The E-Cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on T²ⁿ

2.1 Module Structure for E-Cohomology

We begin this section by recalling E-cohomology from [1], where we slightly modify the definition as in [11]. Let E be a separable real Hilbert space and E+, E- closed subspaces such that E=E+⊕E-. In what follows we denote by H∗ Alexander–Spanier cohomology with compact supports, for which we refer to [14] and the nice survey in [1, Section 1]. Moreover, we let 𝒱 be the set of all finite-dimensional subspaces of E-, which is partially ordered by inclusion and directed.

If U,V,W∈𝒱 are such that W=V⊕U and dim⁡(U)=1, then we can decompose W into two subspaces by setting

where u≠0 is a fixed element in U. Note that the choice of u corresponds to an orientation of the one-dimensional space U, and changing this orientation swaps W+ and W-.

Figure 1

Decomposition of X by W.

We set for a closed and bounded subset X of E,

and note that XW=XW+∪XW- as well as XV:=X∩(E+×V)=XW+∩XW-. (See Figure 1.) If now A⊂X is closed, then we obtain a relative Meyer–Vietoris sequence

In the more general case that W=V⊕U and dim⁡(U)=n≥2, we decompose U into n one-dimensional subspaces U=U1⊕…⊕Un and set Wi=V⊕U1⊕…⊕Ui for 1≤i≤n as well as W0=V. Then the previous construction yields n Mayer–Vietoris homomorphisms

and their composition is a homomorphism Hk⁢(XV,AV)→Hk+n⁢(XW,AW). Hence we have constructed for any q∈ℤ and V,W∈𝒱, V⊂W, a homomorphism

As noted in [1, Proposition 2.2], these maps do not depend on the choice of the one-dimensional subspaces Ui and their orientations. In summary, {Hq+dim⁡(V)⁢(XV,AV),ΔV⁢Wq⁢(X,A)} is a direct system of abelian groups over the directed set 𝒱.

The inclusions ιV,W:XV→XW for V,W∈𝒱 yield an inverse system {Hp⁢(XV),ιV,W∗} over 𝒱. We define for p∈ℤ the group H0p⁢(X) as the inverse limit

In what follows, we denote elements of H0p⁢(X) by [αV]0 if αV∈Hp⁢(XV), and correspondingly elements of HEq⁢(X,A) by [αV]E if αV∈Hq+dim⁡(V)⁢(XV,AV).

Let us point out that H0∗⁢(X) is a ring if we define the product of [αV]0∈H0p⁢(X) and [βV]0∈H0q⁢(X) by

It is readily seen from the naturality of the cup product that this is a sensible definition.

Let now Ω⊂E be closed and bounded and such that X⊂Ω. The inclusions jV:XV→ΩV induce homomorphisms jV∗:Hp⁢(ΩV)→Hp⁢(XV) for V∈𝒱, and it is readily seen that they actually yield a ring homomorphism

Consequently, we obtain the following corollary from Proposition 2.

Henceforth we denote the module product of α∈H0r⁢(Ω) and β∈HEp⁢(X,A) by

We conclude this section with the following crucial definition of a relative cup-length.

In order to keep the definition short we have not defined when actually CL⁡(Ω;X,A)=k for k≥2 as the stated estimate in the final part of Definition 4 is good enough for our purposes (see Theorem 10).

2.2 The E-Cohomological Conley Index and Critical Points

The first aim of this subsection is to introduce the E-cohomological Conley index and to define a module structure for it. Let E be a real separable Hilbert space and L:E→E an invertible selfadjoint operator for which there exists a sequence {En}n∈ℕ of finite-dimensional subspaces of E such that L⁢(En)=En, En⊂En+1 and ⋃n∈ℕEn¯=E. Let U⊂E be open. Following [8], we call a vector field F:U⊂E→E, F⁢(u)=L⁢u+K⁢(u) an ℒ⁢𝒮-vector field if K:U⊂E→E is a locally Lipschitz compact operator. Note that every ℒ⁢𝒮-vector field generates a local flow ηt satisfying

which we call an ℒ⁢𝒮-flow.

Let us now assume that η is a global ℒ⁢𝒮-flow on U, and let us denote by

the maximal η-invariant subset of Ω⊂U.

Let now Ω be an isolating neighborhood of η and S:=Inv⁡(Ω,η).

It was shown in [11, Lemma 2.7] that every isolated invariant set S as above has an index pair.

Note that the space E splits as E=E+⊕E-, where E± are the spectral subspaces with respect to the positive and negative part of the spectrum of L. Henceforth, we denote by HE∗ the E-cohomology with respect to this splitting. The following crucial result was proved in [11, Proposition 2.8].

Hence the next definition is sensible (cf. [11, Definition 2.9]).

If we want to emphasize the isolating neighborhood Ω instead of the isolated invariant set S, we will also write chE⁡(Ω) to denote the E-cohomological Conley index.

When taking the module structure from Section 2 into account, it is readily seen by arguing as in [11, Proposition 2.8] that HE∗⁢(X,A) and HE∗⁢(X′,A′) are actually isomorphic as H0∗⁢(Ω)-modules. Hence we obtain as a consequence of Proposition 7 the following important result.

Consequently, we can define

where (X,A) is any index pair for S such that X⊂Ω. As S is uniquely determined by Ω and the flow η, we will sometimes denote this cup-length by CL⁡(Ω,η) if we want to emphasize η.

Let us now assume that η is the gradient flow with respect to a differentiable functional f:U→ℝ, i.e. the map F:U⊂E→E is of the form F=∇⁡f. As before, we assume that η is global. Let Ω be an isolating neighborhood of η and S=Inv⁡(Ω,η). We denote by Crit⁡(f,Ω) the set of critical values of f|Ω and can now state the main theorem of this paper.

Note that by Theorem 10, the right-hand side in (2.2) is obviously also a lower bound for the number of critical points of f in Ω. We will prove Theorem 10 in Section 4.

3.1 The Analytical Setting

Before proving Theorem 1, let us first recall the analytical setting from [10] (see also [13]). In what follows, we let J be the symplectic standard matrix

which is related to the symplectic form on T2⁢n by

where 〈⋅,⋅〉 denotes the standard Euclidean scalar product.

We now start with the case of ℝ2⁢n and consider the space of smooth loops C∞⁢(S1,ℝ2⁢n) in ℝ2⁢n. If we set ek⁢(t):=et⁢k⁢2⁢π⁢J, k∈ℤ, then any x∈C∞⁢(S1,ℝ2⁢n) is represented by its Fourier series

where xk∈ℝ2⁢n, k∈ℤ. The Sobolev space H12⁢(S1,ℝ2⁢n) is the Hilbert space which is obtained as the completion of C∞⁢(S1,ℝ2⁢n) with respect to the scalar product

There is an orthogonal decomposition

into a 2⁢n-dimensional subspace Z0 and closed infinite-dimensional subspaces Z+ and Z- which correspond to k=0, k>0 and k<0 in the Fourier-series expansion (3.2), respectively. In what follows, we denote by P0, P+ and P- the corresponding orthogonal projections.

Now let H∈C2⁢(S1×ℝ2⁢n,ℝ) be a Hamiltonian such that |H⁢(x)|≤C⋅|x|2 at infinity and such that the second spatial derivative H′′ is globally bounded. We define a functional ΦH:C∞⁢(S1,ℝ2⁢n)→ℝ by the formula

The importance of ΦH comes from the fact that the critical points of ΦH are periodic solutions of the Hamilton equation (3.1). It is easy to see that ΦH extends to H12⁢(S1,ℝ2⁢n), and

where L=∇⁡a=P+-P- is a selfadjoint Fredholm operator and K=-∇⁡b=-j*⁢∇⁡H is a compact map because of the compactness of the adjoint j∗:L2→H12 of the inclusion.

On a general manifold, it is a delicate problem to define spaces H12⁢(S1,M) as H12⁢(S1,ℝ2⁢n) contains non-continuous functions which consequently have no local meaning. However, for a torus one can overcome this problem by using the universal covering ℝ2⁢n→T2⁢n=ℝ2⁢n/ℤ2⁢n. Then smooth Hamiltonians on T2⁢n are in one-to-one correspondence with ℤ2⁢n-invariant smooth Hamiltonians on ℝ2⁢n, where ℤ2⁢n acts on ℝ2⁢n by translations. By a slight abuse of notation, we will denote by H both the Hamiltonian on the torus and the Hamiltonian lifted to ℝ2⁢n. Note that the lifted Hamiltonian on ℝ2⁢n is ℤ2⁢n-invariant and therefore its second spatial derivative is bounded and it obviously satisfies the growth condition mentioned above. Now the corresponding functional ΦH in (3.3) is ℤ2⁢n-invariant as well, and therefore it descends to a functional on the quotient space

3.2 Proof of Theorem 1

We suppose as in the previous subsection that H∈C2⁢(S1×T2⁢n,ℝ) is a given Hamiltonian. Let us note at first that F=∇⁡ΦH in (3.4) is an ℒ⁢𝒮-vector field, even though the operator L is not invertible. Indeed, if we write F=L^+K^:=(L+P0)+(K-P0), where P0 is the orthogonal projection onto the finite-dimensional kernel of L as introduced above, then F is the sum of an invertible selfadjoint operator and a compact map.

As ℳ is a Hilbert manifold, we cannot directly apply the E-cohomological Conley index which we only have defined for flows on open subsets of a Hilbert space. However, if we use a tubular neighborhood, the definition can easily be extended to Hilbert manifolds of the type M×E, where M is a closed manifold and E is a Hilbert space. In the case of ℳ, the construction is as follows. We embed ℳ into E^=ℝ4⁢n×Z+×Z- in such a way that every S1 in T2⁢n=S1×…×S1 is mapped to the unit circle in ℝ2. We consider the open set

of E^, where D0={(x,y)∈ℝ2:0<x2+y2<4} is a punctured disc of radius 2 in ℝ2, and we let π:𝒩→ℳ be the standard projection to T2⁢n on D02⁢n and the identity on Z+ and Z-. The map ΦH can be extended to U by

where ri⁢(x) denotes the polar coordinate in ℝ2 of the projection of x∈U to the i-th component of (ℝ2)2⁢n. Note that the extension is done in such a way that ΨH and ΦH have the same critical points. We denote by K~ the compact operator which is the sum of K^ and ∇⁡(∑i=12⁢n(1-ri⁢(x))2).

Note that ∇⁡ΨH=L^+K~ is an ℒ⁢𝒮-vector field, and the negative and positive spectral subspaces of the selfadjoint isomorphism L^ are given by

Now Theorem 1 can be obtained as follows. Since K is bounded, there is R>0 such that R>∥K⁢(x)∥ for all x∈U. We set

where B⁢(Z±,R) are the closed balls of radius R in Z± and C⊂D0 is a closed annulus containing S1. Note that the boundary ∂⁡X is given by the non-disjoint union

Let now X1,X2,X3 denote the three parts of ∂⁡X in the above order and let η be the flow induced by -∇⁡ΨH as in (2.1). Firstly, if x∈X1 but neither in X2 nor in X3, then there is some t>0 such that η(0,t]⁢(x)⊂X∖∂⁡X. Secondly, if x∈X2, then ∥P+⁢x∥=R and we have

Consequently, the vector field L^⁢x+K~⁢(x) is pointing outwards the sphere ∂⁡B⁢(Z+,R). Hence, if x∈X1∪X2 but x∉X3, then η, which is the flow induced by -∇⁡ΨH, moves x into the interior of X. Finally, if x∈X3, we analogously see that

as ∥P-⁢x∥=R. Hence those x leave X under η. It is now readily seen that

is an index pair for S=Inv⁡(X,η) in the sense of Definition 6, where we have set A:=X3 for simplicity of notation.

To find the E-cohomology of (X,A), let V⊂Z- be of finite dimension. Then

where B⁢(V,R) denotes the ball of radius R in V. Hence we get for k∈ℤ

where S⁢(V,R) denotes the sphere of radius R in V. Moreover, if W⊃V is another finite-dimensional subspace, then the Mayer–Vietoris homomorphism ΔV,Wk mapping

to

is by definition just the suspension isomorphism. Hence we obtain

Finally, to find the cup-length, we note at first that for the isolating neighborhood X, and any finite-dimensional subspace V⊂Z-,

Hence CL⁡(X,η) is just the ordinary cup-length of the torus T2⁢n, which is 2⁢n+1. By Theorem 10, this is a lower bound for the number of critical points of ΦH in X, and so we have proved the Arnold conjecture on T2⁢n.