REIDEMEISTER TORSION AND INTEGRABLE HAMILTONIAN SYSTEMS

ALEXANDER FEL’SHTYN AND HÉCTOR SÁNCHEZ-MORGADO

Abstract. Let N be a 4-dimensional symplectic manifold and H : N → ℝ such that there is a Bott function f : N → ℝ independent of H with {H, f} = 0. We study the relation between Reidemeister torsion of compact energy levels M = {H = const}, and the critical circles and gradient lines of f connecting critical submanifolds of f on M.

0. 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. Reidemeister torsion or R-torsion is a positive number defined as a ratio of determinants concocted from the cochain complex C^{∗}(X, E). Later Milnor identified 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 Reidemeister torsion. The Ray-Singer conjecture was established independently by Cheeger and Müller a few years later.

Recently, Reidemeister torsion has found interesting applications in dynamical systems theory. A connection between Lefschetz type dynamical zeta functions and Reidemeister torsion was established by D.

1991 Mathematics Subject Classification. 58F07.

2 A. FEL’SHTYN AND H. SÁNCHEZ-MORGADO

Fried [6]. The work of Milnor [7] was the first indication that such a connection exists.

In this paper we study R-torsion for 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:

\begin{aligned}ṗ_{i} & = −\frac{∂H}{∂q_{i}} \\ q̇_{i} & = \frac{∂H}{∂p_{i}} \end{aligned}  (0.1)

The three-dimensional level surface M = {H = const} is invariant under the flow defined by the system (0.1). The surface M is called isoenergy surface or energy level. The topological structure of isoenergy surfaces of integrable Hamiltonian systems and the structure of their fundamental groups were described in [4, 3]. Isoenergy surfaces of integrable Hamiltonian systems 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 completely 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 {H, f} = 0 in a neighborhood of M.

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 (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 Δ(γ) be +1 if the outgoing separatrix is orientable, and -1 if it is not. Let ϵ(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. For the gradient flow of f on M, one can construct an index filtration for the collection of critical circles {γ_{i}}, i. e. a collection of compact submanifolds M_{i} of top dimension so that M_{i} ⊂ intM_{i+1}, M_{0} = ∅, M_{i} = M for large i, the flow is transverse inwards on ∂M_{i} and M_{i+1} \ M_{i} is an isolating neighborhood for the critical circle γ_{i}. When E|(M_{i+1}, M_{i}) is acyclic for each i, following the ideas of D. Fried [6] we can compute Reidemeister torsion as [1]

τ(M; E) = ∏_{i} τ(M_{i+1}, M_{i}; E) = ∏_{γ_{i}} | det(I − Δ(γ_{i}) · ρ_{E}(γ_{i}))|^{ε(γ_{i})}(0.2)

This formula means that for an integrable Hamiltonian system on the four-dimensional symplectic manifold, Reidemeister torsion of an isoenergy surface counts the critical circles of the round independent Bott integral on this surface. If E|(M_{i+1}, M_{i}) is acyclic, then det(I − Δ(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 gradient lines of f 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 Matemáticas UNAM and Erwin Schrödinger Institute for Mathematical Physics in Wein, and the second author was visiting CIMAT in Guanajuato, México. The authors are indebted to these institutions for their support and hospitality.

1. 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

0 → V^{0} ⟶^{d} V^{1} → ⋯ → V^{m} ⟶^{d} 0  (1.1)

Let V^{+} = ⨁_{i} V^{2i}, V^{−} = ⨁_{i} V^{2i+1} and

det V = det(V^{−}) ⊗ (det V^{+})^{−1}.

Let Z^{±} = V^{±} ∩ ker d, B^{∓} = d(V^{±}), H^{±} = Z^{±}/B^{±}.

We now define the torsion element τ_{d} ∈ det V ⊗ (det H)^{−1}. Pick ordered relative bases h_{±} for (Z^{±}, B^{±}) and t_{±} for (V_{±}, Z_{±}), then dt_{∓} is a basis for B_{±}. Denote by [h_{±}] the corresponding basis for H^{±}.

τ_{d} = ∧(t_{−}, h_{−}, dt_{+}) ⊗ ∧(dt_{−}, h_{+}, t_{+})^{−1} ⊗ ∧[h_{+}] ⊗ ∧[h_{−}]^{−1} (1.2)

Notation. Consider the cochain complex

0 → V ⟶^{A} W → 0

We have H^{0} = ker A, H^{1} = coker A and denote τ(A) := τ_{d}. When A is an isomorphism, τ(A) is the coordinate free version of det A.

Proposition 1. [5] Let 0 = F_{N+1}^{i} ⊂ F_{N}^{i} ⊂ ⋯ ⊂ F_{0}^{i} = V^{i} be a filtration of the cochain complex (1.1) such that d^{i}(F_{n}^{i}) ⊂ F_{n}^{i+1}. Let {E_{r}, d_{r}} be the corresponding spectral sequence. Then

τ_{d} = τ_{d_{0}} ⊗ ⋯ ⊗ τ_{d_{N}}

Corollary 2. [5] Suppose

0 → (C′, d′) ⟶^{i} (C, d) ⟶^{j} (C′′, d′′) → 0  (1.3)

is an exact sequence of chain complexes and

H : 0 → H^{0}(C′) ⟶^{i^{∗}} H^{0}(C) ⟶ H^{0}(C′′) ⟶^{∂} H^{1}(C′) → ⋯ (1.4)

is the corresponding long exact sequence. For each k choose compatible volume elements in det C′_{k}, det C_{k}, det C′′_{k}, i.e. such that the torsion of (1.3) is 1. Then

τ_{d} = τ_{d′}τ_{d′′}τ_{H}  (1.5)

We now describe the first terms of the spectral sequence {E_{r}, d_{r}}. The filtration defines the associated graded complex G^{i} = ⨁_{n} G_{n}^{i} where G_{n}^{i} = F_{n}^{i}/F_{n+1}^{i}. The coboundary d induces a map d_{0}^{n} : G_{n}^{i} → G_{n}^{i+1} whose cohomology defines the term E_{1} by

E_{1}^{n,q} := H^{n+q}(F_{n}/F_{n+1})  (1.6)

and induces the first differential d_{1}^{n} : E_{1}^{n,q} → E_{1}^{n+1,q} as the coboundary map for the short exact sequence

0 → F_{n+1}/F_{n+2} → F_{n}/F_{n+2} → F_{n}/F_{n+1} → 0

i.e. the map d_{1} in the long exact sequence

→ H^{n+q}(F_{n}/F_{n+2}) ⟶^{k_{n,q}} H^{n+q}(F_{n}/F_{n+1}) ⟶^{d_{1}} H^{n+q+1}(F_{n+1}/F_{n+2}) ⟶^{j_{n+1,q}} (1.7)

The term E_{2}^{n,q} is defined by

E_{2}^{n,q} := \frac{ker(d_{1} : E_{1}^{n,q} → E_{1}^{n+1,q})}{im(d_{1} : E_{1}^{n−1,q} → E_{1}^{n,q})}.  (1.8)

From (1.7) we have E_{2}^{n,q} = im(k_{n,q})/ker(j_{n,q}).

Consider the commutative diagram

H^{n+q}(F_{n}/F_{n+2}) ⟶^{k_{n,q}} H^{n+q}(F_{n}/F_{n+1})  ⟶^{j_{n,q}} H^{n+q}(F_{n−1}/F_{n+1})
δ ↗  ↗ δ_{q+1}
H^{n+q−1}(F_{n−2}/F_{n}) ⟶^{k_{n−2,q+1}} H^{n+q−1}(F_{n−2}/F_{n−1}) ⟶^{j_{n−2,q+1}} H^{n+q−1}(F_{n−3}/F_{n−1}),

The second differential d_{2}^{n−2} : E_{2}^{n−2,q+1} → E_{2}^{n,q} is given as the composite map

im(k_{n−2,q+1})/ker(j_{n−2,q+1}) ⟶^{δ_{q+1}} im(j_{n,q}) ⟶^{j_{n,q}^{−1}} im(k_{n,q})/ker(j_{n,q})(1.9)

Further terms of the spectral sequence E_{r}^{n,q} are obtained as cohomology of the previous term and the differentials d_{r}^{n} : E_{r}^{n,q} → E_{r}^{n+r,q+1−r} are the maps induced by the original d.

Let now K be a finite CW-complex. Let p : K̃ → 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 := K̃ ×_{Γ} ℂ^{m}. Lifting cells to K̃ we obtain a Γ-invariant CW complex structure on K̃. The space of ρ-equivariant cochains

C^{∗}(K; E) = {ξ ∈ C^{∗}(K̃; ℂ^{m}) : ξ ∘ γ = ρ(γ) ∘ ξ ∀γ ∈ Γ}

is preserved by d^{j} : C^{j}(K̃; ℂ^{m}) → C^{j+1}(K̃; ℂ^{m}) and so C^{∗}(K; E), d(K; E) 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

τ_{d(K;E)} ∈ det C^{∗}(K; E) ⊗ (det H^{∗}(K; E))^{−1}.

Order the j-cells σ and choose an oriented lift σ̃ for each σ. This gives an isomorphism C^{j}(K; E) ≅ ⊕_{σ}ℂ^{m} and determines a preferred generator w_{K}^{ρ} of det(C^{∗}(K; E)) up to multiplication by an element of the subgroup

U_{ρ} = {(±1)^{m} det ρ(γ) : γ ∈ Γ} ⊂ S^{1}

The orbit U_{ρ}w_{K}^{ρ} ⊂ det C^{∗}(K; E) is invariant under subdivision, so we can define R-torsion of K at ρ as the U_{ρ} orbit

τ(K; E) = (U_{ρ}w_{K}^{ρ})^{−1} ⊗ τ_{d(K;E)} ⊂ (det H^{∗}(K; E))^{−1} (1.10)

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) with 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

C^{j}(K, L; ℂ) = ⨁_{σ∈K\L} H^{j}(σ, ∂σ; ℂ)

Let K̃ and ρ be as above and let L̃ = p^{−1}(L). We can define the space of relative ρ-equivariant cochains C^{∗}(K, L; E) ⊂ C^{∗}(K̃, L̃; ℂ^{m}) with coboundary d(K, L; E) and then we get a torsion element

τ_{d(K,L;E)} ∈ det C^{∗}(K, L; E) ⊗ (det H^{∗}(K, L; E))^{−1}.

Thus, choosing preferred basis as before we obtain a U_{ρ} orbit

τ(K, L; E) ⊂ det H^{∗}(K, L; E)^{−1}

which is invariant under subdivision.

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.

2. 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

(a) i ≤ k_{1} ⟺ F_{c_{i}} is a minimum circle.

(b) k_{1} < i ≤ k_{2} ⟺ F_{c_{i}} is a minimum torus or Klein bottle.

(c) k_{3} < i ≤ k_{4} ⟺ F_{c_{i}} is a maximum torus or Klein bottle.

(d) k_{4} < i ⟺ F_{c_{i}} is a maximum circle. Choosing numbers A_{0} < c_{1} < A_{1} < ⋯ < c_{l} < A_{l} and letting N_{j} = f^{−1}(−∞, A_{j}] we get an index filtration ∅ = N_{0} ⊂ N_{1} ⊂ ⋯ ⊂ N_{l} = M for the critical sets of f i. e. denoting by ϕ the gradient flow of f, ϕ is transverse inwards on ∂N_{n} and

⋂_{t∈ℝ}ϕ_{t}(N_{j} \ N_{j−1}) = F_{c_{j}}.

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_{l_{n}}. Define the filtration F_{3} ⊂ F_{2} ⊂ F_{1} ⊂ F_{0} of C^{∗}(M) by F_{n} = ker(C^{∗}(M) → C^{∗}(M_{n})). Note that for the corresponding spectral sequence, E_{r}^{n,q} ≠ 0 only for n = 0, 1, 2 and therefore d_{r} = 0 for r ≥ 3. The associated graded complex is given by

G_{n} = F_{n}/F_{n+1} ≅ C^{∗}(M_{n+1}, M_{n})

with 0-differential d_{0} = ⨁_{n} d_{0}^{n}, where d_{0}^{n} : G_{n} → G_{n}, and torsion element τ_{d_{0}} = ⨂_{n} τ_{d_{0}^{n}} where

τ_{d_{0}^{n}} ∈ det C^{∗}(M_{n+1}, M_{n}) ⊗ (det H^{∗}(M_{n+1}, M_{n}))^{−1}.

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])

H^{∗}(M_{n+1}, M_{n}) = ⨁_{M_{n}⊂N_{j}⊂M_{n+1}} H^{∗}(N_{j}, N_{j−1})  (2.1)

τ_{d_{0}^{n}} = ⨂_{M_{n}⊂N_{j}⊂M_{n+1}} τ_{d_{0}^{n,j}}  (2.2)

From (2.1), the computation of the map

d_{1}^{n} : H^{∗}(M_{n}, M_{n−1}) → H^{∗+1}(M_{n+1}, M_{n})

reduces to computing for each i, j with l_{n−1} < i ≤ l_{n} < j ≤ l_{n+1}, its component F_{ij}^{∗} : H^{∗}(N_{i}, N_{i−1}) → H^{∗+1}(N_{j}, N_{j−1}). 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 below.

Proposition 3. If k_{2n} < j ≤ k_{2n+1}, let γ_{j} = F_{c_{j}} and D_{j} = I − Δ(γ_{j})ρ(γ_{j}). Then

H^{k}(N_{j}, N_{j−1}) = \begin{cases}ker D_{j} & if k = n, n + 1 \\ 0 & in other case. \end{cases}  (2.3a)

τ_{d_{0}^{n,j}} = τ(D_{j})^{(−1)^{n}}  (2.3b)

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} = ( \begin{matrix}I−ρ(α_{j}) \\ I±ρ(β_{j}) \end{matrix} ), D_{j}^{∗} = (I ± ρ(β_{j}), ρ(α_{j}) − I), where the +

sign ocurrs precisely when F_{c_{j}} is a Klein bottle. Then

H^{k}(N_{j}, N_{j−1}) = \begin{cases}ker D_{j} = coker D_{j}^{∗} & if |k − n| = 1 \\ ker D_{j} ⊕ ker D_{j} & if k = n \\ 0 & in other case \end{cases} (2.4a)

τ_{d_{0}^{n−1,j}} = 1  (2.4b)

Corollary 4.

E_{1}^{0,q} = H^{q}(M_{1}) = 0 for q ≠ 0, 1, 2.
E_{1}^{1,q} = H^{q+1}(M_{2}, M_{1}) = 0 for q ≠ 0, 1.
E_{1}^{2,q} = H^{q+2}(M, M_{2}) = 0 for q ≠ −1, 0, 1.

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.

Lemma 1. Let c_{i}, c_{i+1} be critical points of G with indices n, n+1. For each q ∈ S(i, j) = S^{s}(c_{i}) ∩ S^{u}(c_{j}) let α_{q}(t) = ϕ_{cot(πt)}(q) : t ∈ [0, 1]. The coboundary map d : H^{n}(K_{i}, K_{i−1}) → H^{n+1}(K_{i+1}, K_{i}) is given by

d = ∑_{q∈S(i,j)} I_{q}ρ(α_{q})  (2.5)

To change the Bott integral in a neighborhood of each critical level set, we will use the following 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.

Proof. Let F (γ) = c. If Δ(γ) = 1, there is a tubular neighborhood U of γ with coordinates (x, y) ∈ B_{ε}(0), θ ∈ ×S^{1} = ℝ/ℤ such that

F (x, y, θ) = c ± x^{2} ± y^{2}

Let ρ : ℝ → [0, 1] be a smooth function with ρ(t) = 0 if t > ε^{2} and ρ(t) = 1 for t < ε^{2}/4. Define

G = \begin{cases}f + δρ(x^{2} + y^{2}) cos θ & on U \\ f & outside U \end{cases}

10 A. FEL’SHTYN AND H. SÁNCHEZ-MORGADO

On U we have

dG(x, y, θ) = 2(±1 + δρ′(x^{2} + y^{2}) cos θ) x dx

+ 2(±1 + δρ′(x^{2} + y^{2}) cos θ) y dy − δρ(x^{2} + y^{2}) sin θ dθ

Thus, if δ is sufficiently small, the only critical points of G in U are w = (0, 0, π) and z = (0, 0, 0) with indices p and p + 1 respectively.

Proposition 6. Let F : M → ℝ be a Bott function and let 𝕋 be a minimum (maximum) torus. Given a small neighborhood U of 𝕋 there is another Bott function G which

1. Agrees with F outside U.

2. Has nondegenerate critical points p, q, r, s ∈ 𝕋 of indices 0, 1, 1, 2 (or 1, 2, 2, 3) 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).

Proof. Consider the case of a mimimal torus. Let F (𝕋) = c. There is a tubular neighborhood U of 𝕋 with coordinates (x, θ, φ) ∈ (−ε, ε) × S^{1} × S^{1} such that F (x, θ, φ) = c + x^{2}.

For each critical point z of F of index 2, W^{u}(z) ∩ {x} × S^{1} × S^{1} is a curve. For each critical point w of F of index 1, W^{u}(w)∩{x}×S^{1} ×S^{1} is a point. Therefore we can choose the coordinates θ, φ such that

(a) (x, 0, 0) ∉ ⋃_{u(z)=2}W^{u}(z)

(b) {x} × S^{1} × {0} ∩ ⋃_{u(w)=1}W^{u}(w) = ∅

(c) {x} × {0} × S^{1} ∩ ⋃_{u(w)=1}W^{u}(w) = ∅

Let ρ be as in the proof of Proposition 5. Define

G = \begin{cases}F + ρ(x^{2})(δ_{1} cos θ + δ_{2} cos φ) & on U \\ F & outside U \end{cases}

On U we have

dG(x, θ, φ) =2(1 + ρ′(x^{2})(δ_{1} cos θ + δ_{2} cos φ)) x dx
−ρ(x^{2})(δ_{1} sin θ dθ + δ_{2} sin φ dφ)

Thus, if δ_{1}, δ_{2} are sufficiently small, the only critical points of G in U are p = (0, π, π), q = (0, π, 0), r = (0, 0, π), s = (0, 0, 0) which have the required indices and

(a) W^{s}(s) ∩ U = (−ε, ε) × {(0, 0)}

(b) W^{s}(q) ∩ U = (−ε, ε) × (S^{1} − {π}) × {0} (c) W^{s}(r) ∩ U = (−ε, ε) × {0} × (S^{1} − {π}) The proof for the maximum torus case is similar.

Applying Proposition 5 to f we obtain a Bott function without critical circles, then using Proposition 6 we obtain a Bott function without critical circles and minimum torus (Klein bottles) and with no gradient lines from a critical point of index i to a critical point of index j ≥ i. Using Proposition 6 again we finally get a Morse-Smale function g that agrees with f outside a neighborhood of each critical set. Denote by C_{j}(g) the set of critical points of g with index j. We have

C_{0}(g) = {w_{1}, …, w_{k_{1}}, p_{k_{1}+1}, …, p_{k_{2}}}
C_{1}(g) = {z_{1}, …, z_{k_{1}}, q_{k_{1}+1}, r_{k_{1}+1}, …,
q_{k_{2}}, r_{k_{2}}, w_{k_{2}+1}, …, w_{k_{3}}, p_{k_{3}+1}, …, p_{k_{4}}}
C_{2}(g) = {s_{k_{1}+1}, …, s_{k_{2}}, z_{k_{2}+1}, …, z_{k_{3}},
q_{k_{3}+1}, r_{k_{3}+1}, …, q_{k_{4}}, r_{k_{4}}, w_{k_{4}+1}, …, w_{l}}
C_{3}(g) = {s_{k_{3}+1}, …, s_{k_{4}}, z_{k_{4}+1}, …, z_{l}}

Proof of Proposition 3. For k_{2n} < i ≤ k_{2n+1} let g(w_{i}) < d_{i} < g(z_{i}) and N_{i}^{∗} = g^{−1}(∞, d_{i}]. Then H^{n}(N_{i}^{∗}, N_{i−1}) = ℂ^{m}w_{i}, H^{n+1}(N_{i}, N_{i}^{∗}) = ℂ^{m}z_{i}, and according to Milnor [8, Lemma 9.4]

τ_{d(N_{i}^{∗},N_{i−1})} = 1, τ_{d(N_{i},N_{i}^{∗})} = 1.

Using Lemma 1 we have the sequence

0 → H^{n}(N_{i}, N_{i−1}) ⟶^{A_{i}} ℂ^{m} ⟶^{D_{i}} ℂ^{m} ⟶^{B_{i}} H^{n+1}(N_{i}, N_{i−1}) → 0 (2.6)

with D_{i} defined in the the statement of Proposition 3. Therefore

H^{n}(N_{i}, N_{i−1}) = ker D_{i}, H^{n+1}(N_{i}, N_{i−1}) = coker D_{i}.

Since sequence (2.6) has torsion τ(D_{i})^{(−1)^{n}}, we obtain by Corollary 2

τ_{0}^{n,i} = τ_{d(N_{i},N_{i}^{∗})}τ_{d(N_{i}^{∗},N_{i−1})}τ(D_{i})^{(−1)^{n}} = τ(D_{i})^{(−1)^{n}}.

For k_{2n_{1}} < i ≤ k_{2n}, let g(p_{i}) < c′_{i} < g(q_{i}), g(r_{i}) < c′′_{i} < g(s_{i}) and N′_{i} = g^{−1}(∞, c′_{i}], N′′_{i} = g^{−1}(∞, c′′_{i}]. Then H^{n−1}(N′_{i}, N_{i−1}) = ℂ^{m}p_{i}, H^{n−1}(N′′_{i}, N′_{i}) = ℂ^{m}q_{i} ⊕ ℂ^{m}r_{i}, H^{n+1}(N_{i}, N′′_{i}) = ℂ^{m}s_{i}. Again from [8, Lemma 9.4]

τ_{d(N′_{i},N_{i−1})} = 1, τ_{d(N′′_{i},N′_{i})} = 1, τ_{d(N_{i},N′′_{i})} = 1.

By Lemma 1 we have the sequence of the triad (N′′_{i}, N′_{i}, N_{i−1}):

0 → H^{n−1}(N′′_{i}, N_{i−1}) ⟶^{A_{i}} ℂ^{m} ⟶^{D_{i}} ℂ^{m} ⊕ ℂ^{m} ⟶^{B_{i}} H^{n}(N′′_{i}, N_{i−1}) → 0, (2.7)

and the sequence of the triad (N_{i}, N′′_{i}, N′_{i}):

0 → H^{1}(N_{i}, N′_{i}) ⟶^{A_{i}} ℂ^{m} ⊕ ℂ^{m} ⟶^{D_{i}^{∗}} ℂ^{m} ⟶^{B_{i}} H^{2}(N_{i}, N′_{i}) → 0, (2.8)

with D_{i} and D_{i}^{∗} defined in the statement of Proposition 3. The first part of the sequence of the triad (N_{i}, N′′_{i}, N_{i−1}):

0 → H^{n−1}(N_{i}, N_{i−1}) ⟶ H^{n−1}(N′′_{i}, N_{i−1}) ⟶ 0,

and sequence (2.7) give

H^{n−1}(N_{i}, N_{i−1}) = H^{n−1}(N′′_{i}, N_{i−1}) = ker D_{i}  (2.9a)
H^{n}(N′′_{i}, N_{i−1}) = coker D_{i}.  (2.9b)

Using (2.7) and (2.8), we have the commutative diagram

D_{i} ↓
ℂ^{m} ⊕ ℂ^{m}
B_{i} ↓  ↘ D_{i}^{∗}
0 → H^{n}(N_{i}, N_{i−1}) → H^{n}(N′′_{i}, N_{i−1}) ⟶^{Δ_{i}} ℂ^{m} → H^{n+1}(N_{i}, N_{i−1}) → 0
↓  ↘
0  H^{n+1}(N_{i}, N′_{i}).

which implies

H^{n}(N_{i}, N_{i−1}) = ker Δ_{i} = B_{i}(coker D_{i} ∩ ker D_{i}^{∗})  (2.10a)

H^{n+1}(N_{i}, N_{i−1}) = coker Δ_{i} = coker D_{i}^{∗} = H^{n+1}(N_{i}, N′_{i}) (2.10b)

Sequence (2.7) has torsion τ(D_{i})^{(−1)^{n−1}} and sequence (2.8) has torsion τ(Δ_{i})^{(−1)^{n−1}}. Therefore, Corollary 2 gives

τ_{d(N′′_{i},N_{i−1})} = τ_{d(N′′_{i},N′_{i})}τ_{d(N′_{i},N_{i−1})}τ(D_{i})^{(−1)^{n}} = τ(D_{i})^{(−1)^{n}}, (2.11a)

τ_{d_{0}^{n−1,i}} = τ_{d(N_{i},N′′_{i})}τ_{d(N′′_{i},N_{i−1})}τ(Δ_{i})^{(−1)^{n}} = τ(D_{i})^{(−1)^{n}}τ(Δ_{i})^{(−1)^{n−1}}. (2.11b)

To compute τ(D_{i}), we recall that D_{i} = ( \begin{matrix}I−ρ(α_{i}) \\ I±ρ(β_{i}) \end{matrix} ). Since ρ(α_{i}), ρ(β_{i}) ∈ U(m) and they satisfy a relation ρ(α_{i})ρ(β_{i}) = ρ(β_{i})^{±1}ρ(α_{i}), there is a splitting

ℂ^{m} = V_{α_{i}} + V_{β_{i}} + ker D_{i},

REIDEMEISTER TORSION AND INTEGRABLE SYSTEMS 13

invariant under ρ(α_{i}), and ρ(β_{i}), such that I − ρ(α_{i}) : V_{α_{i}} ↩ and I ± ρ(β_{i}) : V_{β_{i}} ↩ are nonsingular. Let k_{i}, v_{α_{i}}, v_{β_{i}} be bases of H^{n−1}(N_{i}, N_{i−1}) = ker D_{i}, V_{α_{i}}, V_{β_{i}}. Let

W_{i} = (V_{β_{i}} + ker D_{i}) ⊕ (V_{α_{i}} + ker D_{i}),

then j : W_{i} → H^{n}(N′′_{i}, N_{i−1}) is an isomorphism, and so j[(v_{β_{i}} ∪ k_{i}) × {0}] ∪ j[{0} × (v_{α_{i}} ∪ k_{i})] is a basis of H^{n}(N′′_{i}, N_{i−1}). Thus

τ(D_{i}) = \frac{∧(D_{i}v_{α_{i}}, D_{i}v_{β_{i}}, (v_{β_{i}} ∪ k_{i}) × \{0\}, \{0\} × (v_{α_{i}} ∪ k_{i})) ⊗ ∧k_{i}}{∧(v_{α_{i}}, v_{β_{i}}, k_{i}) ⊗ ∧(B_{i}[(v_{β_{i}} ∪ k_{i}) × \{0\}], B_{i}[\{0\} × (v_{α_{i}} ∪ k_{i})])}
=\frac{∧((I − ρ(α_{i}))v_{α_{i}}, v_{β_{i}}, k_{i}) ⊗ ∧(v_{α_{i}}, (I ± ρ(β_{i}))v_{β_{i}}, k_{i}) ⊗ ∧k_{i}}{∧(v_{α_{i}}, v_{β_{i}}, k_{i}) ⊗ B_{i}^{∗}(∧(v_{β_{i}}, k_{i}) ⊗ ∧(v_{α_{i}}, k_{i}))} . (2.12)

To compute τ(Δ_{i}) we recall that D_{i}^{∗} = (I ± ρ(β_{i}), ρ(α_{i}) − I) and then W_{i} ∩ ker D_{i}^{∗} = ker D_{i} ⊕ ker D_{i}, D_{i}^{∗}(ℂ^{m} ⊕ ℂ^{m}) = D_{i}^{∗}(W_{i}) = V_{β_{i}} + V_{α_{i}}. Thus ker Δ_{i} = B_{i}(ker D_{i} ⊕ ker D_{i}) and coker Δ_{i} = coker D_{i}^{∗} = ker D_{i}. Thus H^{n}(N_{i}, N_{i−1}) ≅ ker D_{i} ⊕ker D_{i}, H^{n}(N_{i}, N_{i−1}) = ker D_{i}, and using k_{i} × k_{i} and k_{i} as their bases we have

τ(Δ_{i}) = \frac{∧((I − ρ(α_{i}))v_{α_{i}}, (I ± ρ(β_{i}))v_{β_{i}}, k_{i}) ⊗ (∧k_{i} ⊗ ∧k_{i})}{B_{i}^{∗}(∧(v_{β_{i}}, k_{i}) ⊗ ∧(v_{α_{i}}, k_{i})) ⊗ ∧k_{i}}.

(2.13)

From equations (2.11), (2.12), (2.13) we have τ_{d_{0}^{n−1,i}} = 1.

3. R-torsion and connecting gradient lines 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

∅ = K_{−1} ⊂ K_{0} ⊂ L_{1} ⊂ P_{1} ⊂ K_{1} ⊂ L_{2} ⊂ P_{2} ⊂ K_{2} ⊂ K_{3} = M such that L_{1} ⊂ M_{1} ⊂ L_{2}, P_{1} ⊂ M_{2} ⊂ P_{2} and

\begin{aligned}⋂_{t∈ℝ}ψ_{t}(K_{i} \ K_{i−1}) & = C_{i}(g) i = 0, 1, 2, 3. \\ ⋂_{t∈ℝ}ψ_{t}(L_{i} \ K_{i−1}) & = C_{i}(g) ∩ M_{1} i = 1, 2. \\ ⋂_{t∈ℝ}ψ_{t}(P_{i} \ L_{i}) & = C_{i}(g) ∩ (M_{2} \ M_{1}) i = 1, 2. \\ ⋂_{t∈ℝ}ψ_{t}(K_{i} \ P_{i}) & = C_{i}(g) \ M_{2} i = 1, 2. \end{aligned}

We have

H^{i}(K_{i}, K_{i−1}) = ⨁_{index(x)=i} ℂ^{m}x.  (3.1)

Note that all the components G_{xy}^{∗} : ℂ^{m}x → ℂ^{m}y of the maps d^{K} in the cochain complex

0 → H^{0}(K_{0}) ⟶^{d^{K}} H^{1}(K_{1}, K_{0}) ⟶^{d^{K}} H^{2}(K_{2}, K_{1}) ⟶^{d^{K}} H^{2}(M, K_{2}) → 0, (3.2)

are given as in equation (2.5) of Lemma 1.

Theorem 1. For critical circles γ_{i} the components F_{ij}^{k} are induced by the maps

• (c.c) G_{w_{i}w_{j}}^{n}, G_{z_{i}z_{j}}^{n+1} for k_{2n} < i ≤ k_{2n+1}, k_{2n+2} < j ≤ k_{2n+3}, n = 0, 1 • (c.t) G_{w_{i}q_{j}}^{1}, G_{w_{i}r_{j}}^{1}, G_{z_{i}s_{j}}^{2} for k_{2} < i ≤ k_{3} < j ≤ k_{4}.

For critical tori F_{c_{i}} the components F_{ij}^{k} are induced by the maps

• (t.c) G_{p_{i}w_{j}}^{0}, G_{q_{i}z_{j}}^{1}, G_{r_{i}z_{j}}^{1} for k_{1} < i ≤ k_{2} < j ≤ k_{3}.

Proof. Consider the diagrams

\begin{aligned}H^{0}(M_{1}) & ⟶^{d_{1}} & H^{1}(M_{2}, M_{1}) & ⟶^{d_{1}} & H^{2}(M, M_{2}) \\ & & ↓ & & ↑ \\ ↓ & & H^{1}(M_{2}, L_{1}) & & H^{2}(M, P_{2}) \\ & & ↓ & & ↓ \\ H^{0}(K_{0}) & & H^{1}(P_{1}, L_{1}) & & H^{2}(K_{2}, P_{2}) \\ & ↘ d & ↓ \\ & & H^{1}(P_{1}, K_{0}) & & ↓ \\ & & ↑ \\ & & H^{1}(K_{1}, K_{0}) & ⟶^{d^{K}} & H^{2}(K_{2}, K_{1}) \end{aligned}  (3.3)

and

\begin{aligned}H^{1}(M_{1}) & ⟶^{d_{1}} & H^{2}(M_{2}, M_{1}) & ⟶^{d_{1}} & H^{3}(M, M_{2}) \\ ↓ & & ↑ \\ H^{1}(L_{1}) & & H^{2}(P_{2}, M_{1}) \\ & & ↑ & & ↑ \\ ↑ & & H^{2}(P_{2}, K_{1}) \\ & & ↓ \\ H^{1}(L_{1}, K_{0}) & & H^{2}(P_{2}, L_{2}) & & H^{3}(M, K_{2}) \\ ↓ & & ↓ & d^{K} ↗ \\ H^{1}(K_{1}, K_{0}) & ⟶^{d^{K}} & H^{2}(K_{2}, K_{1}) \end{aligned} (3.4)

where the downwards maps are injective and the upwards maps are surjective. By (3.3) the maps d_{1} in

H^{0}(M_{1}) ⟶^{d_{1}} H^{1}(M_{2}, M_{1}) ⟶^{d_{1}} H^{2}(M, M_{2})

are induced by the components H^{0}(K_{0}) → H^{1}(P_{1}, L_{1}) and

H^{1}(P_{1}, L_{1}) = ⨁_{k_{2}<i≤k_{3}} ℂ^{m}w_{i} → H^{2}(K_{2}, P_{2}) = ⨁_{k_{4}<i} ℂ^{m}w_{i}⊕ ⨁_{k_{3}<i≤k_{4}} ℂ^{m}q_{i}⊕ℂ^{m}r_{i}

of the maps d^{K} in (3.2). By (3.4) the maps d_{1} in

H^{1}(M_{1}) ⟶^{d_{1}} H^{2}(M_{2}, M_{1}) ⟶^{d_{1}} H^{3}(M, M_{2})

are induced by the components

H^{1}(L_{1}, K_{0}) = ⨁_{i≤k_{1}} ℂ^{m}z_{i}⊕ ⨁_{k_{1}<i≤k_{2}} ℂ^{m}q_{i}⊕ℂ^{m}r_{i} → H^{2}(P_{2}, L_{2}) = ⨁_{k_{2}<i≤k_{3}} ℂ^{m}z_{i}

and H^{2}(P_{2}, L_{2}) → H^{3}(M, K_{2}) of the maps d^{K} in (3.2).

We now consider the term (E_{2}, d_{2}) of the spectral sequence defined by (1.8). By Corollary 4, we can have nonzero maps d_{2} : E_{2}^{n,q} → E_{2}^{n+2,q−1} only for n = 0, q = 0, 1, 2.

E_{2}^{0,q} = ker(d_{1} : H^{q}(M_{1}) → H^{q+1}(M_{2}, M_{1})), q = 0, 1.E_{2}^{0,2} = H^{2}(M_{1})

E_{2}^{1,q} = ker(d_{1} : H^{q+1}(M_{2}, M_{1}) → H^{q+2}(M, M_{2}))/d_{1}(H^{q}(M_{1})), q = 0, 1

E_{2}^{2,−1} = H^{1}(M, M_{2}), E_{2}^{2,q} = H^{q+2}(M, M_{2})/d_{1}(H^{q+1}(M_{2}, M_{1})), q = 0, 1.

Theorem 2.

(0) The map d_{2} : E_{2}^{0,0} → E_{2}^{2,−1} is induced by the component

H^{0}(K_{0}) → ⨁_{k_{3}<i≤k_{4}}ℂ^{m}p_{i} of d^{K} : H^{0}(K_{0}) → H^{1}(K_{1}, K_{0}).

(1) The map d_{2} : E_{2}^{0,1} → E_{2}^{2,0} is induced by the component

⨁_{i≤k_{1}} ℂ^{m}z_{i} ⊕ ⨁_{k_{1}<i≤k_{2}} ℂ^{m}q_{i} ⊕ ℂ^{m}r_{i} → ⨁_{k_{4}<i} ℂ^{m}w_{i} ⊕ ⨁_{k_{3}<i≤k_{4}} ℂ^{m}q_{i} ⊕ ℂ^{m}r_{i}

of d^{K} : H^{1}(K_{1}, K_{0}) → H^{2}(K_{2}, K_{1}).

(2) The map d_{2} : E_{2}^{0,2} → E_{2}^{2,1} is induced by the component

⨁_{k_{1}<i≤k_{2}}ℂ^{m}s_{i} → H^{3}(M, K_{2}) of d^{K} : H^{2}(K_{2}, K_{1}) → H^{3}(M, K_{2}).

Proof. By (1.9), the map d_{2} : E_{2}^{0,q} → E_{2}^{2,q−1} is given as the composition

ȷ_{q−1}^{−1} ∘ δ_{q}|im k_{q} : im k_{q} → H^{q+1}(M, M_{2})/im j_{q−1}

where k_{q} : H^{q}(M_{2}) → H^{q}(M_{1}), δ_{q} : H^{q}(M_{1}) → H^{q+1}(M, M_{1}) is the coboundary map and j_{q−1} : H^{q+1}(M, M_{2}) → H^{q+1}(M, M_{1}) defines the isomorphism ȷ_{q−1} : H^{q+1}(M, M_{2})/ ker j_{q−1} → im j_{q−1}. Consider the diagrams

\begin{aligned}E_{2}^{0,0} & ⟶^{δ_{0}|im k_{0}} & im j_{−1} & ⟵^{ȷ_{−1}} & E_{2}^{2,−1} \\ ∩ & & ∩ & & ↑ \\ H^{0}(M_{1}) & ⟶^{δ_{0}} & H^{1}(M, M_{1}) & ⟵^{j_{−1}} & H^{1}(M, M_{2}) \\ & & & & ↓ \\ & & & & H^{1}(M, P_{1}) \\ ↓ & & & & ↓ \\ & & & & H^{1}(K_{1}, P_{1}) \\ & & & & ↓ \\ H^{0}(K_{0}) & & ⟶^{d^{K}} & & H^{1}(K_{1}, K_{0}) \end{aligned}  (3.5)

\begin{matrix}E_{2}^{0,1} & ⟶^{δ_{1}|im k_{1}} & im j_{0} & ⟵^{ȷ_{0}} & E_{2}^{2,0} \\ ∩ & & ∩ & & ↑ \\ H^{1}(M_{1}) & ⟶^{δ_{1}} & H^{2}(M, M_{1}) & ⟵^{j_{0}} & H^{2}(M, M_{2}) \\ ↓ & & & & ↑ d^{1} \\ H^{1}(L_{1}) & & & & H^{2}(M, P_{2}) \\ ↑ & & & & ↓ \\ H^{1}(L_{1}, K_{0}) & & & & H^{2}(K_{2}, P_{2}) \\ ↓ & & & & ↓ \\ H^{1}(K_{1}, K_{0}) & & ⟶^{d^{K}} & & H^{2}(K_{2}, K_{1}) \end{matrix} (3.6)

\begin{matrix}E_{2}^{0,2} & ⟶^{δ_{2}|im k_{2}} & im j_{1} & ⟵^{ȷ_{1}} & E_{2}^{2,1} \\ ∩ & & ∩ & & ↑ \\ H^{2}(M_{1}) & ⟶^{δ_{2}} & H^{3}(M, M_{1}) & ⟵^{j_{1}} & H^{3}(M, M_{2}) \\ ↑ \\ H^{2}(L_{2}) \\ ↑ & & & & ↑ \\ H^{2}(L_{2}, K_{1}) \\ ↑ \\ H^{2}(K_{2}, K_{1}) & & ⟶^{d^{K}} & & H^{3}(M, K_{2}) \end{matrix} (3.7)

where as in Theorem 1 the downwards maps are injective and the upwards maps are surjective. By (3.5) and H^{1}(K_{1}, P_{1}) = ⨁_{k_{3}<i≤k_{4}}ℂ^{m}p_{i}, we

get (0). By (3.6), H^{1}(L_{1}, K_{0}) = ⨁_{i≤k_{1}}ℂ^{m}z_{i} ⊕ ⨁_{k_{1}<i≤k_{2}}ℂ^{m}q_{i} ⊕ ℂ^{m}r_{i} and

H^{2}(K_{2}, P_{2}) = ⨁_{k_{4}<i}ℂ^{m}w_{i} ⊕ ⨁_{k_{3}<i≤k_{4}}ℂ^{m}q_{i} ⊕ ℂ^{m}r_{i} we get (1). By (3.6) and

H^{2}(L_{2}, K_{1}) = ⨁_{k_{1}<i≤k_{2}}ℂ^{m}s_{i}, we get (2).

4. Examples

We consider instances of the motion of a rigid body or a gyrostat. The Euler-Poisson equations describing the motion of a rigid body about a fixed point in a gravitational field are

\begin{aligned}Aω̇ & = Aω × ω + ν × C \\ ν̇ & = ν × ω. \end{aligned}  (4.1)

where ω is the angular velocity in the body frame, the matrix A is the moment of inertia that we may assume diagonal, C is the center of mass and ν is a unitary vector along the vertical axis in the fixed frame. More generally the equations of motion of a gyrostat in a gravitational field are

\begin{aligned}Aω̇ & = (Aω + λ) × ω + ν × C \\ ν̇ & = ν × ω. \end{aligned}  (4.2)

The system (4.2) has the integrals

f_{1} = |ν|^{2}, f_{2} = ⟨Aω + λ, ν⟩, H = \frac{1}{2}⟨Aω, ω⟩ + ⟨ν, c⟩.

We embed the system (4.2) in the dual e(3)^{∗} ≅ ℝ^{6} of the Lie algebra of the euclidian group. N_{g,λ} = {(ν, Aω + λ) : f_{1}(ν, ω) = 1, f_{2}(ν, ω) = g} is a coadjoint orbit and so a symplectic leaf of the Poisson manifold e(3)^{∗}. The system (4.2) is transformed into the hamiltonian system

\begin{matrix}Ṙ = & \frac{∂H}{∂S} × S + \frac{∂H}{∂R} × R \\ Ṡ = & \frac{∂H}{∂S} × R. \end{matrix}  (4.3)

Example 1. The Kovalevskaya case of the equations (4.1) corresponds to A_{1} = A_{2} = 2A_{3}, C = (c, d, 0), that we normalize to A_{1} = c = 1, d = 0. In this case 4.3 admits the independent integral

K = ((S_{1}^{2} − S_{2}^{2})/2 − R_{1})^{2} + (S_{1}S_{2} − R_{2})^{2}

with {H, K} = 0.

For h > g^{2} + 1/2g^{2}, M = {(R, S) ∈ N_{g,0} : H(R, S) = h} is the 3dimensional real projective space and so π_{1}(M) = ℤ_{2}. Let ρ : π_{1}(M) → U(1) be the nontrivial representation. The critical sets of K : M → ℝ are two minimum circles m_{1}, m_{2} with ρ(m_{i}) = 1, one orientable and two nonorientable saddle circles r_{1}, r_{2}, r_{3} with ρ(r_{j}) = −1, and one maximum circle n with ρ(n) = −1. The family of tori starting at m_{i}nnnnnnn changes to a family F_{i} of tori when crossing r_{i} (i = 1, 2). The families F_{1} and F_{2} come together to become one family when crossing r_{3}.

Since Δ(r_{i}) = −1, i = 1, 2, Δ(r_{3}) = 1, we have

H^{k}(M_{1}) = \begin{cases}ℂ ⊕ ℂ & if k = 0, 1 \\ 0 & if k ≠ 0, 1 \end{cases}

H^{k}(M_{2}, M_{1}) = \begin{cases}ℂ ⊕ ℂ & if k = 1, 2 \\ 0 & if k ≠ 1, 2 \end{cases}

H^{∗}(M, M_{2}) = 0

τ_{d_{0}^{0,i}} = τ_{d_{0}^{1,i}} = 1 (i = 1, 2), τ_{d_{0}^{1,3}} = \frac{1}{2}, τ_{d_{0}^{2}} = 2 ⇒ τ_{d_{0}} = 1

We now change the Bott integral K to a Morse Smale function f 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 f 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.

Therefore d_{1} : H^{0}(M_{1}) → H^{1}(M_{2}, M_{1}) = ( \begin{matrix}2 & 0 \\ 0 & 2 \end{matrix} ) and d_{1} : H^{1}(M_{1}) →

H^{2}(M_{2}, M_{1}) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ). Thus E_{2}^{∗∗} = 0 and τ_{d_{1}} = 2 · 2/1 = 4.

Example 2. The Stretensky case of equations (4.2) corresponds to A_{1} = A_{2} = 4A_{3}, C = (c, d, 0), λ = (0, 0, l), that we normalize to A_{1} = c = 1, d = 0. In this case (4.2) is integrable only for g = 0. An independent integral with {H, K} = 0 is

K = (S_{3} + l)(S_{1}^{2} + S_{2}^{2}) − S_{1}R_{3}

The manifold M = {(R, S) ∈ N_{0,3/4} : H(R, S) = 0.97} is the connected sum of two copies of S^{2} × S^{1} and so π_{1}(M) is free with two generators α_{1}, α_{2}. The critical sets of K : M → ℝ are one contractible minimum circle m, one nonorientable and two orientable contractible saddle circles r_{1}, r_{2}, r_{3}, and three maximum circles n_{1} ≅ α_{1}, n_{2} ≅ α_{2}, n_{3} ≅ α_{1}α_{2}. The family of tori starting at m changes to a family F of tori when crossing r_{1}. When F crosses r_{2}, r_{3} simultaneously, it splits into three families of tori ending at the three maximum circles. Let ρ : π_{1}(M) → U(p) be a representation such that det(I −ρ(α_{i})), det(I −ρ(α_{1}α_{2})) ≠ 0. We have

H^{k}(M_{1}) = \begin{cases}ℂ^{p} & if k = 0, 1 \\ 0 & if k ≠ 0, 1 \end{cases}

H^{k}(M_{2}, M_{1}) = \begin{cases}ℂ^{p} ⊕ ℂ^{p} & if k = 1, 2 \\ 0 & if k ≠ 1, 2 \end{cases}

H^{∗}(M, M_{2}) = 0

τ_{d_{0}^{0}} = τ_{d_{0}^{1,i}} = 1 (i = 2, 3), τ_{d_{0}^{1,1}} = \frac{1}{2}

τ_{d_{0}^{2}} = det(I − ρ(α_{1})) det(I − ρ(α_{1})) det(I − ρ(α_{1}α_{2})).

We now change the Bott integral K to a Morse Smale function f to compute d_{1} : H^{k}(M_{1}) → H^{k+1}(M_{2}, M_{1}), k = 0, 1. The function f has critical points w, z on m with indices 0, 1 respectively, and critical points η_{i}, ζ_{i} on r_{i} with indices 1, 2 respectively. There are two orbits β_{i}, γ_{i} connecting ζ_{i} to z with ρ(β_{i}) = ρ(α_{i}), ρ(γ_{i}) = I, and two orbits δ_{i}, ε_{i} connecting η_{i} to w with ρ(δ_{i}) = ρ(α_{i}), ρ(ε_{i}) = I. There-

fore d_{1} : H^{0}(M_{1}) → H^{1}(M_{2}, M_{1}) = ( \begin{matrix}I−ρ(α_{1}) \\ I−ρ(α_{2}) \end{matrix} ), and d_{1} : H^{1}(M_{1}) →

H^{2}(M_{2}, M_{1}) = ( \begin{matrix}I−ρ(α_{1}) \\ I−ρ(α_{2}) \end{matrix} ) Thus

E_{2}^{s,q} = \begin{cases}H^{1+q}(M_{2}, M_{1})/d_{1}(H^{q}(M_{1})) ≅ ℂ^{p} & if s = 1, q = 0, 1 \\ 0 & in other case \end{cases} ,

(4.4)

and then

H^{q}(M) = \begin{cases}E_{2}^{1,q−1} & if q = 1, 2 \\ 0 & in other case \end{cases} .  (4.5)

Let h = \{h_{1}, …, h_{p}\} be a complementary basis to im ( \begin{matrix}I−ρ(α_{1}) \\ I−ρ(α_{2}) \end{matrix} ) and denote by [h_{1}], [h_{2}] the corresponding bases of H^{1}(M), H^{2}(M). Then

τ_{d_{1}} = \frac{∧[h_{2}]}{∧[h_{1}]}.

References

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Institute für Mathematik, E.-M.-Arndt-Universität Greifswald, Jahnstrasse 15a, D-17489 Greifswald, Germany.

E-mail address: felshtyn@rz.uni-greifswald.de

Instituto de Matemáticas, UNAM, Ciudad Universitaria C. P. 04510, Cd. de México

E-mail address: hector@matem.unam.mx