Morse cohomology in a Hilbert space via the Conley Index

Main theorem of this paper states that Floer cohomology groups in a Hilbert space are isomorphic to the cohomological Conley Index. It is also shown that calculating cohomological Conley Index does not require finite dimensional approximations of the vector field. Further directions are discussed.


pair (X,Y ) define cohomology groups by
where CY is a cone on Y . In the case when X is a compact Hausdorff space and Y is its closed subset H n (X, Y ) coincides with Alexander -Spanier cohomology group and H n (X, Y ) = [X/Y, K(F, n)].
One can also define cohomology groups with compact supports of a locally compact Hausdorff space U by H n c (U ) = H n (U + , * ) = [U, K(F, n)] c , where U + denotes the one-point compactification of U and [, ] c denotes homotopy classes of compactly supported maps. Throughout rest of the paper, we take F = Z 2 .
We are now ready to give an overview on what E-cohomology is. Let E be a Hilbert space with a splitting E = E + ⊕ E − where each of E + and E − is either infinite dimensional or trivial. We say that {E n } n∈N is an approximating system for E if 1. E n is finite dimensional subspace of E for every n; 2. there is an inclusion i n,n : E n → E n for every n > n ; 3. n E n = E.
We recall the definition of E-cohomology in two extremal cases: when E + = {0}, E − = l 2 and when E + = l 2 , E − = {0}. For l 2 we take a canonical approximation system induced by the spaces of finite sequences. However, one can prove (see [G-G], [Abb97]) that the definition does not depend on the choice of the approximation system. Let us first consider the case of E = {0} ⊕ l 2 . Take a closed and bounded set X ⊂ E. We define finite codimensional cohomology in the following way ( [G-D]). Put E n = {(x 1 , x 2 , ...) ∈ E : x k = 0 for k > n} E n = {(x 1 , x 2 , ...) ∈ E n : x n ≥ 0} E n = {(x 1 , x 2 , ...) ∈ E n : x n ≤ 0} and X n = X ∩ E n ,X n = X ∩Ê n ,X n = X ∩Ě n .
SinceX n+1 ∩X n+1 = X n andX n+1 ∪X n+1 = X n+1 the Mayer-Vietoris sequence for a triad (X n+1 ,X n+1 ,X n+1 ) gives a homomorphism Definition 2.1. Finite codimensional cohomology groups on l 2 are defined to be Notice that if k > 0 then dim E n + k > dim X n . Thus, above groups can be nontrivial only for negative k. We have chosen a convention which is compatible with [Abb97] and opposite to that in [G-D]. This would be more convenient when we deal with the case when both E + and E − are nonzero.
As the simplest nontrivial example take X = S(E) i.e. X is a unit sphere in E. Then X n = S(E n ), H n−1 (X n ) = F (= Z 2 ) and all the maps δ n are isomorphisms. Thus Let us also emphasize that if X is compact (in particular if it is contained in a finite dimensional subspace) then all the E-cohomology groups are trivial. For a general separable Hilbert space E = {0} ⊕ E − we can take an isomorphism with l 2 (i.e. choose an approximating system) and repeat the construction. This simple concept can be generalized in many directions. K. Gȩba and A.Granas ([G-G]) proved well definiteness for any generalized cohomology theory. For example, taking cohomotopy groups instead of cohomology groups gives us stable cohomotopy groups. In addition, they proved that the resulting theory is always a generalized cohomology theory on the Leray-Schauder category. Since the morphisms of the Leray -Schauder category are compact fields, one could try to apply above techniques to the fixed point theory. However, what would like to have an easy prove that cohomology of sphere does not depend on the radius of sphere. This cannot be done only using maps of the form Id + K. First of all, notice that above cohomology groups are trivially invariant under translations. A.Abbondandolo proved that they are also invariant under the flow deformations. This allows us to compare spheres of different radius and also, which is more important, to use Morse theory and the Conley Index techniques. Another feature of [Abb97], [K-Sz] and [Sz] is a generalization to so-called middle dimension cohomology i.e. the case when both E + and E − are of infinite dimension. Before introducing that, let us consider second extremal example: E + = l 2 , E − = {0}. In this case E -cohomology groups are defined by H k E (X) = lim(H k (X m , δ m ). where δ m is induced by the inclusion of X m into X m+1 . One can easily see that if X is a sphere in E m then H k E (X) is nontrivial (and equal to Z 2 ) only if k = m − 1. In fact, it is true that if X is locally compact then H * E is isomorphic to the compactly supported cohomology mentioned above. Now the middle dimensional cohomology groups are defined to be where X m,n = E m ⊕E n , δ n : X m,n → X m,n+1 is the map from the Mayer -Vietoris sequence and δ m : X m,n → X m+1,n is the map induced by inclusion. Again, this definition does not depend on the approximating system.
Above considerations suggest that one should think of E cohomology as cohomology of finite codimension cohomology with respect to E − and cohomology with compact supports with respect to E + .
We would like to emphasize the fact that E-cohomology groups satisfy axioms of a generalized cohomology theory (see Theorem 0.2 in [Abb97]). All the results presented below can be obtained by these axioms without the knowledge of the precise construction of E-cohomology groups. For the sake of completeness, let us recall the homotopy invariance, strong excision axiom and long exact sequence for a triple.
where L is a linear automorphism of E and K maps bounded sets into precompact sets.
We also say that E-morphisms Φ and Φ from (X,A) to (Y,B) are E-homotopic if there exists an E-homotopy joining them i.e. a continuous map Ψ : where L t is a linear automorphism of E and K maps bounded sets into precompact sets.
Above definitions allows us to state: • (Strong excision) if X and Y are closed and bounded subsets of E and i : ( • (Long exact sequence) For a triple X ⊂ Y ⊂ Z) of closed and bounded sets we have a long exact sequence In the proof of Proposition 3.1 we would also need two following lemmas ( [Abb97]).
the isomorphism being induced by the inclusion map.
Lemma 2.2. Let B be a closed subset of A. If there exists an E-homotopy the isomorphism being induced by the inclusion map.

Conley Index.
Let f ∈ C 2 (E, R) be a function of the form where L is a self-adjoint isomorphism, ∇b(x) is globally Lipschitz and D 2 b(x) is compact for every x ∈ E. Then operator L gives a splitting of E into E + and E − corresponding to positive and negative eigenspaces respectively. We would like to work with flows generated by the minus gradient equations i.e.ẋ = −∇f (x).
We define the cohomological Conley Index in a Hilbert space E to be the E-cohomology of an index pair in E. This is a different approach than in [Izy] and [Man] because it does not use finite dimensional approximations of the vector field. We compare those two approaches after proving Proposition 3.1.
Definition 3.1. Let N be an isolating neighborhood of an invariant set S. We call a closed and bounded pair (N 1 , N 0 ) an index pair for S if 1. N 0 is positively invariant relative to N 1 , 2. S ⊂ intN 1 \ N 0 and 3. if γ ∈ N 1 , t > 0 and γ · t ∈ N , then there exists t such that γ · [0, t ] ⊂ N 1 and γ · t ∈ N 0 . Moreover, we say that an index pair is regular if the function Unless otherwise stated, we assume that all the index pairs are regular.
Definition 3.2. We define the cohomological Conley Index of S (denoted by ch * (S)) to be Above definition only makes sense if we prove the independence of a choice of index pairs. This is stated in the following proposition.
Proposition 3.1. Let (N 1 , N 0 ), (N 1 ,N 0 ) be two regular index pairs contained in the same isolating neighborhood N . Then Proof of the Proposition 3.1 can be divided into three steps.
step 1: For every t > 0 there is an isomorphism step 2: For every t > 0 there is an isomorphism step 3: There exists T > 0 such that ) and the inclusions induce isomorphisms of E-cohomology groups.

Proof.
Step 1: From the excision axiom we have Step 2 can be done in a similar way as Step 1.
Step 3: TakeN = cl(N \ N 1 ). For every x ∈N there exists T x such that x · (−T x ) ∈N . In fact, we will show that there exists T 1 which satisfies above condition for every x ∈N . In a finite dimensional case this is just a consequence of the compactness ofN . Suppose we have a sequence (y n ) ⊂N such that y n · (−2n, 0) ⊂N . Put x n := y n × (−n). Then both sets x n · (−n, 0) and x n · (0, n) are contained inN . If x n k → x 0 then x 0 · (−∞, 0) ⊂N and x 0 ∈N and we have arrived at a contradiction.
Lemma 3.1. If x n · (−n, n) ⊂N for every n then (x n ) contains a convergent subsequence.
Suppose that (x + n ) ⊂ E + does not have a convergent subsequence. Then there exists > 0 such that |x + n − x + m | > for every n = m. Take s, T 1 > 0 such that N ⊂ B(0, s) |e T1L x| > 3s |x| for every x ∈ E + . Then for n, m > T 1 we have and so |K(x m , T 1 ) − K(x n , T 1 )| > s for every n, m > T 1 . However, K(·, T 1 ) is compact and we have a contradiction. As a consequence, we can choose a convergent subsequence (x + n l ). In a similar way, from (x − n l ) we can take a convergent subsequence (x − n k ) and this gives us a convergence of (x n k ).
By the same argument we can find T 2 > 0 such that for every x ∈ N 0 we have x · T ∈ N 0 . Take T = max{T 1 , T 2 , T 1 , T 2 } where T 1 , T 2 correspond to the pair (N 1 ,N 0 ). Then The proof that above inclusions induce isomorphisms on the cohomology groups runs as in the finite dimensional case (see [Smol]).
Remark 3.1. We want to emphasize that for an isolated invariant set S there exists a regular index pair. Let U be an isolating neighbourhood and define G T (U ) = |t|<T N · (−t, t). Then there exists T > 0 such that G T (U ) ⊂ int U . Suppose the converse, i.e. G n (U ) ⊂ int U for every n. Take x n ∈ G n (U ) \ int U i.e. x n · (−n.n) ⊂ U . By Lemma 3.1 there exists a convergent subsequence x n k → x 0 ∈ S ⊂ int U . A contradiction. This proves that G T (U ) ⊂ int U for some T > 0. For such an U one can construct a regular index pair (see Theorem 5.5.13 in [Ch]).
Let us compare our definition of the cohomological Conley to the one which uses finite dimensional approximations of the vector field ( [Izy]). For a compact K define K n : E → E by K(x) = P n • K • P n (x).
Let S be an invariant for the flow generated by F = L+K and letN be its isolating neighborhood. Then F is related by the continuation to F n0 = L + K n0 for sufficiently large n 0 . Let (N, L) be an index pair for the approximation i.e. for the finite dimensional flow generated by F n0 |En 0 : coincide with the cohomological Conley index defined in [Izy]. Since E-cohomology does not depend on the index pair, those two approaches coincide in general.

Main Theorem.
Let us recall that we are interested in a flow generated by the minus gradient vector field for a function f ∈ C 2 (E, R) of the form where L is a self-adjoint isomorphism, ∇b(x) is globally Lipschitz and D 2 b(x) is compact for every x ∈ E.
Let S be a compact isolated invariant set containing only non-degenerate critical points x 1 ,...,x n and orbits connecting them. For a non-degenerate critical points x we define an E-index by where V is the negative eigenspace of D 2 f (x), E + and E − are respectively positive and negative eigenspaces of L. Suppose further that the transversality condition holds i.e. if ind E y − ind E x = 1 the stable manifold of y and unstable of x intersect transversally.
Main Theorem. We have where HF * (S) denotes Floer cohomology.
D.Salamon used analogous theorem in a following way (see [Sal]). Take  Thus, this is just another proof that Morse theory recovers singular cohomology groups.
Let us first give the main ideas of the proof. Take two non-degenerate critical points x and y of a relative index 1 and a connecting orbit C. By the transversality,Ŝ = {x, y, C} is an isolated invariant set. Now choose a triple (N 2 , N 1 , N 0 ) in such a way that the pairs (N 2 , N 0 ), (N 2 , N 1 ), (N 1 , N 0 ) are index pairs for the invariant setsŜ, {y} and {x} respectively.
We have a long exact sequence One can show that the cohomological Conley Index for S is trivial i.e. all the groups H n E (N 2 , N 0 ) are trivial. Thus, for every k we have an isomorphism A.Abbondandolo computed (see Proposition 14.6 in [Abb97]) the cohomological Conley Index for a non-degenerate critical point.
(let us remind that we work with coefficients F = Z 2 ). The only nontrivial morphism in (1) is an isomorphism between H k+1 E (N 2 , N 1 ) and H k E (N 1 , N 0 ) where k = ind E x. By the compactness of S and the transversality condition, we have finitely many orbits C 1 , C 2 , . . ., C m connecting y and x. TakeŜ = {x, y, C 1 , . . . , C m }. By the additivity (see [McC]), the Conley connections matrix is sum of isomorphisms from Z 2 to itself so it is an algebraic count modulo 2. This is exactly the Floer boundary operator. Since ch * ({x}) Z 2 one can think of ch * ({x}) as of a generator of the Floer chain group C indE x .
Here are some technical details of the above construction. We would like to prove that the Conley Index ofŜ = {x, y, C} is trivial. Let us examine a special case. Suppose C is contained in one dimensional subspace E 1 and Lemma 4.1. The Conley Index ofŜ is trivial.
Proof. We follow an approach of C.McCord ( [McC]) i.e. we use a series of continuations. Choose a small isolating neighborhood N of S . First continue the vector field F (x, y) = (F x (x, y), F y (x, y)) to F 1 (x, y) = (F x (x, 0) + D y F x (x, 0)y, F y (x, 0) + D y F (x, 0)y) = (F x (x, 0) + D y F x (x, 0)y, D y F (x, 0)y) and then to F 2 (x, y) = (F x (x, 0), D y F (x, 0)y). Now put a(x) = F x (x, 0), M = max x∈[0,1] a(x) and continue F 2 (x, y) to Notice that inv(F 3 , S) = ∅ and thus the Conley Index is trivial. Now we would like to find an E-homotopy which reduces a general case to the above one (compare section C in [G]).
Let M be a compact C 1 submanifold of a Hilbert space E.
Lemma 4.2. There exists a finite dimensional subspace T of E such that the orthogonal projection P T onto T maps M diffeomorphically onto P T (M ).
Proof. For every x ∈ M there is an open neighbourhood U x such that U x is diffeomorphic to the open neighbourhood of 0 in T x M via the exponential map. Choose a finite subcover U x1 ,U x2 , ..., U x k and put T = span{T xi M : i = 1, . . . , k}.
The orthogonal projection P T : M → T is a local embedding. Thus, for a given x ∈ M , there is only a finite number of points y 1 ,y 2 , ..., y p such that P T x = P T y i . Define T x to be the space spanned by T and y 1 − x, y 2 − x, ... , y p − x and let P x be the orthogonal projection onto T x . It is easy to see that there is an open neighbourhood V x of x such that y ∈ V x , z ∈ M and P x z = P x y imply z = y. Again, choose a finite cover V x1 ,V x2 ,...,V xq and put T = span{T i : i = 1, . . . , q} Proof. By Lemma 4.2 we can find a finite dimensional space such that P T : M → T is an injection. For x ∈ M define φ x : M → E by φ x (y) = P T (y − x) + x Then φ x is an imbedding and φ x (x) = x. Let U x1 , ..., U x k be a cover of M and {ν i } be a subordinated partition of unity. Define φ : M → E by φ(x) = ν i (x)φ xi (x). Take T 1 to be the space spanned by T and x 1 , ..., x k . Then φ(M ) ⊂ T . Define η 0 : P T (M ) → T ⊥ by η(P T x) = x − φ(x). Since P T (M ) is a C 1 submanifold of T we can extend η to a C 1 map on T . Define Ψ(x, y) = x − tη(P T x).
Two critical points together with an orbit between them is a compact submanifolds of E. Thus we can apply above lemma to M =Ŝ. Suppose Φ(Ŝ, 1) is contained in a finite dimensional space T 1 . Choose one dimensional subspace E 1 ⊂ T 1 and a diffeomorphism h of T 1 which takeŝ S onto (−1, 1) ⊂ E 1 . Extend h to E by the identity on T ⊥ . This reduces a general case to the one in Lemma 4.1.

Further Directions.
Some of the Floer theories come with additional symmetry. One expects an analogous theorem to the main theorem of this paper for the equivalent Floer cohomology and the equivariant Conley Index. In a special case of an S 1 -action, this would prove the conjecture that the Monopole Floer cohomology and the Seiberg-Witten Floer cohomology are isomorphic (HM * (Y ) HSW * (Y )). Another direction, that one would like to investigate, is the case of Hilbert (Banach) manifolds. Let us just recall that recently intensively explored Lagrangian intersection Floer theory (see [FOOO]) is a Floer theory on a Banach manifold.