Reidemeister torsion and Integrable Hamiltonian systems

In this paper we compute the Reidemeister torsion of a isoenergetic surface for the integrable Hamiltonian system on the four-dimensional symplectic manifold. We use the spectral sequence defined by the filtration and following Witten-Floer ideas we bring into play the orbits connecting the critical submanifolds.


Introduction
Reidemeister torsion is a very important topological invariant which has useful applications in knot theory, quantum field theory and dynamical systems. In 1935 Reidemeister [9] classified up to P L equivalence the lens spaces S 3 /Γ where Γ is a finite cyclic group of fixed point free orthogonal transformations. He used a certain new invariant -the Reidemeister torsion-which was quickly extended by Franz, who used it to classify the generalized lens spaces S 2n+1 /Γ. Let X be a compact smooth manifold. A representation ρ : π 1 (X) → U(m) of the fundamental group defines a flat C m bundle E over X. When the twisted cohomology H * (X; E) vanishes, the representation ρ and the flat bundle E are called acyclic. The Reidemeister torsion or R-torsion is a positive number which is a ratio of determinants concocted from the π 1 (X)-equivariant chain complex of the universal covering of X. Later Milnor identified the Reidemeister torsion with the Alexander polynomial, which plays a fundamental role in the theory of knots and links.
In 1971, Ray and Singer [10] introduced an analytic torsion associated with the de Rham complex of forms with coefficients in a flat bundle over a compact Riemannian manifold, and conjectured it was the same as the Reidemeister torsion. The Ray-Singer conjecture was established independently by Cheeger and Müller a few years later.
Recently, the Reidemeister torsion has found interesting applications in dynamical systems theory. A connection between the Lefschetz type dynamical zeta functions and the Reidemeister torsion was established by D. Fried [6]. The work of Milnor [7] was the first indication that such a connection exists.
In this paper we study the Reidemeister torsion of isoenergy surfaces of an integrable Hamiltonian system. Let N be a four-dimensional smooth symplectic manifold and consider the Hamiltonian system with smooth Hamiltonian H, which in Darboux coordinates has the form: The three-dimensional level surface M = {H = const} is invariant under the flow defined by the system (0.1). The surface M is called an isoenergey surface or a constant-energy surface. The topological structure of isoenergy surfaces of integrable Hamiltonian systems and the structure of their fundamental groups were describeed in [4,3]. Isoenergy surfaces of integrable Hamiltonians system possess specific properties which distinguish them among all smooth three dimensional manifolds. Namely, they belong to the class of graph-manifolds introduced by Waldhausen [12]. Since N is orientable (as a symplectic manifold), the surface M is automatically orientable in all cases. Suppose that the system (0.1) is complete integrable (in Liouville's sense) on the surface M. This means, that there is a smooth function f (the second integral), which is independent of H and with Poisson bracket Definition. We shall call f : M → R a Bott function if its critical points form critical nondegenerate smooth submanifolds of M. This means that the Hessian d 2 f of the function f is nondegenerate on the planes normal to the critical submanifolds of the function f .
A.T. Fomenko [4] proved that a Bott integral on a compact nonsingular isoenergy surface M can have only three types of critical submanifolds: circles, tori or Klein bottles. The investigation of concrete mechanical and physical systems shows [4] that it is a typical situation when the integral on M is a Bott integral. In the classical integrable cases of the solid body motion (cases of the Kovalevskaya, Goryachev-Chaplygin, Clebsch, Manakov) the Bott integrals are round Morse functions on the isoenergy surfaces. A round Morse function is a Bott function all whose critical manifolds are circles. Note that critical circles of f are periodic solutions of the system (0.1) and the number of this circles is finite. Suppose for the moment that the Bott integral f is a round Morse function on the closed isoenergy surface M. Let us recall the concept of the separatrix diagram of the critical circle γ. Let x ∈ γ be an arbitrary point and N x (γ) be a disc of small radius normal to γ at x. The restriction of f to the N x (γ) is a normal Morse function with the critical point x having a certain index u(γ) = 0, 1, 2. A separatrix of the critical point x is an integral trajectory of the field −grad f , called a gradient line, which is entering or leaving x. The union of all the separatrices leaving the point x gives a disc of dimension u(γ) and is called the outgoing separatrix diagram (disc). The union of incoming separatrices gives a disc of complementary dimension and is called the separatrix incoming diagram (disc). Varing the point x and constructing the incoming and outgoing separatrix discs for each point x, we obtain the incoming and outgoing separatrix diagrams of the circle γ. Let ∆(γ) be +1 if the outgoing separatrix is orientable, and -1 if it is not. Let ǫ(γ) = (−1) u(γ) . Let ρ E : π 1 (M, p) → U(E p ) be the holonomy representation of the hermitian bundle E over M; E p is the fiber at the point p.  [6] we can compute the Reidemeister torsion as [1] This formula means that for the integrable Hamiltonian system on the four-dimensional symplectic manifold, the Reidemeister torsion of the isoenergy surface counts the critical circles of the second independent Bott integral on this surface. If E|(M i+1 , M i ) is acyclic, then det(I − ∆(γ i ) · ρ E (γ i )) = 0 for each i. Since in many classical integrable cases there are contractible critical circles it is interesting to study the situation when not all E|(M i+1 , M i ) are acyclic. In this paper we carry out this study and in fact we consider the general situation when the Bott integral has critical tori and Klein bottles. We use the spectral sequence defined by the filtration and following Witten-Floer ideas we bring into play the orbits connecting the critical submanifolds. A similar approach was developed in [11] for Morse-Smale flows. Parts of this article were written while the first author was visiting the Instituto de Matematicas Universidad Nacional Autonoma de Mexico and Erwin Schrödinger International Institute for Mathematical Physics in Wien .The first author is indebted to these institutions for their invitations, support and hospitality.

R-torsion and Spectral Sequences
Let W be a finite dimensional vector space with basis w = {w 1 , . . . , w n }, then ∧w = w 1 ∧ · · · ∧ w n is a generator of det W = ∧ n W . If dim V = 0 set det V = C.
Consider a cochain complex of finite dimensional vector spaces Notation. Consider the cochain complex is an exact sequence of chain complexes and is the corresponding long exact sequence. For each k choose compatible volume elements in det We now describe the first terms of the spectral sequence {E r , d r }. The filtration defines the associated graded complex n whose cohomology defines the term E 1 by and induces the first differential d n 1 : E n,q 1 → E n+1,q 1 as the coboundary map for the short exact sequence The term E n,q 2 is defined by . (1.8) From (1.7) we have ker(d 1 ) =im(k n ) and im(d 1 ) = ker(j n−1 ), and thus E n,q 2 = im(k n )/ker(j n−1 ).

Consider the commutative diagram
The second differential d n−2

2
: E n−2,q+1 2 → E n,q 2 is given as the composite map im(k n−2 )/ker(j n−3 ) Further terms of the spectral sequence E n,q r are obtained as cohomology of the previous term and the differentials d n r : E n,q r → E n+r,q+r−1 r are the maps induced by the original d.
Let now K be a finite CW-complex. Let p :K → K be the universal covering and ρ : Γ → U(m) be a representation of the fundamental group Γ of K which defines a flat vector bundle E :=K × Γ C m . Lifting cells toK we obtain a Γ-invariant CW complex structure onK. The space of ρ-equivariant cochains forms a subcomplex. Its cohomology H * (K; E) is called the ρ-twisted cohomology of K. As usual H * (K; E) is subdivision invariant and we have a torsion element Order the j-cells σ and choose an oriented liftσ for each σ. This gives an isomorphism C j (K; E) ∼ = ⊕ σ C m and determines a preferred generator w ρ K of det(C * (K; E)) up to multiplication by an element of the subgroup which is invariant under subdivision. When ρ is acyclic, i.e. when H * (K; E) = 0, we have det H * (K; E) = C and we can identify τ (K; E) as an element of C * /U ρ . Since U ρ ⊂ S 1 , all elements in τ (K; E) have the same modulus which we still denote by τ (K; E).
The previous definitions can be extended to relative pairs. Let L be a subcomplex of K. For each j we have the relative space of cellular j-cochains LetK and ρ be as above and letL = p −1 (L). We can define the space of relative ρ-equivariant cochains C * (K, L; E) ⊂ C * (K,L; C m ) with coboundary d(K, L; E) and then we get a torsion element Thus, choosing preferred basis as before we obtain a U ρ orbit Remark 1. Another name for the twisted cohomology is cohomology with local coefficients. One chooses a point on each cell of K and a path from a fixed point to each chosen point. In this way any path c between chosen points defines a closed path γ c and then a matrix ρ(c) := ρ(γ c ) which gives the relation between the coefficients at the ends of the path.

R-torsion and critical submanifolds
The foliation of the isoenergy surface M by Liouville tori is given by the level sets of the Bott integral f −1 (c) for c a regular value. The bifurcation of Liouville tori ocurr at the sets F c = f −1 (c) ∩ Crit(f ) for c a critical value. We will make the following assumption wich is satisfied in the generic case.
Assumption. We will assume that there are no gradient lines of the Bott integral f connecting saddle circles i.e. circles with index 1.
We will substitute the Bott integral for another Bott function , still denoted by f and not necessarily an integral, giving the same foliation by Liouville tori and such that its critical values c 1 < · · · < c l , are ordered in the following way for the critical sets of f i. e. denoting by φ the gradient flow of f , φ is transverse inwards on ∂N n and Fix a representation ρ : π 1 (M) → U(m). All cochain complexes and cohomology groups will have coefficients in the flat bundle defined by ρ. Let l 0 = 0, l 1 = k 2 , l 2 = k 3 , l 3 = l and M n = N ln . Define the filtration F 5 ⊂ · · · ⊂ F 1 ⊂ F 0 of C * (M) by F n = ker(C * (M) → C * (M n )). The associated graded complex is given by Since there are neither gradient lines connecting two minimum (maximum) critical submanifolds nor gradient lines connecting two saddle circles (by assumption), we have (see [11]) . To do so, we will use the trajectories of the gradient flow of f , but we will modify f in the neighborhood of each critical level set in order to apply Lemma 1 below giving such a map in the Morse function case. This modification is just a technical device to choose some of the orbits connecting the critical submanifolds to describe the maps F * ij . We will give a proof of the following Proposition using Lemma 1.
If k 2n−1 < j ≤ k 2n and α j , β j are generators of the fundamental group of F c j , let D j = I−ρ(α j ) I±ρ(β j ) , D * j = (I ± ρ(β j ), ρ(α j ) − I), where the + sign ocurrs precisely when F c j is a Klein bottle. Then = 0 for q = 0, 1, 2. Let G : M → R be a Morse Smale function and let c 1 < · · · < c N be its critical points. For A 0 < c 1 < · · · < c N < A N and K i = G −1 (∞, A i ] we get a filtration K 0 ⊂ · · · ⊂ K N . The orientation of M and grad G define an orientation of L a = G −1 (a) for each regular value a. Giving an orientation to the unstable subspace E u (x) for each critical point x of G, and using the orientation of M we also get an orientation of E s (x). Then we have orientations of W u (x) and W s (x). Let x, y be critical points of G of indices n, n + 1 respectively and let a be a regular value with G(x) < a < G(y). Then S u (y) = W u (y) ∩ L a and S s (x) = W s (x) ∩ L a are oriented transverse submanifolds of L a with dimensions n and and 2 − n respectively. Therefore S s (x) ∩ S u (y) is a finite set. For each q ∈ S s (x) ∩ S u (y) denote by I q the intersection number.
The proof given by Floer in [2] for the untwisted version of the following Lemma can be readily adapted. The only new ingredients are the matrices ρ(α) for nonclosed paths α used to define cohomology with local coefficients.
To change the Bott integral in a neighborhood of each critical level set, we will use the folllowing Propositions.
Proposition 5. Let F : M → R be a Bott function and let γ be a critical circle of index n. Given a small neighborhood U of γ there is another Bott function G which agrees with F outside U and has nondegenerate critical points w, z ∈ γ of indices n, n + 1 and no other critical points in U.

3.
There are no gradient lines either from a critical point of index 1 to any of q, r, s or from a critical point of index 2 to s (either from p to a critical point of index 1 or from p, q, r, s to a critical point of index 2).

REIDEMEISTER TORSION AND INTEGRABLE HAMILTONIAN SYSTEMS 13
To compute τ (∆ i ) we recall that D * i = (I ± ρ(β i ), ρ(α i ) − I) and then and using k i × k i and k i as their bases we have We now come to the description of the components F * ij of d 1 . Let ψ t be the gradient flow of g. One can construct an index filtration We have Note that all the components G * xy : C m x → C m y of the maps d K in the cochain complex    (1) The map d 2 : E 0,1 2 → E 2,0 2 is induced by the component We now change the Bott integral f to a Morse Smale function g to be able to compute d 1 : H k (M 1 ) → H k+1 (M 2 , M 1 ), k = 0, 1. Note that W u (r i ), i = 1, 2 is a Möbius strip Σ i with ∂Σ i = m i . The function g has critical points w i , z i on m i with indices 0, 1 respectively, and critical points η i , ζ i on r i with indices 1, 2 respectively. There is one orbit α i connecting ζ i to z i and two orbits β i , δ i connecting η i to w i , with ρ(α i ) = 1, ρ(β i ) = 1 ρ(δ i ) = −1. Thus G w i η i : C → C = 2 and G z i ζ i : C → C = 1.