aa r X i v : . [ m a t h . S G ] S e p GEOMETRIC CYCLES IN FLOER THEORY
MAX LIPYANSKIY Introduction
Historical Overview.
In the mid 80’s, Andreas Floer obtained a positive solu-tion to Arnold’s conjecture on the minimal number of fixed points of a Hamiltoniansymplectomorphism. For this purpose, Floer introduced a new homology theory forthe loop space of a symplectic manifold. His theory is an infinite dimensional versionof Morse theory applied to the symplectic action functional on the loop space. Thecritical points of this functional correspond to fixed points of the symplectomorphism,while the boundary operator counts the dimension zero moduli spaces of (perturbed)holomorphic curves connecting the critical points [5]. Under suitable hypothesis, Floerwas able to show that the resulting homology theory is well defined and independentof the perturbation data necessary to construct the theory. Moreover, Floer showedthat the homology theory was isomorphic to the singular homology of the underlyingsymplectic manifold and thus proved the the Arnold conjecture. In subsequent work,Floer generalized his theory to other contexts such as the more general problem of La-grangian intersections as well as an analogous theory for the Chern-Simons invariantfor connections on a 3-manifold. In these cases, a simple topological interpretation ofthe resulting groups is not available. The groups encode deep geometric informationabout the relevant configuration space which cannot be reduced to the ”classical”topology of that space.From a foundational standpoint, the definition of the Floer homology groups is per-haps not satisfactory. The relevant functionals are usually not Morse-Smale and thushave to be perturbed in some manner to even define the groups. As a result, onehas to then show that the groups are indeed independent of the chosen perturba-tion. Furthermore, a rather delicate analysis of the compactification of the modulispace of trajectories is necessary to establish even the most basic properties of thetheory; for instance, the fact that the chain of groups generated by critical pointsindeed form a complex. The situation is of course analogous to the finite dimensionalstory. One may take as the definition of homology of a compact manifold the chaincomplex associated to a Morse-Smale function. However, establishing even the basicproperties, such as functoriality under mappings, is quite nontrivial. On the otherhand, with singular homology theory at one’s disposal, Morse homology becomes aneffective and illuminating way of computing the homology groups. The central goal f the present work is to find an appropriate analogue of singular homology in theFloer context. It should be emphasized, however, that while in the finite dimensionalsituation singular homology provides a way of avoiding the analytic machinery that isnecessary for setting up Floer’s theory, the theory developed in the current work restsheavily on the use of Sobolev spaces and appropriate nonlinear Fredholm operatorsbetween them. This is, perhaps, a reflection of the fact that although many results inFloer theory have purely topological interpretations, ultimately the theory deals withthe qualitative behavior of solutions to certain elliptic partial differential equations.Let’s briefly describe, what is to our knowledge, the earliest evidence for the exis-tence of such a theory. In the late ’80s, Atiyah [1] and others, observed that, from thepoint of view of relative Donaldson invariants, one may view Floer’s theory as a theoryof ”semi-infinite cycles”. We retell his observation in the language of Seiberg-Wittentheory. Consider a closed Riemannian 4-manifold X with a spin C -structure and spinorbundle W (see section ?? for the full definitions). Let M ( X ) be the moduli spaceof solutions to the SW equations modulo the gauge group action. As is well known(see for example [4]), when b + > X = X + ⊔ Y X − decomposes along a 3-manifold Y intotwo compact 4-manifolds with boundary. Let B ( Y ) denote the configuration space ofpairs ( B, Ψ), where B is a Clifford connection and Ψ is a section of the spinor bundleover Y , modulo the gauge group action. We have restriction maps R ± : M ( X ± ) → B ( Y )At least on the point-set level, one has M ( X ) = M ( X + ) × B ( Y ) M ( X − )In other words, modulo the action of the gauge group, solutions on X correspondto solutions on X ± that agree on the boundary. Standard elliptic boundary valuetheory implies that M ( X ± ) are in fact Hilbert manifolds. Therefore, one mighthope to interpret the fibre product M ( X + ) × B ( Y ) M ( X − ) in the smooth category.Speculating even further, one might hope for the existence of Floer groups HF + ( Y )and HF − ( Y ) with an intersection pairing HF + ( Y ) ⊗ HF − ( Y ) → H ∗ ( B ( Y ))where H ∗ denote the singular homology functor. Abstracting this situation, given asmooth map σ : P → B ( Y )where P is some Hilbert manifold, we are led to the following problems:1. What properties should such maps have to have finite dimensional intersections?2. What properties should such maps have to have compact intersections? he answer to the first problem is well known and involves the notion of a polarizedHilbert manifold. Very loosely, one may think of a polarization of a Hilbert manifoldsuch as B ( Y ) as an equivalence class of local splittings of the tangent bundle: T B ( Y ) = T + B ( Y ) ⊕ T − B ( Y )As it turns out, B ( Y ) comes with a natural choice of polarization for which we have π − ◦ D R + : T M ( X + ) → T − B ( Y )Fredholm and π + ◦ D R + : T M ( X + ) → T + B ( Y )compact. Similarly, π + ◦ D R − : T M ( X − ) → T + B ( Y )is Fredholm while π − ◦ D R − : T M ( X − ) → T − B ( Y )is compact. The fact that M ( X + ) × B ( Y ) M ( X − ) is finite dimensional is an immediateconsequence of Fredholm theory. Therefore, to ensure finite dimensional intersectionsit is reasonable to require our cycles to respect the polarization. The resolution of thesecond problem is significantly more subtle and is the main subject of this work. Asfar as we know, the first attempt to do so is due to Tom Mrowka and Peter Ozsvathand the present dissertation owes a considerable debt to their original insight. Acknowledgement.
We wish to thank Tom Mrowka for supervising the author’sthesis which is the basis of this work. In addition, we thank Peter Kronheimer andDennis Sullivan for useful conversations. Finally, we would like to thank the SimonsCenter For Geometry and Physics for their hospitality while this work was beingcompleted.1.2.
An Outline of the Contents.
Here we briefly describe the contents and pro-vide some motivation for the constructions that follow.As the basic structure we will be considering a Hilbert space B with a polariza-tion T B ∼ = T + B ⊕ T − B as well as a functional L : B → R . In section 2 we layout the axioms for a map σ : P → B from a Hilbert manifold P to define a cycle.The motivation comes from the strong L proof of compactness for Seiberg-Wittenmoduli spaces as presented in [4]. On a more technical note, we will define and usethe notion of locally cubical (lc) manifolds. It appears to be a useful and technicallysimple structure to work with.In section 3 we will discuss our main example coming from Symplectic Geometry. Theexample concerns the space of loops in C n . We prove the required L -compactnesstheorem. In section 4 we construct a family of perturbations for cycles ensuring thatintersections can be arranged to be transverse. The construction is quite similar to erturbing manifolds in finite dimensions and also rests on the Sard-Smale theorem.One must simply check that the perturbation does not take us out of the category ofcycles. It is perhaps worth remarking that unlike in traditional Floer theory, whereperturbations involve changing the metric, complex structure or the hamiltonian func-tion, the perturbations here do not alter the geometric data. Therefore, our theory isdefines for a wide class of functionals that can potentially be highly degenerate. Thisis illustrated in our proof of the existence of periodic orbits for loops in C n . In thiscase the functional is degenerate and yet we are able to extract the relevant geometricinformation.In section 5 we define a general class of maps (called correspondences) Z → B × B that give rise to maps on the Floer groups via fibre products. The definition is generalenough to include not only moduli spaces on a cobordism W but also the diagonalmap B → B × B . The definition is a little technical but is forced on us by theconsiderations that follow.Section 6 is the technical heart of the theory. Our goal is to prove that the triv-ial cobordism induces the identity on the Floer groups. This is established by findinga cobordism between the correspondence coming from the moduli space of solutionson the cylinder and the diagonal map. If M t denotes the solutions on a cylinder oflength t , we can form the disjoint union ⊔ t ∈ (0 , M t . We complete this manifold byadding the diagonal B → B × B at t = 0. This is achieved by arguing that on asmall cylinder, a solution is specified by the appropriate spectral projections to theboundary. For this we need to apply the contraction mapping theorem and in viewof the nonlinear terms need rather precise estimates. This establishes that we have aHilbert manifold that with boundary M ⊔ B . However, the restriction maps definedon the cylinder extend only weakly to the diagonal map as we approach the t = 0boundary. Given a cycle P → B we show that by changing coordinates, we canassume that the difference map R − σ : ⊔ t ∈ (0 , M t × P → B is C up to the bound-ary. Such a coordinate change preserves the lc-structure but not the smooth structureon ⊔ t ∈ (0 , M t × P → B . This is the principal motivation for introducing lc-manifolds.In section 7 we illustrate the general theory by reproving that for a general classof hamiltonian functions H : C n → R , there exists a nontrivial periodic orbit. Su-perficially, the proof is similar to the one in [2]. However, our proof is based on theunregularized gradient flow and does not use minimax methods. It gives a rathernatural interpretation of the cycles appearing in the construction. We end with acouple of technical appendices. . Main Construction lc-Manifolds of Depth ≤ . In this work it will be important to work withHilbert manifolds with corners and some rather weak smoothness between differentstrata. In an appendix, we will introduce a rather technical notion of locally-cubicalmanifolds or lc-manifolds for short. For the sake of the reader, in this section wesimply write down the definitions for the simplest nontrivial case. In the terminologyof lc-manifolds, this is a depth one lc-manifold. For the purposes of defining thebordism groups this is sufficient and illustrates all the essential technical difficulties.Therefore, we decided to first give the definition in this special case. While manyof the propositions will be stated for general lc-manifolds, on first reading one maysimply restrict to the case described below.
Definition 1.
An lc-manifold of depth one is a Hausdorff space P , with a distin-guished closed subset called its boundary ∂P ⊂ P . We assume both ∂P and P − ∂P are Hilbert manifolds. Furthermore, each point p ∈ ∂P has a neighborhood U ⊂ ∂P and an open embedding f : U × [0 , ǫ ) → P such f ( u,
0) = u while f | U ×{ } and f | U × (0 ,ǫ ) are diffeomorphisms. Let us call such a map f : U × [0 , ǫ ) → P an lc-chart. Let P be an lc-manifold ofdepth one and B some Hilbert manifold. Definition 2.
A continuous map σ : P → B is lc-smooth if the following hold:1. σ is smooth on ∂P and P − ∂P
2. Each point p ∈ ∂P has an lc-chart U × [0 , ǫ ) such that, in the chart coordinates, σ along with its first derivative in the U direction is continuous on U × [0 , ǫ ) . Given an lc-smooth map σ : P → B , we denote by Dσ p : T P → T B the differentialrestricted to the open stratum on which p lives.2.2. Floer Spaces.
We will assume all our Hilbert manifolds to be separable. Let B be a Hilbert manifold. For our examples, however, it suffices to consider the caseof a Hilbert space. We have the following notion of polarization: Definition 3.
A polarization of B is a direct sum decomposition T B = T + B ⊕ T − B Definition 4. A Floer space ( B , L ) is a polarized Hilbert manifold together with acontinuous function L : B → R . In addition, we assume B is equipped with a coarserweak topology. Definition 5. A chain σ : P → B where P is an lc-manifold is an lc-smooth mapsatisfying the following axioms: Axiom . On im( σ ) , L is bounded below and lower semi-continuous for the weaktopology. xiom . Given a weakly converging sequence σ ( x i ) with limit y , if lim( L ( σ ( x i )) = L ( y ) then some subsequence of x i converges strongly on P . Axiom . Any subset S ⊂ im( σ ) on which L is bounded is precompact for theweak topology. Axiom . Π − ◦ Dσ p : T P → T − B is Fredholm, Π + ◦ Dσ p : T P → T + B is compactfor each p ∈ P . Remark. A σ satisfying Axiom 4 is said to be a semi-infinite map . Example.
Take a Hilbert space H = H + ⊕ H − split into two infinite dimensionalsubspaces with its usual strong/weak topologies and L ( v + , v − ) = | v − | − | v + | . Thepolarization is given by the splitting. P = H − with the inclusion map defines a cycle. Definition 6.
A chain σ : P → B has index k if the linearized map Π − ◦ Dσ : T P → T − B has index k at each point of P − ∂P . Remark.
Note that index ( σ | ∂P ) = index ( σ ) −
1. Indeed, since in an appropriatelc-chart around a point p ∈ P , σ becomes σ : V × [0 , ǫ ) → B the differential of σ in the v -variables is continuous on V × [0 , ǫ ). Therefore, the indexof Dσ on V × (0 , ǫ ) is exactly one greater than the index Dσ | V . Definition 7.
Two chains σ : P → B , σ : P → B are said to be isomorphic ifthere exists a diffeomorphism f : P → P such that σ ◦ f = σ . Floer Bordism.
The easiest invariant to define is a Floer Bordism Group:
Definition 8.
A cycle is a chain of depth 0. In other words, ∂P = ∅ Definition 9.
Let Ω F k ( B , L ) be the Z -vector space generated by isomorphism classesof cycles of index k . Disjoint union is the additive structure. Furthermore, [ P ] = 0 if σ : P → B extends to a chain of depth one σ ′ : W → B with ∂W = P and σ ′| ∂W = σ .Let Ω F ∗ ( B , L ) = L k Ω F k ( B , L ) . Definition 10.
Given ( B , L ) as above, let − B be the polarized Hilbert space obtainedby switching T + B and T − B and let ( − B , −L ) be the Floer space obtained by switchingthe sign of L . The motivation for our definition of chain is the following result:
Lemma 1.
Given cycles σ : P → B and τ : Q → − B , their intersection σ ∩ τ = P × B Q is compact. roof. On the image of the intersection, L is bounded above and below. Therefore,Axiom 3 implies that this image is weakly precompact. Furthermore, since on the im-age L is both lower and upper semi-continuous, it is continuous in the weak topology.Axiom 2 implies every sequence x i ∈ σ × B τ must have a convergent subsequence. (cid:3) Let us make the following perturbation hypothesis which will be verified for exam-ple we consider.
Existence of Perturbations:
Given cycles σ : P → B and τ : Q → − B thereexists a chain F : P × [0 , → B with F | P × = σ and F | P × transverse to τ . Further-more, if σ is already transverse to τ , without changing F | P × , we may alter F to betransverse to τ as well. Theorem 1.
Given cycles σ and τ as above, having transverse intersection, theirfibre product σ × B τ is a closed manifold mapping to B . The fibre product gives awell-defined map Ω F k ( B , L ) × Ω F l ( − B , −L ) → Ω k + l ( B ) where Ω k + l ( B ) denotes ordinary lc-bordism with Z -coefficients.Proof. Note that we can view σ × B τ as ( σ × τ ) − (∆) where ∆ is the diagonal in B × B . By assumption, σ × τ is transverse to ∆ and thus σ × B τ is a smoothfinite dimensional lc-manifold. To calculate the dimension, note that locally σ × B τ is ( σ − τ ) − (0). Up to compact perturbation, the linearized operator has the form (cid:18) Π − ◦ Dσ − Π + ◦ Dτ (cid:19) Thus, the dimension of σ × B τ is ind (Π − ◦ Dσ )+ ind (Π + ◦ Dτ ). Finally, if F : W → B is an chain with ∂F = σ , we have ∂ ( F × B τ ) = ∂F × B τ = σ × B τ when F is transverse to τ . Therefore, the fibre product descends to a map onΩ F ∗ ( B , L ). (cid:3) Remark.
In the context of Floer theory discussed in this work one can modify thedefinition of Ω F ∗ ( B , L ) so that the fibred product lies in the usual smooth bordismgroups, rather than the lc-bordism groups. However, since our primary interests is inhomology rather than bordism we do not develop this here.3. Loop Space of C n Semi-Infinite Cycles for the Action Functional on L / ( S , C n ) . We nowturn to our main example. Let B be the Hilbert space of L / loops on C n . Explicitly, f a loop γ is decomposed in Fourier series γ ( θ ) = X n c n e inθ then the square of the L / -norm of γ is X n | c n | | n | + | c | Given a smooth function H : C n → R we define the action functional by: L H ( γ ) = Z h− J ˙ γ ( θ ) , γ i − H ( γ ( θ )) dθ where J is the standard complex structure in C n . The formal L -gradient of L H is: ∇L H ( γ ) = − J ∂ θ γ − ∇ H ( γ )Since B is a linear space, we may define the polarization by the splitting T B = T + B ⊕ T − B where T + B is spanned by the positive eigenvectors of − J ∂ θ and T − B by the nonpositive eigenvectors of − J ∂ θ . We have constructed a Floer space and thushave an associated Floer group Ω F ∗ ( B , L H ).From now on, assume H is smooth with H = 0 near 0 and H ( x ) = (1 + ǫ ) | x | for | x | large. Fix a unit vector e + ∈ T + B . We construct cycles for this Floer space.Following [2], we have distinguished subsets:Σ τ = { γ | γ − + se + , || γ − || L / ≤ τ, ≤ s ≤ τ } and Γ α = { γ ∈ T B + , || γ || L / = α } It is elementary to show (see [2]) that for τ ≫ L H | ∂ Σ τ ≤ α > β > L H | Γ α ≥ β . Note that Σ τ ∩ Γ α = { αe + } transversely. Lemma 2. Σ τ is a cycle for ( − B , −L H ) and Γ α is a cycle for ( B , L H ) .Proof. The proofs are nearly identical so let us focus on Γ α . The key observation isthat L H ( γ + ) = Z h− J ˙ γ + ( θ ) , γ + i − H ( γ + ( θ )) dθ = 12 || γ + || L / − Z H ( γ + ( θ )) dθ From this it follows that the action functional on γ + essentially coincides with the L / norm. Indeed, we may write H ( x ) = H c ( x )+(1+ ǫ ) | x | where H c has compact support.Given γ + ∈ Γ τ we have || γ + || L / bounded uniformly. Therefore, L H is bounded.Lower semicontinuity follows from the fact that the L / norm can only drop in aweak limit and the fact that H c is continuous for the weak topology. Given that L H does not drop implies the L / does not drop which in turn implies convergence. (cid:3) . Perturbations
To ensure transverse intersection of cycles we need to be able to perturb them witha sufficiently large parameter space at the same time ensuring that the perturbedmap is still a chain. We show how to construct such perturbations for the loop space.
Definition 11.
Let P ⊂ B = L / ( S ; C n ) be the unit ball in the L -norm. From thecompactness of the inclusion L ⊂ C we have that every sequence v i ∈ P has a C convergent subsequence. Let ρ be a positive bump function equal to 1 on [ − ,
1] and 1 /x outside [ − , F : σ × P → B by F ( x, v ) = σ ( x ) + ρ ( || σ ( x ) || L / ) v We have the following theorem:
Theorem 2.
Given a chain σ : P → B the map F : P × P → B ( Y ) satisfies:1. F ( x,
0) = σ ( x ) DF ( x,v ) has dense image for all ( x, v ) Π − ◦ D x F is Fredholm, Π + ◦ D x F and D v F are compact4. Given a compact lc-manifold K ⊂ P , F | P × K is again a semi-infinite chain.Proof. Claim: F satisfies all the requirements of the theorem.Part 1: Clear from the construction.Part 2: Note that DF ( x,v ) (0 , w ) = ρ ( || σ ( x ) || L / ) w and since L ( S ; C n ) ⊂ B is compact with dense image and ρ ( || σ ( x ) || L / ) > DF has dense image and D v F is compact.Part 3: DF ( x,v ) ( y,
0) = Dσ x ( y ) + T ( y ) v where T : T P → R is the linear map given by T ( y ) = Dρ ( y ) Therefore,viewed as amap on T P , DF ( y,
0) differs from Dσ by an at most rank one map from which part3 follows.Part 4: Observe that L ( F ( x, v )) − L ( σ ( x )) is bounded independent of x . For this,write L ( γ ) = Q ( γ, γ ) + G ( γ ) here Q is a bilinear form and G is a bounded function. Let a = σ ( x ) and b = vρ ( || a || L / ). To bound L ( F ( x, v )) − L ( σ ( x )) we need to bound Q ( a, b ) and Q ( b, b ).This follows from the definition of the perturbation.Also, observe that if σ ( x i ) are weakly L / convergent L ( ˜ F ( x i , v i )) − L ( σ ( x i )) isin fact strongly convergent, after passing to a subsequence. This follows from thefact that b i is C precompact. This implies L ( F ( x i , v i )) drops exactly when L ( σ ( x i ))drops. Let us check that P × K → B satisfies all the axioms. Given a weaklyconvergent sequence σ ( x i ) + c i · v i note that c i · v i converge strongly since K is L / precompact. Therefore, σ ( x i ) is weakly convergent and lower semi-continuity fol-lows. If L does not drop in the limit, we must have x i precompact and thus ( x i , v i )precompact as well. (cid:3) We can now use the perturbations to put cycles in general position:
Theorem 3.
Given a cycle σ : P → ( B , L ) and a cycle τ : Q → ( − B , −L ) thereexists a cobordant cycle σ ′ : P → ( B , L ) such that σ ′ is transverse to τ . Furthermore,given two such transverse cycles σ ′ and σ ′′ there exists a cobordism Σ : P × [0 , → ( B , L ) transverse to τ , with ∂ Σ = σ ′ − σ ′′ .Proof. The argument follows the standard route. The map F × τ → B × B istransverse to ∆ ⊂ B × B . The map ( F × τ ) − (∆) → P is Fredholm. Hence, applyingSmale’s extension of Sard’s theorem [3], we have that for generic p ∈ P the map F ( , p ) × τ is transverse to ∆. Any to such generic values p, q may be connected byan arc γ : [0 , → P transverse to ( F × τ ) − (∆) → P (cid:3) Correspondences
Definitions.
We now explain how to obtain maps between bordism groups ofFloer spaces:
Definition 12.
A correspondence ( Z, f ) ∈ Cor (( B , L ) , ( B , L )) is a map f : Z → B × B where Z is a Hilbert manifold (possibly with boundary) satisfying the follow-ing axioms: Axiom ′ . On im( f ) , L − L is bounded below and lower semi-continuous forthe weak topology. Axiom ′ . If L ( π ( z i )) is bounded above and π ( z i ) is a weakly precompact se-quence then π ( z i ) weakly precompact. Axiom ′ . Given π ( z i ) is weakly convergent to x , if lim f ∗ ( L − L )( z i ) = ( L −L )( x ) and π ( z i ) is converges strongly then z i converges strongly (up to a subse-quence). xiom ′ . ( π +0 , π − ) ◦ Df : T Z → T + B ⊕ T − B is Fredholm. Given a boundedsequence v i ∈ T Z , if π +0 ( Df )( v i ) is weakly convergent, π +1 ( Df )( v i ) is precompact. Axiom ′ . Df : T Z → T B is dense. Df | ∂ ( Z ) : T Z → T B is also dense. Example 1.
The diagonal map ∆ : B → B × B is a correspondence. Theorem 4.
Given a chain σ : P → B and a correspondence f : Z → B × B ,the fiber product π ◦ f : P × B Z → B is chain in ( B , L ) .Proof. Axiom 1: −L + L > C and L > C imply L > C . Given a sequence( x i , z i ) ∈ P × B ( Y ) Z with a weakly convergent sequence f ( z i ), we havelim inf( −L ( f ( z i ))) + L ( f ( z i )) ≥ −L ( f ( z ∞ )) + L ( f ( z ∞ ))and lim inf L ( f ( z i )) ≥ L ( f ( z ∞ ))imply lim inf L ( f ( z i )) ≥ L ( f ( z ∞ )).Axiom 2: If lim L ( f ( z i )) = L ( f ( z ∞ )) Axiom 1 ′ implies that L can only rise inthe limit. However, since σ ( x i ) are assumed weakly convergent as well, we havelim L ( σ ( x i )) = L ( σ ( x ∞ )) and thus σ ( x i ) = π ( f ( z i )) is strongly convergent so Ax-iom 3 ′ implies z i strongly convergent as well.Axiom 3: L ( f ( z i )) < C implies L ( f ( z i )) < C and thus σ ( x i ) is weakly precom-pact. Axiom 2 ′ implies that π ( z i ) is weakly precompact as well.Axiom 4: We use the following lemma Lemma 3.
Given a linear Fredholm map T : W → V ⊕ V such that Π ◦ T issurjective, T | ker(Π ◦ T ) → V is Fredholm with the same index.Proof. We have ker( T ) = ker( T | ker(Π ◦ T ) ). Surjectivity of Π ◦ T implies the dimensionof cokernel coincides as well. (cid:3) Take a polarization of T B ⊕ T B with projections (Π + i , Π − i ). We apply this lemmato the map F : T Z × T P → T + B ⊕ T − B ⊕ T − B with F ( z, p ) = (Π +0 ( Df ( z )) − Π +0 ( Dσ ( v ))) ⊕ (Π − ( Df ( z )) − Π − ( Dσ ( p ))) ⊕ Π − ( Df ( z ))Axiom 6 ′ implies the above lemma applies since im Df ⊕ Dσ in T B is closed anddense. To calculate the index deform through Fredholm operators to˜ F ( z, p ) = Π +0 ( Df ( z )) ⊕ Π − ( Dσ ( p )) ⊕ Π − ( Df ( z )) hus, with respect to the given polarization,dim( Z × B ( Y ) P ) = ind ( Df ) + ind ( Dσ ) (cid:3) Lemma 4.
A correspondence F : Z → B × B of index k without boundary inducesa map: Ω F k ( F ) : Ω F ∗ ( B , L ) → Ω F ∗ + k ( B , L ) Proof.
Given a cycle σ : P → B we have ∂ ( σ × B F ) = ∂ ( σ ) × B F since ∂F = ∅ This shows that Ω F k ( F ) commutes with the boundary operator. (cid:3) The Trivial Cobordism L Compactness for a Holomorphic Cylinder.
In this section we explainhow a holomorphic cylinder gives rise to a correspondence. In fact, we will demon-strate that this correspondence induces identity on Floer bordism. The intuitionbehind the proof is the observation that a trivial cobordism is the analogue of agradient flow in finite dimensions and letting the flow time shrink to zero inducesthe identity map on the underlying manifold. In our infinite dimensional setting theinitial value problem is not well-defined and thus our homological argument is meantas a substitute notion.Take S to be the standard circle of length 2 π . Let Z T = [0 , T ] × S be the cylinderwith coordinates ( t, θ ) and complex structure j ( ∂ t ) = ∂ θ . Let H : C n × S → R be a Hamiltonian with associated vector field X H = J ◦ ∇ H . Given an L -map u : Z T → R n we define the energy to be E ( u ) = 12 Z T Z π | u t | + | u θ − X H ( u, t ) | dθds The upward gradient flow of ∇L H is: ∂ t u ( t, θ ) = ∇L H ( u ( t, θ )) = − J ∂ θ u ( t, θ ) − ∇ H ( u ( t, θ ))We write this as a perturbed J -holomorphic curve equation as:(1) ∂ t u ( t, θ ) + J ( ∂ θ u ( t, θ ) − J ◦ ∇ H ( u ( t, θ ))) = 0Therefore, for a u : [0 , T ] × S → C n satisfying the perturbed holomorphic curveequation we have: L H ( u ( T, · )) − L H ( u (0 , · )) = E ( u )Consider X H ( u, t ) = c · u + X H c ( u, t ) where X H c ( u, t ) : R n × S → R n is C and compactly supported and c ∈ √− R . Lemma 5. X H c ( u, t ) is continuous in u for the L topology. roof. Since X H c ( u, t ) has compact support, we have | X H c ( v, t ) − X H c ( v ′ , t ) | ≤ C | v − v ′ | Therefore, | X H c ( v, t ) − X H c ( v ′ , t ) | ≤ C | v − v ′ | Given u , u : S → R n , we have Z | X H c ( u , t ) − X H c ( u , t ) | dt ≤ C Z | u − u | dt = C || u − u || L (cid:3) Theorem 5.
Assume c / ∈ √− Z . Given a sequence with E ( u i ) < C , the u i are uni-formly L bounded and thus weakly precompact. Given a weakly convergent sequence u i , we have E ( u ∞ ) ≤ lim inf E ( u i ) . If lim E ( u i ) = E ( u ∞ ) , the u i converge stronglyin L to u ∞ . Finally, if c ∈ √− Z , the theorem applies if we furthermore assume || u i (0 , θ ) || L < C ′ .Proof. We first prove the theorem when X H c = 0. When c / ∈ √− Z we have anisomorphism ∂ θ + c : L ( S ) → L ( S )Thus, 12 Z T Z π | u t | + | u θ − X H ( u, t ) | dθds = const · || u || L Thus, the energy is equivalent to the L norm from which everything follows. When c ∈ √− Z a special argument is needed. Since we have Z T Z π | u t | dθdt < C we get || u (0 , θ ) | L − | u ( τ, θ ) | L | ≤ τ / || u t || L for all τ ∈ [0 , T ]. This, together with the L bound on u (0 , θ ), implies an L boundon u . The bound on u θ follows since we have bounds on u t and X H ( u ) = c · u . Now,assume u i are weakly convergent. We may rewrite the energy as12 Z T Z π | u i,t | + | u i,θ | + | c · u i | + 2 h u i,θ , c · u i i dθds Weak L convergence of u i implies R T R π h u i,θ , c · u i i dθds converges to Z T Z π h u ∞ ,θ , c · u ∞ i dθds and thus lower semicontinuity follows from that of the L norm. The case X H c = 0is a slight modification of the argument. We observe that | ( u θ − c · u ) − X H c ( u, t ) | = | u θ − c · u | + | X H c ( u, t ) | + 2 h u θ − c · u, X H c ( u, t ) i ence | ( u θ − c · u ) − X H c ( u, t ) | ≥ | u θ − c · u | + | X H c ( u, t ) | − |h u θ − c · u, X H c ( u, t ) i| ≥| u θ − c · u | + | X H c ( u, t ) | − | u θ − c · u | / − | X H c ( u, t ) | = 1 / | u θ − c · u | − | X H c ( u, t ) | Since we have an L ∞ bound on X H c ( u, t ), a bound on E ( u ) is the same as a boundon 12 Z T Z π | u t | + | u θ − c · u | dθds In addition, since X H c ( u, t ) is continuous for the L topology, the lim i →∞ E ( u i ) = E ( u ∞ ) exactly whenlim i →∞ Z T Z π | u i,t | + | u i,θ − c · u i | dθds = 12 Z T Z π | u ∞ ,t | + | u ∞ ,θ − c · u ∞ | dθds (cid:3) Remark.
In the case c ∈ √− Z the assumption on u may seem artificial. How-ever, this assumption is exactly met when describing the axioms of a correspondence.Therefore, the holomorphic maps on a cylinder will always give rise to a correspon-dence under the above assumptions.Let M T be the moduli space of solutions to equation 1 on a cylinder of length T .Restriction maps to 0 × S and T × S induce maps R : M T → B , R T : M T → B . Lemma 6.
Given
T > , Z T together with the restriction maps define a correspon-dence R × R T : M T → B × − B Proof.
This follows from the compactness theorem above together with the uniquecontinuation property. (cid:3)
APS Boundary Value Problem.
Consider D = ∂ t + L where L = J ∂ θ is afirst order elliptic differential operator acting on C n valued maps on S . We have theAPS boundary value problem:( D ǫ , Π − L ◦ r ǫ − Π + L ◦ r ) : L ([0 , ǫ ] × S , C n ) → L ([0 , ǫ ] × Y, C n ) ⊕ L / ( Y, C n )Where Π + is the spectral projection to the nonnegative part of the spectrum of L and Π − is the projection to the negative part.Thus, given α ∈ L ([0 , ǫ ] × S , C n ) the boundary data is specified by β = β +0 + β − ǫ with β +0 = − Π + L ◦ r ( α ) and β − ǫ = Π − L ◦ r ǫ ( α ) where r denotes restriction. We havethe following lemma: emma 7. D ǫ is an isomorphism with inverse P ǫ ⊕ Q ǫ where P ǫ ⊕ Q ǫ : L ([0 , ǫ ] × S , C n ) ⊕ L / ( S , C n ) → L ([0 , ǫ ] × S , C n ) There exists
C > such that || P ǫ || ≤ C and || Q ǫ || ≤ C , independent of ǫ .We have || P ǫ ( a ) | ∂ ([0 ,ǫ ] × S ) || L / ≤ C || a || L and || Q ǫ ( b ) | ∂ ([0 ,ǫ ] × S ) − b || L / ≤ Z [0 ,ǫ ] × S | ∂ t Q ǫ ( b ) | Proof. P ǫ : Take { φ λ } an orthonormal eigenbasis for L ( S , C n ). We may write anyelement in L ([0 , ǫ ] × ( S , C n ) as a sum P λ g λ ( t ) φ λ with g λ ∈ L . For λ > P ǫ ( g λ φ λ ) = φ λ Z t e − λ ( t − τ ) g λ ( τ ) dτ = h ( t ) φ λ We compute the L norm of h ( t ) φ λ . Since λ > ǫ ) we can use λ R ǫ | h | + R ǫ | ∂ t h | to compute the square of the L norm of h ( t ) φ λ .We have Z [0 ,ǫ ] × Y | D ǫ ( hφ λ ) | = λ Z ǫ | h | + Z ǫ | ∂ t h | + λ | h ( ǫ ) | This bounds the L norm of h ( t ) φ λ as well as the L / norm of h ( ǫ ) φ λ in terms of || gφ λ || L . Q ǫ : We have an explicit formula for the inverse: Q ǫ ( φ λ ) = − e − tλ φ λ , λ > Q ǫ ( φ λ ) = e − ( t − ǫ ) λ φ λ , λ ≤ λ >
0. We have D ǫ ◦ Q ǫ = 0. The L -norm of Q ǫ ( φ λ ) isbounded by 1 since e − tλ ≤
1. We have ∂ t ( Q ǫ ( φ λ )) = − λQ ǫ ( φ λ ) thus, Z [0 ,ǫ ] × S | ∂ t ( Q ǫ ( φ λ )) | = λ (1 − e − λǫ )2Since ∂ t ◦ Q ǫ = L ◦ Q ǫ and the L / -norm of φ λ is | λ | / this establishes the desiredbound. Finally, note that || φ λ − Q ǫ ( φ λ ) |{ ǫ }× S || L / = λ (1 − e − ǫλ ) ≤ λ (1 − e − ǫλ ) = 2 Z | ∂ t ( Q ǫ ( φ λ )) | (cid:3) In applications, we will have a nonlinear term for which we use the following lem-mas: emma 8. Given f ∈ L ([0 , ǫ ] × S , C n ) , vanishing on one of the ends, we have || f || L ≤ C || f || L with C independent of ǫ .Proof. The argument reduces to that of a function on R with support in the rectangle[0 , × [0 , ǫ ]. We assume f vanishes on say { } × [0 , ǫ ] and [0 , × { } . We have | f ( x, y ) | ≤ Z | ∂ f ( x ′ , y ) | dx ′ ≤ ǫ / ( Z | ∂ f ( x ′ , y ) | dx ′ ) / Similar estimate with ∂ implies | f ( x, y ) | ≤ ǫ Z | ∂ f ( x ′ , y ) | dx ′ Z | ∂ f ( x, y ′ ) | dy ′ Integrating, gives Z | f ( x, y ) | dxdy ≤ ǫ Z | ∂ f ( x, y ) | dxdy Z | ∂ f ( x, y ) | dydx ≤ ǫ ( Z |∇ f ( x, y ) | dxdy ) (cid:3) Lemma 9.
Given β ∈ L / ( Y, E ) and v ∈ L ([0 , ǫ ] × Y, E ) we have || Q ǫ ( β ) + P ǫ ( v ) || L ≤ C || β || L / + C || v || L Proof.
Decompose v = v + + v − into positive (nonpositive) eigenvectors of L . We have || P ǫ ( v ) || L ≤ || P ǫ ( v − ) || L + || P ǫ ( v + ) || L By construction, each of these two terms vanishes on an end of the cylinder. Thus,since || P ǫ ( v − ) || L ≤ C || v || L and || P ǫ ( v + ) || L ≤ C || v || L the conclusion holds for the v term.For Q ǫ ( β ) we need to investigate terms of 3 types. Let β = β − + β + + β wherethe decomposition corresponds to breaking up b into the positive, negative and zeroeigenspaces. For β , we have Q ǫ ( β ) = β Since there are only finitely many eigenvec-tors of L with zero eigenvalue, we can bound the L -norm of Q ǫ ( β ) by C || β || L / . For β + we note that although Q ǫ ( β + ) = − P λ e − tλ c λ φ λ does not vanish on the endpointsit extends to an L function on [0 , ∞ ] × S . In fact, by the calculation in the firstlemma, the L -norm of the extension is bounded by the L / norm of β + . The lemmaabove applies since this extension vanishes at ∞ . The argument for β − is similar. (cid:3) Lemma 10. || Q ǫ || ( L / : L ) is uniformly bounded in ǫ and approaches 0 weakly as ǫ → .Proof. Choose any δ >
0. We have β = P ki =1 c i φ λ i + β ′ where || β ′ || L / < δ/C . As ǫ →
0, we have || Q ǫ ( φ λ i ) || L → C norm of Q ǫ ( φ λ i ) is bounded by that of φ λ i the length of the cylinder is going to zero. Thus || Q ǫ ( β ) || L ≤ P ki =1 || c i Q ǫ ( φ λ i ) || L + δ and we can choose ǫ so small that P ki =1 || c i Q ǫ ( φ λ i ) || L ≤ δ (cid:3) emma 11. We may write X H ( x ) as K ( x ) · x where K is a function with K (0) = 0 and | K ( x ) | ≤ C | x | . Furthermore, | K ( x ) · x − K ( y ) · y | ≤ C ( | x | + | y | ) | x − y | .Proof. Let G ( x ) = X H ( x ). Since H ( x ) = 0 when x is near 0, we have G ( x ) = Z DG ( tx ) dt · x Let K ( x ) = R DG ( tx ) dt . Since DG (0) = 0 and | DG ( tx ) | ≤ C | t || x | , we have K (0) =0 and | K ( x ) | ≤ C | x | as desired. Note that | K ( x ) − K ( y ) | ≤ C | x − y | since | DG ( tx ) − DG ( ty ) | ≤ C | tx − ty | We estimate: | K ( x ) · x − K ( y ) · y | ≤ | K ( x ) · x − K ( x ) · y | + | K ( x ) · y − K ( y ) · y | ≤ C ( | x | + | y | ) | x − y | (cid:3) Lemma 12.
Given functions α , β on the cylinder S × [0 , ǫ ] , we have || X H ( α ) − X H ( β ) || L ≤ C ( || α || L + || β || L ) || α − β || L where C is independent of ǫ .Proof. Integrating the inequality of the previous lemma we get: Z | X H ( α ) − X H ( β ) | dθdt ≤ C Z ( | α | + | β | ) | α − β | dθdt ≤ C ( || α || L + || β || L ) || α − β || L (cid:3) Borrowing notation from the section on the shrinking cylinder, we deduce that v X H ( Q ǫ ( β ) + P ǫ ( v ))is a contraction mapping for small enough ǫ and a fixed β . Let us denote this mapby F ǫ ( β, v ). Slightly more generally, we may consider F ǫ ( β, v ) = g ǫ − X H ( Q ǫ ( β ) + P ǫ ( v ))where g ǫ ∈ L ([0 , ǫ ] × S , C n ). F ǫ ( β, ) is contraction mapping with the same con-stants. To ensure F ǫ ( β, ) maps a ball of radius 1 / C to itself we must suppose that g is sufficiently small.Consider the map G ǫ ( β, v ) = ( β, v − F ǫ ( β, v )). For ǫ small, the previous lemma allowsus to conclude that the existence of an inverse H ǫ ( β, v ) with G ǫ ( β, H ǫ ( β, v )) = ( β, v ). urthermore, H ǫ ( β, → ǫ →
0. We would like to conclude the same for thederivative:
Lemma 13.
Let D H ǫ ( β, be the derivative with respect to the β variable at the point ( β, . We have | D H ǫ ( β, | → as ǫ → .Proof. Pick b ∈ T β L / ([0 , ǫ ] × S , C n ). F ǫ ( β, H ǫ ( β, H ǫ ( β,
0) implies D F ǫ ( β,H ǫ ( β, ( b ) + D F ǫ ( β,H ǫ ( β, ( D H ǫ ( β, ( b )) = D H ǫ ( β, ( b )The desired result will follow if we can estimate the LHS. From the definition, D F ǫβ,v ( b ) = ∇ K ( Q ǫ ( β )) Q ǫ ( b ) · ( P ǫ ( v ) + Q ǫ ( β )) + K ( P ǫ ( v ) + Q ǫ ( β )) · ( Q ǫ ( b ))and D F ǫβ,v ( w ) = ∇ K ( P ǫ ( v ))( P ǫ ( w )) · ( P ǫ ( v ) + Q ǫ ( β )) + K ( P ǫ ( v ) + Q ǫ ( β )) · ( P ǫ ( w ))Plugging in v = H ǫ ( β,
0) and w = D H ǫ ( β,
0) and using that Q ǫ ( β ) and P ǫ ( H ǫ ( β, → ǫ → (cid:3) Adding the Collar.Lemma 14.
Given b ∈ B , decompose b as b +0 + b − ǫ . There exists sufficiently small ǫ > such that there is a unique small energy holomorphic curve γ with b +0 + b − ǫ asthe mixed boundary value.Proof. This follows immediately from the arguments of the previous subsections. In-deed, let P ǫ = P ǫ ⊕ P + ǫ as above. If v is the unique small fixed point of the map v g − X H ( Q ǫ ( b ) + P ǫ ( v ))then ( ∂ t + J ∂ θ )( P ǫ ( v ) + Q ǫ ( b )) + X H ( Q ǫ ( b ) + P ǫ ( v )) = g Note that since g is some fixed section with L norm and L norm going to zero as ǫ → (cid:3) We complete ∪ t ∈ (0 , M t to form a lc-manifold ∪ t ∈ [0 , M t as follows. The 0th stratumis ∪ t ∈ (0 , M t while the 1st stratum is M ` ∆. A sequence z i ∈ ∪ t ∈ (0 , M t is saidto converge to z ∈ ∆ if R ( z i ) converges in L / to z . Lemma 14 implies that thecompleted space has the structure of an lc-manifold. In fact, the completed manifoldis smooth, but the extension of the restriction map to the diagonal is not smooth.We will see how to deal with this in a later section. .4. Verifying the Axioms.
In this section we verify that the completed correspon-dence satisfies the first 3 axioms of a correspondence. This ensures that for a chain σ , we will have that σ × B ∪ t ∈ [0 , M t satisfies the first 3 axioms of a chain. In otherwords, those axioms that deal with the convergence properties. The existence of alc-structure on σ × B ∪ t ∈ [0 , M t will be handled in a separate section. Note, that it ispossible to modify the definition of a correspondence so that ∪ t ∈ [0 , M t is a genuinecorrespondence and thus σ × B ∪ t ∈ [0 , M t automatically has such an lc-structure. Wechoose to avoid this more general definition since it seems to obscure matters and willnot be used in the future.We will first deduce a uniform L bound on configurations. Suppose for this sec-tion that E ( γ i ) is bounded and R ( γ i ) weakly converges (this hypothesis is satisfiedin all of the axioms we need to check). We may also assume that we have a sequence ofsolutions γ i on cylinders of shrinking length as all the other cases have been covered.Energy bounds give us uniform bounds on L -norm of γ i by theorem 5 Lemma 15.
We have a uniform bound on ||R ǫ ( γ i ) || L / . Assume, R ( γ i ) is L / convergent and E ( γ i ) → . We have R ǫ ( γ i ) L / -convergent and γ i L -convergent.Proof. For strong compactness, we assume E ( γ i ) → R ( γ i ) converges. By thearguments on weak compactness, convergent of R ( γ i ) implies the same of R ǫ ( γ i ).Note, since the energy is approaching zero, eventually the sequence is in the domain ofthe contraction mapping theorem and thus lies in the collar. Therefore, the endpointsuniquely parameterize the solutions γ i and strong convergence follows from that ofthe endpoints. (cid:3) Observe that our discussion establishes the following:
Corollary 1. If E ( γ i ) < C and R ( γ i ) is uniformly bounded, we have || γ i || L isuniformly bounded We are in good shape to verify the axioms:Axiom 1 ′ : L − L is bounded below by 0 since energy L is nonincreasing on tra-jectories. To establish lower semi-continuity, we claim that in fact any sequence( R ( γ i ) , R ǫ ( γ i )) as above weakly converges (up to a subsequence) to a diagonal ele-ment. We can assume ( R ( γ i ) , R ǫ ( γ i )) is strongly L convergent. We have: ||R ( γ i ) − R ǫ ( γ i ) || L ≤ Z ǫ || dγ i /dt || L ≤ ǫ / || γ i || L ≤ Cǫ / since || γ i || L is uniformly bounded. This implies the claim and thus the lower semi-continuity of L − L . Axiom 2 ′ and 3 ′ have been verified in the previous lemma. .5. Concluding the Proof.
Given a cycle σ : P → B we verify that P × B ∪ t ∈ [0 , M t is a cobordism between P and P × B M . Given that P is transverseto ∪ t ∈ (0 , M t , the axioms of a cycle for P × B ∪ t ∈ [0 , M t were verified in the precedingsections. What needs to be checked is that it has the structure of a lc-manifold.The potential problem occurs near the diagonal where the restriction maps convergein C to the inclusion map. More precisely, in the collar coordinates ( b + , b − , t ), R ( b + , b − , t ) = b + + e tL b − + G ( t, b ) where G ( t, b ) as a function of b converges to 0in C topology as t →
0. Note that e tL | B − is a family of compact operators convergingto the identity in the weak topology. Thus, we don’t have C convergence for therestriction map.We will work locally, so assume that P is a ball around the origin in Hilbert space.Let b = σ (0). Near b , B is an affine space modeled on T − B ⊕ T + B . By assumption, π − ◦ Dσ is Fredholm. Applying the inverse function theorem to π − ◦ σ , we can findcoordinates for σ so that σ ( p ) = b + f ( p ) + A ( p ) where Df p is compact at all p and A is an linear Fredholm map A : T P → T − B In these coordinates, the map R − σ : P × ∪ t ∈ [0 , M t → B can written as( p, b + , b − , t ) → ( e tL − b − + b + − A ( p ) + e tL b − − f ( p ) + G ( t, b + b )Pick a left inverse A − for A . Thus, A ◦ A − − I has finite rank on T − B . Define achange of coordinates by ( b + , b − , p, t ) ( b + , b − , ˜ p, t )with ˜ p = p − A − ◦ e tL b . This is a homeomorphism with inverse taking ˜ p to p =˜ p + A − e tL b − . Note that for each fixed t ≥
0, the map is a diffeomorphism. With thenew coordinates the map R − σ becomes(˜ p, b + , b − , t ) → ( e tL − b − + b + − A (˜ p + A − e tL b − )+ e tL b − − f (˜ p + A − e tL b − )+ G ( t, b + b )This can be simplified to(˜ p, b + , b − , t ) → ( e tL − b − + b + − A (˜ p ) + K ◦ e tL b − − f (˜ p + A − e tL b − ) + G ( t, b + b )where K = I − A ◦ A − is a finite rank operator. Since the change of coordinates is ahomeomorphism, the continuity of the map up to the boundary still holds. We claimthat the differential in the ˜ p and b variables converge as t → t = 0.Computing the differential at (˜ p, b, t ) we have:( δ ˜ p, δb + , δb − ) δb + − A ( δ ˜ p )+ K ◦ e tL δb − − Df p ( δ ˜ p + A − e tL δb − )+ DG t,b + b ) ( δb + + δb − )We want this differential to converge to( δ ˜ p, δb + , δb − ) δb + + Kδb − + Df p ( δ ˜ p + A − δb − ) n view of the compactness of Df p and K as well as the fact that e tL is self-adjoint,the claim is a consequence of the following lemma proved in the appendix: Lemma 16.
Given a uniformly bounded weakly converging sequence of operators A i : V → W between Hilbert spaces and a strongly convergent sequence of compactoperators K i : W → U , K i ◦ A i converge strongly to K ∞ ◦ A ∞ provided A ∗ i convergeweakly to ( A ∞ ) ∗ . Thus, we have found coordinates where the difference R − σ is lc-smooth so theinverse function theorem with parameter implies ( R − σ ) − (0) is a manifold withboundary. Finally, we need to verify that the projection to the other end is smoothin the new coordinates. This time the map is( p, b + , b − , t ) e − tL ( b + + b +0 ) + b − + b − + G ( t, b )where again G ( t, b ) as a function of b converges to 0 in C topology as t →
0. Notice,however, restricted to the fiber product b + = π + ( σ ( p )) and thus, restricted to the fibreproduct, the map may be written using the ˜ p coordinates as(˜ p, b + , b − , t ) e − tL ( π + ( σ (˜ p + A − e tL b − )) + b +0 ) + b − + b − + G ( t, b )The derivative is:( δ ˜ p, δb + , δb − ) e − tL ( π + ◦ Dσ p ( δ ˜ p + A − e tL δb − )) + δb − + DG t,b ) ( δb )Since π + ◦ Dσ p is compact for any p , we have e − tL ◦ π + ◦ Dσ p compact as well. Thefollowing lemma (also proved in the appendix) implies e − tL ◦ π + ◦ Dσ p is converging: Lemma 17.
Given a uniformly bounded weakly converging sequence of operators A i : V → W between Hilbert spaces and a strongly convergent sequence of compactoperators K i : U → V , A i ◦ K i converge strongly to A ∞ ◦ K ∞ . From the lemma we conclude that e − tL ◦ π + ◦ Dσ p is a convergent sequence ofcompact operators, hence we can apply the previous lemma to conclude that e − tL ◦ π + ◦ Dσ p ◦ A − e tL is converging as well. This completes the proof of the existence ofan lc-structure on P × B ∪ t ∈ [0 , M t The Existence of a Critical Point
One of the central themes in [2] is how the existence of a critical point of the func-tion L H leads to a variety of applications in symplectic geometry. For example, thecelebrated nonsqueezing theorem is a rather direct consequence of the existence ofa symplectic capacity which in turn is defined crucially using the existence of crit-ical points of L H . In [2] this is established by using compactness properties of theregularized gradient flow of L H as well as the Leray-Schauder degree theory. In thissection we will demonstrate how the existence of a critical point can be establishedusing the unregularized gradient by appealing to the theory developed in this work.Recall that we have the decomposition T B = T + B ⊕ T − B . We may identify T B ith B . Fix a unit vector e + ∈ T + B . We assume H is smooth with H = 0 near 0and H ( x ) = (1+ ǫ ) | x | for | x | large. Recall the definition of Γ α and Σ τ from section 3 . σ : P → B , let F t ( σ ) = σ × B M t The shrinking argument applies in this case to show that the new cycle is cobordantto the original one.
Theorem 6.
There exists a critical point x of L H with L H ( x ) ≥ β .Proof. We argue by contradiction. Assume no such critical point exists. Then L H ( F t (Γ α )) > C for any C > t is sufficiently large. This is a familiarargument from Morse theory. By contradiction, if there exists a sequence of increas-ingly long gradient flow lines that start on Γ α and have bounded energy. There thenwould be a subsequence of trajectories of some fixed length with energy convergingto 0. By compactness, such a subsequence would converge to a critical point x with L ( x ) >
0. This is a contradiction. On the other hand, we then would have F t (Γ α ) × B Σ τ = ∅ since L H restricted to Σ τ is bounded above. This is impossible since F t (Γ α ) is cobor-dant to Γ α by a cobordism staying away from points where L H ≤ ∂ (Σ τ ), while Γ α intersects Σ τ transversely in a point. (cid:3) Appendix: Weakly Convergent Operators
Definition 13.
A sequence of operators A i : V → W between Hilbert spaces is saidto converge weakly, if there exists a bounded operator A ∞ : V → W such that for any v ∈ V we have A i ( v ) → A ∞ ( v ) . Lemma 18.
Given a uniformly bounded weakly converging sequence of operators A i : W → U between Hilbert spaces and a compact operator K : V → W , A i ◦ K convergestrongly to A ∞ ◦ K .Proof. Taking the new sequence A i − A ∞ we can assume A ∞ = 0. By contradiction,suppose there exists a sequence v i with | v i | = 1 and | A i ◦ K ( v i ) | ≥ C >
0. Since K iscompact, the elements w i = K ( v i ) have a converging subsequence w j with limit w ∞ .By assumption, lim i ( A i ( w ∞ )) = 0. Since A i are uniformly bounded we have A j ( w j ) = A j ( w j − w ∞ ) + A j ( w ∞ )Since A j are uniformly bounded and we havelim j | A j ( w j − w ∞ ) | ≤ const · lim j | w j − w ∞ | = 0 nd thus lim j A j ( w j ) = 0 contradicting the fact that | A j ( w j ) | = | A j ◦ K ( v j ) | ≥ C > (cid:3) Similarly we have:
Lemma 19.
Given a uniformly bounded weakly converging sequence of operators A i : V → W between Hilbert spaces and a compact operator K : W → U . Assume A ∗ i isalso weakly converging with limit ( A ∞ ) ∗ . We have, that K ◦ A i converge strongly to K ◦ A ∞ .Proof. Apply the previous first lemma to A ∗ i and K ∗ . (cid:3) Finally, combining the previous two lemmas we obtain:
Lemma 20.
Given a uniformly bounded weakly converging sequence of operators A i : V → W and A ′ i : V ′ → W ′ and a strongly convergent sequence of compact operators K i : W → V ′ , A ′ i ◦ K i ◦ A i converge strongly to A ′∞ ◦ K ∞ ◦ A ∞ .Proof. We have A ′ i ◦ K i ◦ A i − A ′ i ◦ K ∞ ◦ A i = A ′ i ◦ ( K i − K ∞ ) ◦ A i and | A ′ i ◦ ( K i − K ∞ ) ◦ A i | ≤ | A ′ i || K i − K ∞ || A i | Thus, since A i and A ′ i are uniformly bounded, it suffices to assume K i = K ∞ . Onthe one hand, the previous lemmas imply that the uniformly bounded sequence ofcompact operators T i = K ◦ A i is strongly convergent. Now, apply the same argumentto A ′ i ◦ T i . (cid:3) Appendix: lc-Manifolds
Spaces Stratified By Hilbert Manifolds.Definition 14.
A second countable Hausdorff space P has a stratification byHilbert manifolds of depth k , if P k ⊂ P k − . . . P = P where for each i , P i is closed in P and the open stratum P i − P i +1 is Hilbert manifold. A stratumsmooth map f : P → X where X is a Hilbert manifold is a continuous map smoothon each open stratum. Such a map is said to be transverse to a submanifold Y ⊂ X if it is transverse on each stratum. Note that the product of P and Q , for any two such spaces, is also stratified byHilbert manifolds. .2. Locally Cubical Hilbert Manifolds.
Let ~t ( k ) = [0 , k . We view ~t ( k ) is astratified space in the natural way. Let ~t denote a typical coordinate in ~t ( k ). Attimes, by abuse of notation, we let ~t ( k ) denote a neighborhood of the origin in [0 , k Definition 15.
Given a space P stratified by Hilbert manifolds and an open set V in P i − P i +1 , a locally cubical Hilbert manifold chart about V is an embedding (ofstratified Hilbert manifolds) f : V × ~t ( i ) → U ⊂ P where U is open in P and f ( v,
0) = v . Definition 16. A locally cubical Hilbert manifold (or lc-manifold for short) isa stratified Hilbert space P with a cover by locally cubical charts as above with nofurther compatibility assumptions (other than those impossed by the being embeddingsof stratified Hilbert manifolds). Lemma 21.
The product of two lc-manifolds is a lc-manifold.Proof.
Since ~t ( k ) × ~t ( k ′ ) = ~t ( k + k ′ ) in a canonical way, a cover is specified by chartsof the form V × V ′ × ~t ( k + k ′ ). (cid:3) Definition 17. A smooth map σ from a lc-manifold P to a Hilbert manifold X is astratum smooth map such that each point has at least one chart V × ~t ( i ) where σ hasthe form σ ( v, ~t ) with σ is smooth in the v coordinates and, along with its v -derivative,continuous in the ~t coordinates. Remark.
The reason for restricting to lc-maps as opposed to say smooth maps frommanifolds with corners will become apparent when dealing the shrinking cylinder ar-gument in the section 6.
Remark.
Given a smooth map σ : P → X as in the previous lemma and a smoothmap f : X → Y of Hilbert manifolds the composition f ◦ σ : P → Y is also smooth. Lemma 22.
Given a smooth map σ : P → X as in the previous lemma and a closedsubmanifold Y ⊂ X such that σ is transverse to Y , σ − (∆) is an lc-manifold.Proof. Note that σ − (∆) is naturally a space stratified by Hilbert manifolds. In achart, we are reduced to the following local situation. Given σ : V × ~t ( k ) → W where V, W are Hilbert spaces and f is smooth in the v variables and, along with itsfirst v -derivative, continuous in the ~t variables. Assume σ ( v ) = σ ( v,
0) is has 0 asa regular value. Then, locally there is a stratum preserving smooth homeomorphism σ − (0) × ~t ( k ) → σ − (0). This, in turn, follows from the inverse function theorem withdependence on a parameter. (cid:3) eferences [1] M Atiyah
New Invariants of 3- and 4-Dimensional Manifolds , Proceedings of Symposiain Pure Mathematics, volume 48.[2]
H Hofer
Symplectic invariants and Hamiltonian dynamics , Birkh¨auser Verlag, 1994.[3]
S Smale
An infinite dimensional version of Sard’s theorem , Amer. J. Math., pg 861-866,1965.[4]
P Kronheimer and T Mrowka
Monopoles and Three-Manifolds , New MathematicalMonographs (No. 10), Cambridge University Press, 2007.[5]
A Floer
The unregularized gradient flow of the symplectic action , Comm. Pure Appl.Math., vol 41 pg. 775-813, 1988.
Simons Center for Geometry and Physics, Stony Brook University, Stony Brook,NY 11794
E-mail address : [email protected]@gmail.com