( I T ,H ) , k w N k W , ( I T ,H ) , k w N k W , ( I T ,H ) o c. (50)Choosing c larger if necessary we can assume that c T . From (49) we deduce that k ( ι − ι N ) w N k C ( I T ,H ) = 1 (51)or k ( ι − ι N ) w N k C ( I T ,H ) = 1 . (52)We first discuss case (51). In this case there exists s ∈ I T such that k (id − π N ) w N ( s ) k H = 1 . For s ′ ∈ I T with | s ′ − s | c we have | s ′ − s | c k w N k W , ( I T ,H ) k (id − π N ) w N k W , ( I T ,H ) . We can thus estimate k (id − π N ) w N ( s ′ ) k H > k (id − π N ) w N ( s ) k H − (cid:12)(cid:12)(cid:12)(cid:12) Z s ′ s k (id − π N ) ∂ σ w N ( σ ) k H dσ (cid:12)(cid:12)(cid:12)(cid:12) > − k (id − π N ) w N k W , ( I T ,H ) p | s ′ − s | > − = . In particular, k (id − π N ) w N ( s ′ ) k H > πN from which we deduce that k w N k L ( I T ,H ) > r π N c = πN c which contradicts (50) as soon as N > c π . This contradiction shows that case (51) cannotoccur, and case (52) leads to a contradiction in a similar way. This shows that ι N convergesto ι as N → ∞ . In particular, ι arises as the limit of compact operators and is thereforecompact itself. This finishes the proof of the lemma. (cid:3) Proof of Theorem B.1, for k = 2 . Under the hypothesis of Theorem B.1, and in view ofequation (43), the proof of Proposition B.2 reveals that there exists a bounded subset U ⊂ L ( I T − , H ) ∩ W , ( I T − , H ) ∩ W , ( I T − , H )such that w ν | I T − ∈ U for all ν ∈ N . Hence by Lemma B.3 a subsequence of w ν | I T − converges in C ( I T − , H ) ∩ C ( I T − , H ) to w , that clearly solves (42). (cid:3) References [1] A. Abbondandolo and W. Merry. Floer homology on the time-energy extended phase space. Toappear in
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E-mail address : [email protected] Felix Schlenk, Institut de Math´ematiques, Universit´e de Neuchˆatel
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