L., — - - : ^ w i : IC/93/305

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

SYMPLECTIC TOPOLOGYOF INTEGRABLE HAMILTONIAN SYSTEMS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL.

SCIENTIFICAND CULTURALORGANIZATION

Nguyen Tien Zung

MIRAMARE-TRIESTE

IC/93/305

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SYMPLECTIC TOPOLOGYOF INTEGRABLE HAMILTONIAN SYSTEMS

Nguyen Tien ZungInternational Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

We study the topology of integrable Hamiltonian systems, giving the main attentionto the affine structure of their orbit spaces, In particular, we develop some aspects ofFomenko's theory about topological classification of integrable non degenerate systems,and consider some relations between such systems and "pure'' contact and symplerticgeometry. We give a notion of integrable surgery and use it to obtain some interestingsymplectic structures.

M1RAMARE TRIESTE

August 1993

1 Introduction

In classical mechanics and theoretical physics, there are many well-known differ-

ential equations, which can be written in the form of {or reduced to) integrable

Hamiltonian systems. Examples are many cases of motion of a rigid body, Toda

equations, solitons,... Generally speaking, one would like to inderstand such sys-

tems in various aspects: algebraic, dynamical, and topological. Since all well-known

systems are of algebraic nature, algebraic geometry plays a, fundamental role in find-

ing mechanisms of integrability and the algebraic structure of integrable systems.

There are numerous authors working on algebraically integrable systems. Dynamical

and topological structure of integrable systems is also extremely interesting, since it

gives a qualitative picture about integrable systems as smooth dynamical systems.

On the other hand, symplectic geometry, which grew out from classical mechan-

ics, has become a beautiful branch of mathematics. It has many relations with other

subjects, e.g. classical mechanics and theoretical physics, complex arid algebraic ge-

ometry, and it has many interesting problems on its own.

This work arises from several simple observations about the topological and

geometrical structure of integrable systems, and its relations with "pure" symplectic

and contact geometry, and affine geometry. In particular, I found (in a somewhat

roundabout way) that the affine structure of the orbit space is the most important

thing in the topology of integrable systems. I came to these observations while

trying to understand what is essential in the topology of integrable systems, and

to develop Fomenko's theory about topological classification of integrable systems

[F]. Later I learned about many papers dealing with topology of integrable systems.

In particular, besides Fomenko's school, Lerman, Umanskii, Eliasson, Condevaux,

Dazord, Molino, Delzant and others wrote several recent important papers on the

subjects, and in this work I tried to incorporate their results and ideas. Then I

found a very beautiful book by Michele Audin [Aul], from which I learned a lot of

symplectic geometry and Hamiltonian torus action. Torus actions are a very special

case of integrable systems, where all the singularities are elliptic, and where the

geometry looks very elegant. As regards "pure" symplectic and contact geometry,

the recent works by Gromov and Eliashberg influenced me most. In this we work

use holomorphic disks (Eliashberg's result) to give a partial answer to the cobordism

problem in integrable systems.

We will often restrict ourself to the case of 2 degrees of freedom, i.e. systems on

4-dimensional symplectic manifolds. The reason is that it is the smallest dimension

when we have a "non-trivial" symplectic geometry. Another reason is that many

results i'i the case of dimension 4 ate valid for higher dimensions with the same

proof. !n the end of each section we will discuss higher-dimensional cases.

Our lain results of this work can be stated roughly as follows:

Theorem 1 (Propositions 3.1, 3.2, 4.1) There is a canonical form for codimension

1 singularities, and for focus-focus singularities.

Theorem 2 (Proposition 5.1) There is an obstruction for a 2-complex to be an

orbit space of some integrable system.

Theorem 3 (Propositions 5.4, 5.5, 5.9) There are some sufficient conditions for

the existence and for the uniqueness of integrable surgery.

Theorem 4 (Proposition 6.1) For any given finitely generated group there is an

(non-degenerate) integrable Hamiltonian system on a closed symplectic 4-manifold

with that group as the fundamental group.

Theorem 5 (Theorem 7.2, with Boisinov) The formal cobordism group of inte-

grable systems restricted on isoenergy 3-manifolds is trivial

Theorem 6 (Propositions 9.4, 9.5, 9.6) An integrable system restricted on a

isoenergy 3-manifold admits a compatible locally ^'-invariant contact structure if

and only if it is almost convex. It admits a (not necessary locally ^'-invariant)

compatible contact structure only if a similar condition is satisfied.

Theorem 7 (Corollary 9.13) There exist integrable Hamiltonian systems re-

stricted on isoenergy 3 manifolds, which are not a boundary of any integrable system

on a compact symplectic 4-manifold.

I feel that the notion of integrable surgery is one of the most important contri-

butions, if there is any, of this work. As applications of integrable surgery, we give a

!*_j»_.jJilL M .

new proof of Delzant's theorem (cf. [De]), and an analogue of Gompf's construction

[Go] for the case of integrable systems (Theorem 4)

This work is organized as follows:

In section 2 we give some preliminary notions which we will use throughout the

work. Section 3 deals with codimension 1 singularities of integrable systems. Fist

we give some results in case of 2 degrees of freedom, and then generalize them for

any dimension. In section 4 we give a notion of nondegenerate systems and study

the behavior of the affine structure of the orbit space near singularities. In section

5 we discuss general orbit spaces, namely existence of integrable systems with given

affme-structured orbit spaces, and give a notion of "integrable surgery". We give

some examples of integrable surgery, including a simple proof of Delzant's theorem.

In Section 6 we use integrabie surgery to produce examples of non-degenerate inte-

grable systems on symplectic closed 4-manifolds with any given finitely generated

group as its fundamental group (Gompf's construction), in Section 7 we discuss

about Fomenko's theory of classification of integrable systems restricted on a isoen-

ergy level, and then consider the cobordism problem of such systems. In Section

8 we define some new invariants for integrable systems restricted on an isoenergy

level. Unlike Fomenko's invariants, these new invariants are "geometrical", i.e. they

depend on the symplectic form. We will use these invariants in the next section. In

Section 9 we discuss about compatible contact structures, their role in the cobordism

problem, and relations with "pure" contact geometry. In particular, we give some

sufficient and necessary conditions for a system restricted on an isoenergy surface to

admit a compatible contact structure. The last section is about some non-standard

symplectic structures on ft4 and R2n.

2 Preliminaries

Let us introduce some basic notions from symplectic and contact geometry, with the

notations we will use throughout the work. There are now many available textbooks

and monographs where one can find background on symplectic geometry. We refer

to [AG, Au, F2, Wei] for symplectic geometry, and [F2, GSJ for applications in

physics.

A symplectic manifold is a smooth even-dimensional manifold MJ" with a non-

degenerate closed differential 2-form u> on it: Anw ^ 0. In this case u> is called

a symplectic form (or a symplectic structure). A sub-manifold N of a symplectic

manifold (JW2",u) is called isotropic if w restricted on N is identically zero. Since

UJ vanishes on JV, the dimension of N can not be greater than n. If the dimension

of N is equal to n, then N will be called a Lagrangian submanifold. A foliation by

Lagrangian submanifolds is called a Lagrangian foliation.

To every smooth function H from a symplectic manifold to the real numbers,

H : (M2n,u>) -4 R, we can associate a vector field XH, sometimes denoted by v, by

the following equation:

dH = -iXHu

XH is called a Hamiltonian vector field corresponding to the Hamiitonian func-

tion H. Differential equation and dynamical system given by Xfj are called Hamil-

tonian equation and Hamiltonian system, respectively.

By Darboux's theorem, locally there exist coordinates (p,-,q,) in which ui has the

canonical form w = Yj dptf\dq,. If (p., <7i) are the global coordinates in the Euclidean

space R2l>, then what we have is called a standard symplectic form in R2". In these

coordinates the Hamiltonian equation will look like:

f pi=\q,=

pi=-dH/dqi

q,=

A function F : (M2n,u>) —f fl is called a first integral for a Hamiltonian H (or a

Ha.miHonian system XH), if F is preserved by the flow of XH- Equivalently, F and

H are commutative, i.e. their Poisson bracket is identically equal to zero:

{F,H} i~u{XF,XH) = 0

Notice that H is obviously a first integral for XH-

A Hamiltonian system on a symplectic 2n-manifotd is called integrable by Li-

ouville, if there exists a family of n first integrals, F\ = H, Fi,..., Fn, which are

pairwise commutative, and which are functionally independent almost everywhere:

dFi A dF2 A . . . A dFn ± 0.

There is an analogue of Darboux's theorem for integrable systems, which is called

Arnold-Liouville theorem (cf. [Ail]). It says the following: Suppose a connected

component of a level set {F\ = consti,..., Fn = constn} is compact and nonsingular.

Then it is an n-torus, and its tubular neighborhood has the form Dn x T" with the

coordinates (p,, q±) (i = 1,... , n, qi — modi), in which the symplectic form u and the

Hamiltonian H have the following canonical form:

w = X ] d j > i A rf*'H = H&u • • • 'P»)

The tori {p< = const} are called Liouville tori. Note that these tori are invariant

under the Hamiltonian system v = XH. If the system is non-resonant, i.e. if for

almost every {<j,} the orbits of XH are dense in the corresponding Liouville torus,

then Liouville tori are smallest closed invariant subsets for almost every {qt}, and

the Lagrangian foliation by Liouville tori is determined uniquely by the system. If

the second derivative of H considered as a function of {p,} is nondegenerate almost

everywhere, then the system will be non-resonant.

From now all we will assume that all the systems under consideration are non-

resonant whenever necessary. In other words, the foliation by Liouville tori is

uniquely determined by the system. Of course, in general, this foliation has sin-

gularities at the points where the first integrals are dependent. From now on when

we say "Lagrangian foliation" we will always mean a singular Lagrangian foliation.

Of course one has to impose some conditions on the singularities. We will talk more

about singularities later. Conversely, given an arbitrary Lagrangian foliation, then

any function which is invariant on the tori of that foliation will be automatically

integrable, because the Hamiltonian vector field of that function will be tangent

to the foliation. Thus one can talk about integrable Hamiltonian systems in the

language of Lagrangian foliations.

There is another language to describe integrable systems: via Poisson actions.

Abstractly, a Poisson action is just a Hamiltonian group action for some Abelian

Lie group. If Fi,..., Fn are the commutative first integrals, then A"F, , . . . , Xpn are

commutative llamiltonian vector fields, and they are generators for an ftn-Poisson

action.

Sometimes it is more comfortable to work with Lagrangian foliations or Poisson

actions, than with Hamiltonian functions. Thus by abuse of language, sometimes

under the word "integrable system" we mean a Lagrangian foliation, or an Poisson

action.

An atlas of a manifold is called affine if all the transformation maps are affine.

A equivalent class of affine atlases is called an affine structure on a given mani-

fold. A curve which is locally a straight line everywhere in an affine structure is

called a geodesic with respect to this affine structure (see e.g. [CDMj). An affine

structure is called flat if all the transformation maps for some atlas are of the shift

type X 4 X + a. An affine structure is called integral if for some atlas all the trans-

formation maps have integral linear part, i.e. the matrix of the linear part has

integral coefficients. As a corollary of the Arnold-Liouville theorem, there are two

affine structures arisen from an integrable system (see e.g. [DD]): the first one is on

every Liouville torus, which is given by the coordinates (%). This affine structure

is actually flat: it equals the quotient of Rn by Zn. The tlamiltonian system XH

is constant on every Liouville torus in the sense that Xn is invariant under shifts

(<?i) ~~* (q,) +const. It is clear since both the symplectic form w and the Hamiltonian

H are invariant under such transformations. The second affine structure, not less

important, is the affine structure on the "base space" Dn, which is given by the

coordinates p,. It means that if there is another local system of coordinates (p;,</,')

which satisfies the conditions in Liouviile-Arnold theorem, then (p') can be obtained

from (pi) by an affine transformation. This second affine structure is integral. The

reason is that the Hamiltonian system Xp, of every coordinate p, is consi nt on

Liouville tori and has a periodic flow with (minimal) period 1.

A contact structurcis a distribution, say f, by spaces of codimension 1 on an odd-

dimensional manifold, say Q2n~l, which is totally non-integrabte. In local terms, it

means that if a is a differential non-singular one-form with kero = £, then aAA""'a

is different from zero everywhere. Note that this condition does not depend on the

choice of Q. If a can be defined globally, then we have a strong contact structure.

A submanifold of a contact manifold is called isotropic if it is tangent to the

Legendrian distribution. A Legendrian submanifold in a contact manifold Q2n~l is

an isotropic submanifold of dimension n— 1 (which is the maximal possible din ^nsion

for isotropic submanifoids). A submanifold in a contact manifold is called transversal

if it is transversal to the contact distribution.

A germ of a symplectic structure, or shortly, a symplectic germ, on an odd-

dimensional orientable manifold Q2"'1 is an equivalent class of symplectic structures

on Q2"~* x ( —e, +t), c small. Here two symplectic structures are called equivalent if

they coincide near Q x {0} (may be after a djffeomorphism preserving Q x {0}). A

symplectic manifold (M, u) induces on every its orientable hypersurface a symplectic

germ, also denoted by ui. More generally, we can talk about symplectic germs on

a (2n - k)-dimensional manifold Q2"~k, which come from symplectic structures on

Q2"-* x Dk, for which Q2n~k = Q2n~k x {0} is coisotropic.

(Q3n~\uj) will denote an orientable manifold Q2"~l with a symplectic germ w

on it. Given (Q2"'1,^), there is a natural Hamiltonian system t) on it, defined by

the Hamiltonian Q5""1 x ( — t, +c) —$• R, which is a projection on the second factor.

Note that, up to reparametrization, v is uniquely defined by (Q2"^1 , w). Sometimes

we will write (Q2"~',UJ, v) to emphasize that there is a Hamiltonian system on Q.

If ((J2™"1, )) is a hypersurface in a symplectic manifold, then the above denned

Hamiltonian system on it is called a characteristic flow, and the line field tangent

to that flow is called a characteristic line field.

A compatible (strong) contact structure on (Q2n~} ,UJ,V) is a contact structure

defined by an one-form a, which is positively transversal to the Hamiltonian vector

field: Q(UI) > 0. If an orientable hypersurface Q in a symplectic manifold admits a

compatible contact structure, then Q is said to be of contact type (ef. Eliashberg...).

In the case n — 2, Legendrian submanifoids have dimension 1. Thus they are

called Legendrian curves. A contact 3-manifold (or more generally, any contact (4fc —

1)- dimensional contact manifold) is automatically orientable: the form a A da, for

any local contact form a, gives rise to a unique orientation on Q3. If the orientation

on Q is already given, then we will consider only contact forms which give rise to

the same orientation on it. We will also call such contact forms "positive".

A function F from a smooth compact manifold Q to the real numbers R is called

a Bott function (see e.g. [Bot]) if all the maximal connecter singular sets of F are

smooth submanifoids of Q, and the second derivative of F is non- degenerate on the

normal bundle of that singular manifolds. So Bott functions are a generalization of

Morse functions.

Example 2.1

On the standard symplectic space (/f = {ii,yi,X2,y2},wo) there is an integrable

system, called a Hopf system, which is given by the Haroiltonian

The functions Fx = {(xl + y2) and F2 = 5(^2+ !/j) a r e first integrals for H, an they

are action coordinates for the Arnold-Liouville theorem. They are Bott functi ns on

every isoenergy level {H = const > 0}. The induced system on every 3- sphere S3 =

{H — const > 0} is also called a Hopf system (exactly here arises the terminology

"Hopf system", in analogy with Hopf fiber bundle S'1 -)• S1). By abuse of language,

the foliation by Lagrangian tori {F\ — const, F2 = const} is also called a Hopf

system. Hopf system is a. simplest example of integrable systems, but perhaps the

most important. All the known examples of integrable systems in physics has a local

Hopf system somewhere. Analogously, higher dimensional Hops system is given in

the standard symplectic space (/?2",LJ0) by the Hamiltonian H — ^(x\ + y\) + . . . +

3 Codimension one singularities

Suppose we are given an integrable system with two degrees of freedom, i.e. a Poisson

/^-action on a symplectic 4-manifold (M4,w) generated by 2 vector fields Xfj and

Xp, where H : {M*,u)) —} ft is an integrable Hamiltonian and F is an additional first

integral. We will always assume that the common level sets {H = const, F = const)

are compact, and thus they are tori wherever non-singular.

Here we are interested in the singularities of this Poisson action. A singular

value of the moment map (H,F) : M4 —• R2 is a value {/to,/o} for which there is a

preimage xo € M4 such that dH A dF|I0 = 0. Suppose now that we have a singular

point Xo, i.e. a point with the above property. Look at the connected component of

the level set {H = h0, F = f0}, which contains x0, and denote it by A'. In general,

N is not a torus. JV will be called a singularity of codimension 1 if there is a linear

combination of H and F, say aH + bF with some real constants a and 6, which is

non-singular on N: adH + bdF\x j£ 0 Vi £ N. By changing the generators of the

Poisson action, we will assume that dH ^ 0 on JV.

Consider the isoenergy surface Q = {H = ha}, and set UQ(N) C Q to be a

smooth saturated neighborhood of N. We will call N a non-degenerate (or Bott)

singularity if the function F restricted on UQ(N) is a Bott function, with the only

singular level at F = f0. In this case F is called a Bott first integral for H (locally

near /V).

Fomenko [F2] has classified all such codimension 1 non-degenerate singularities

up to topological equivalence. Let us briefly recall his results here.

Since F is preserved by the non-singular vector field XH, F cannot have isolated

singularities in UQ(N): its connected singular sets must contain at least one orbit

of XH. Since F is a Bott function, every its connected singular set must be diffeo-

morphic to either a circle S1, a torus T2 or a Klein bottle K2. (Note that it can

not be diffeomorphic to other surfaces because there is a non-singular vector field

Xfj sitting on it). In case it is T2, we have N = T2 and UQ{N) is diffeomorphic

to Dl x T2. Here D1 denotes an interval. In case it is K2, we have N — K2 and

UQ(N) is diffeomorphic to the quotient of D1 x T2 by some involution. In Fomenko's

theory, the cases Xs and A*2 are often ignored, since the foliation on UQ(N) is in

fact non-singular in such cases (up to Z^-coverings).

In the rest of this section we will study in more detail the case where alt the

singular sets are circles. In this case Fomenko [F] shows that N consists of singular

circles plus annuli: if we take out all singular curves from N, it will become a

disconnected sum of a non-negative number of annuli. The main reason why they

are annuli is the fact that the Poisson R2 action is locally free on them.

If N consists of just one circle, then it is called an elliptic singularity. Otherwise

it is called a hyperbolic one, and singular circles on it are called singular hyperbolic

orbits. In case of elliptic singularity, UQ(N) is just a solid torus, with that singular

10

m- w

curve lying in the center. The hyperbolic case is much more complicated. In any

case, UQ(N) will be a Seifert fibered space [F2].

We will construct here an explicit Seifert S'-foliation on UQ(X), Mure precisely,

we have:

Proposition 3.1 (cf. [Zl]j Suppose ;V is a hyperbolic singularity in the a6oi;e no-

tatiotis. Thfn Ifurr arr amnoth functions a.bfrvm a tubular neighborhood U,i .V) =

D' xWg(.V) c ;!/' to l{ fvch that:

u)il« A ,111 A ,//.' = (lh A. , / / / A ,//.• = 0

h, Tilt Jl r ,/tnfrntnl hi) .V ( , . / /+ tr) i* jitvimlir t>f )itnod I in H\,i.\ i. r.,, r pt at .«,mt

.inKjuhir rurrt* iriin* il mnij Ixirt \>< noil l/i.

! ' h < - i i s s c - r l i o i i a ) m e n u s t h a t t h o v e c t o r f i e l d . V ( ! l W + i / . - | i s r o l i n c H r t o X u a n d A ' ; . - :

i t l i e ; , r ; u t h e l e a v e s o f t h e L a g r a i i g i a n f o l i a t i o n g i v e n t ) y t h e s y s t e m . ' I h f a s s e r t i o n

I J I > im- r f i ]1 ic-s i t i a t . V ( , , ; / + i / . - , $'ivc> r i s e t o a S e i f e r i > ' ' - f i l i r < i t i o n o f ^ f . V ) .

I1 ignro 1

Proof. (Figure 1) Look at one of the annuli contained in ,V, say .1 C ;V. Then it

has a flat structure generated by XH, A>, and there is a uniquely determined (up to

the sign) pair or numbers (%*,) such that uaXH + &oA> generates a periodic flow of

period one on A. Take a point x0 £ A, and denote by 5|o the orbit of a0XH + boXF

11

going through x0. Since dH A dF 0 at x0, for every x near to xa in M4, there is

a uniquely determined circle going through x, which is flat in the Lagrangian torus

(or annulus) containing I, and which lies in a tubular neighborhood of S^. On

the torus (or annulus) containing x, there is a unique pair of numbers (a(x),b(x))

(again up to the sign), such that the flow generated by a(x)Xn + b(x)Xp is periodic

of period 1, and its orbits are homotopic to S*. From the construction it is clear that

a{x),b(x) depend smoothly on x. Furthermore, they are invariant on the Liouville

tori and annuli, thus da A dH A dF = db A dH A dF = 0. Still we have to proved

that a and 6 are well-defined: if we define (a, 6) on some torus or annulus by going

from different annuli of N, the results will be the same. Let xa and x'o lie in different

annuli of N, x be a point near to x0 and x' near to i'o, x and x' belong to the same

torus (ar annulus). Then since S\Q and S]., are disjoint, Si and S\, are also disjoint.

It means that they are homotopic in the torus (or annulus) containing them. Thus

they give rise to the same pair (a,i) (up to the sign, but in fact the signs can be

made to be the same).

The flow of a(x)X[j + b(x)XF is of period exactly 1 outside singular hyperbolic

orbits. It remains to show that this flow has period at least 1 /2 at singular hyperbolic

orbits. But it is also clear since topologically, because F is a Bott function, for every

hyperbolic orbit either this orbit is a boundary component for some local annulus

("orientabSe case", Figure 2a), or twice of it is so ("non-orientable case", Figure 2b)

(cf. [F2]). •

Remark. The first assertion of the proposition means that a and 6 are functionally

dependent on H and F. If we were working in the category of analytic functions, it

would yield, for example, that a can be written as an analytic function of variables

H and F. In the smooth category a weaker statement is true: a can be written as

a smooth function of // and F plus a small correction term of any given order.

In case there are orbits of period 1/2, we can make a double covering of our

WM(JV) by setting

UM(N)=UM(N)xs, S\

Here the action of S1 = [z £ C : \z\ = 1} on UM{N) 'S defined as above, and the

12

Figure 2

action of .<i'1 on $' is given by t o : := t'2z.

We r<in lift everything, from the symplectic forns to the singularities, from U,\t(N)

to UM[S), On/V.i/(;Y) the flow will have period I everywhere. Than bv taking dou-

ble cuvennji and making things Z2-equivariant, whenever necessary, we ran always

restricted ourselves lo ihc rase when the above flow is uf period 1 everywhere in

U\i(S), hi other words, all the hyperbolic singular orbits can be assumed to be

"orienlable".

Note that the assertions of Proposition 3.1 arc also valid for tlie < ase of elliptic

singularities, rf. [El. LI.'J. In the hyperbolic case, the periodic How we have con-

st ructecl is unique ii[) to the change of time to inverse. In the elliptir case, however,

it is uot unique.

Lei's look now' for a canonical form near ;V for the symplectic form and the

int.egra.ble system. Here we will assume that the flow defined in Proposition 3.1 is

of period 1 everywhere.

Denote a(i)X}i + b{i}Xy by (,. First note that, by Arnokl-Liouville theorem, f

is a Hamiltonian vector field (since outside the singular points it is constructed ex-

actly as in \rnold-Liouville theorem). Denote by xx the Hamiltodiau function (with

•'i( -V) = 0) associated with £ (since outside the singular points it. is constructed ex-

13

actly as in Arnold-Liouville theorem). Then of course dxt A dH A dF = 0. Moreover,

locally near N, x\ considered as a Hamiltonian aho admits an additional Bott firstintegral, For example, one can check that the function F' — b^H — aaF is a Bott

function on the level sets of x\.

Now note that, since UQ(N) has non-empty boundary, the 5'fibration given by £

on it is trivial: UQ(N) is equivalent to the direct product 5 ' x P2 for some orientable

2-dimenaional surface P2 with non-empty boundary. Thus there is a smooth map

Xl '• M\f{N) -* S* = Rj2 such that dx2{Z) = i. The function Xi gives rise to a

symplectic vector field XX2. Denote by tj the following normalization of XX1:

Then we have that U>(TJ, f) = 1. Since every thing is ^-invariant, r; is also ^-invariant.

In other words, rj and f are commutative.

Let P2 be the surface UM(N) n { i , = 0} D {x2 = 0} and ( i 3 , n ) : P2 -> B?

be an immersion. Then extend ( I 3 , I 4 ) to be a map U\t{N) -* R2 by making it

invariant under TJ,^. This way we have a composition of (an appropriately chosen)

UM(N) into product

UM(N) = £>' x 51 xP1

associated with the system of coordinates (xi,xi{mod[),x3,ifi). Standard calcula-

tions give us the following

Proposition 3.2 In the above coordinates (xi,x^,x3,x4) the Hamiltonian Junction

H does not depend on xi, and the symplectic UJ has the following canonical form:

(+) w = dx, A dx-i + 7r*(u/i),

where UJ^ is some area form on P2, and n is a projection. D

Conversely, if w has such a canonical form, then in general any Hamiltonian

function which is invariant under £ = d/dx2 will be integrable: {xlf H) = 0, since

xx, = d/dx2.

As a corollary of the above proposition, we obtain the following fact: The topo-

logical structure of a Lagrangian foliation by Liouville tori restricted on the level set

14

t

H = hB near /V is determined by the topological behavior of F on the 2-dimensional

surface P2 above. So the (singular) S'-foliation by level sets of F on P2, which has

only one singular leaf, characterizes the topological foliation by Lagrangian tori on

the local isoenergy submanifold {H = h0}. In the terminology of Fomenko [F,BMF],

a codimension 1 singularity (of the Lagrangian foliation restricted on {H = ho}) is

called a "letter-atom". Thus we recover the fact in [F] that a letter-atom is charac-

terized by a S'-foliation on a surface P*, if all the critical curves are "orientable",

and by that ^'-foliation plus an invariant orientable involution on P2, in case there

are "non-integrable" hyperbolic singular periodic orbits (or, equivalently, by the S1

foliation on the quotient of P2 by that involution, plus some marked points). In other

words, letter-atoms can be viewed as surface singularities. Some letter-atoms with

small complexity have special names in [BMF]. For example, the letter A means

an elliptic singularity. P2 in this case is a disk foliated by concentric circles.The

letter B means a simplest orientable hyperbolic singularity: P2 in this case is a

thickened 8-figure. Figure 3 illustrates some more letter-atoms, which often appear

in well-known integrable systems.

In practice it happens most often that the singularities are stable. The idea of

stability is perhaps due to Fomenko and Bolsinov. Let's make more precise what

is it, since we will need this notion later. Suppose we have a normal form (*•) for

some hyperbolic singularity. Fix i j = 0 and look at the restriction of the additional

integral F on every P2^ = P2 x {xi}. Thus on every P£ we have a singular foliation

by circles which are level sets of F. If this foliation is non-degenerate (i.e. F is a

Bott function with saddle singularities on P^) and remains homeomorphically the

same when xx changes (i.e. there is a foliation-preserving homeomorphism from P^

to P , which is near to the obvious "projection" which changes only x\ to x[), then

the singularity is called stable. For elliptic singularities one may make the same

definition, but elliptic singularities are automatically stable. We have:

Proposition 3.3 Suppose we have a non-degenerate singularity with the canonical

form as in Proposition 3.2. Then the following properties are equivalent:

(1) This singularity is stable.

(2) The Bott integral F can be chosen so that F = 0 at all the singular periodic orbits

15

at any energy level H = h (in some small neighborhood UM(N). In particular, at

every point in any singular orbit we have dF = 0.

(S) The letter-atom given by this singularity on euery energy level hi = h does not

depend on h (in some small neighborhood UM{N)).

Proof. (1) => (2). Recall that dx, =adH + bdF, therefore dx, A dF = adHhdF,

Thus if O is a singular point of F restricted to the energy level set // = //(O), then

O will be also a singular of F restricted to the level set Xi = Xj(O). From (1) we

have that F takes the same value, denoted g(xi), at all the singular points on every

level set of i i . Set F' = F — g{x\). Then on every level X\ — canst F' is a Bott

function and F' = 0 at all the singular points. Thus F' = 0 also at all the singular

points on the energy levels H = const. We want to prove that F' is a Bott function

on a energy level set H = h for any constant h near to ha. We will show it at the

singular points of F on {H = h] first. Fix such a point, say O. O is a nondegenerate

singular point of F on {*i = ii(O), thus it lies in a smooth curve of singular points

of F restricted to various levels sets of x\. This smooth curve is a non-degenerate

singular submanifold of F', i.e. F' is non-degenerate in the transversal direction.

Remember that at 0 we have dti -fi 0, dxi ^ 0 , and they are colinear. It follows that

F' restricted to {H = h} is non-degenerate at 0- In particular, dF l\dbi ^ 0 at any

point near to O and different from O on {H — h}. On the other hand, recall that

dn = a dH +bdF, therefore dH/\dF' = dHAdF-dHAdg = (l-bdg/dx^dHArfF.

Thus (1 — bdg/dxy) ^ 0 at O, and since it is invariant on Lagrangian tori, it is

different from zero in the maximal closed leaf containing O. It follows that F' is

Bott on {H = h], since F is Bott.

The proof of the other directions is similar.a

From Proposition 3.2 one can see that stable singularities are "generic": they

can be achieved on every given energy level by a small C™ perturbation. In Section

8 and 9 of this work, we will assume codimension 1 singularities to be stable, for

simplicity.

Observe that while it is natural to assume that the system has only non-degenerate

codimension 1 singularities on some fixed isoenergy submanifold, it is not natural

16

to assume that our system has only non-degenerate codimension 1 singularities ev-

erywhere in an 1-parameter family of isoenergy submaiifolds, The idea is clear

from the singularity theory: it is not a general position that all the functions in

a 1-parameter family of functions are Morse functions. Thus we must accept also

degenerate singularities somewhere. It leads us to the following

Definition 3.4 A degenerate closed one-dimensional orbit 7 is called simply degen-

erate if it satisfies the following two conditions:

1. The symplectic form can be written in the above canonical form (*)

2. In the above symplectic coordinates (may be after a double covering of a neighbor-

hood off), F can be chosen in the class of all smooth functions invariant on Liouville

tori so that it is a Bolt function on all the plants {xj = 0, H — const ^ /io}, and on

{H = ho] either is Bott or has the image point of the projection of 7 on {x? — 0}

as a unique degenerate singular point (there may be other singular points which are

non-degenerate).

The second condition needs some explanation: If A' is still a Bott function when

restricted to the level set {// = 0}, then the image point of the projection of 1 on

{x? = 0} belongs to a smooth curve of singular points, and F is quadratic in the

transversal direction to this curve.

Let us mention that the conditions in our definition are weaker than that ones of

Lerman and Umanskii [LU]. In other words, simply degenerate closed one-dimensional

singular orbits in the sense of [LU] are also simply degenerate in our sense. Moreover

there are examples, in rigid body motion equations say, where there are orbits that

are simply degenerate in our sense, but not in the sense of [LU]. In the language

of moment maps, roughly speaking, simply-degenerate orbits correspond to some

of the "beaks" in bifurcation diagrams. Only simplest beaks correspond to simply-

degenerate orbits in the sense of [LU]). Take the Kovalevskii top for example (see e.g

[Os]). There are several simply degenerate orbits, but only few of them are simply

degenerate in the sense of [LU].

Propositions 3.1 and 3.2 have a direct generalization in higher dimensions. We

will state it here, without much proof.

17

Suppose we have a Poisson /{"-action on a symplectic manifold (MJn,uj), which

is generated by n commuting functions F\...,F", We will assume that all the

common level sets of F1,..., F" are compact, thus theit connected components are

n-dimensional tori outside singularities. Suppose at some point x0 £ M we have

that dFi A . . . A dFn-i ^ 0 but <ff\ A . . . A dFn-, A dFn = 0. Let N be the connected

component of the level set {Ft = Fj(xo), • • •, Fn = Fn(xo)} which contains xQ. As

before, N is called a codimension 1 singularity. It is called non-degenerate if the

function Fn is a Bott function on UQ{N) = {F, = F ((x0) , . . . , Fn_i = F^,(x0)} n

UM{N), where UM{N) is a small tubular neighborhood of N in M {see e.g [El, F2]).

The topological classification of torus foliation on Uq(N) in this case is also carried

out in [F2] and it looks much like the case of 2 degrees of freedom. In particular,

nondegenerate singular submanifolds of Fn in UQ(N) must have dimension at least

n — \ because it is invariant under the locally free Poisson /^""'-action generated

by Fi,..., Fn-i- Here we will assume that al of them have dimension n — 1 (since

dimension n can be ignored by the same reason as before). It is known that in this

case all the singular submanifolds are n\-dimensional tori and UQ( N) is, may be after

a double covering, diffeomorphic to a product of T71"1 with some orientable surface

P1 with boundary, and Fn is a lift of some function from P2. Again, depending on the

behavior of Fn on P2, one says the singularity is hyperbolic or elliptic respectively.

Proposition 3.5 Suppose N is a nondegenerate codimension I singularity m the

above notations. Then there are smooth functions a,j,(i = 1,.. ,,n — 1, j — l,...,n)

from a tubular neighborhoodUM(N) = Dn~l x Uq{N) C M2" to R such that:

a)daij A dH A dF = 0 Vi, j

b)For every i = l , . . . ,n — 1, the flow generated by f, = ^(a,,F,+...+a,nrn| is Ffamil-

tonian and periodic of period I in UM{N). These flows commute and give rise to

a hamiltonian Tn~l group action on (a saturated neighborhood) UM(N). This ac-

tion is free everywhere, except at singular hyperbolic orbits of the original Poisson

Rn-action (which are diffeomorphic to Tn~]), where it can have Z? as its stabilizer.

The proof of this proposition is completely analogous to that of Proposition 3.1,

so we will omit it here. 0

18

_ jj j j i , ȣ. . j^ .

As before, we can assume that the above torus action (which is unique in the

hyperbolic case and not so unique in the elliptic case) is free, by taking a double

covering if necessary. The existence of a Xn~i group action, though iocally, is some-

thing very good, since one can apply the theory of torus group action here (see e.g.

[Aul]). Such torus actions are also known to be useful in constructing generalized

action-angle coordinates (see e.g. [Ne]).

P ropos i t i on 3.6 There exist "canonical coordinates" (xi,yi,... ,xntyn) in (an ap-

propriately chosen) UM{N), where yt,... , yn_i are defined modulo 1 (action coordi-

nates), and (Jn.j/n) defines an immersion from F 2 to R2, suck that:

a) these coordinates give a natural diffeomorphism fromUf,j(N) to D"~' xTn~l x P2

b) The symplectic form ui has the form

it as Ui — ui2 = d(Yl" ' a,dx,+0), where 0 is some 1-form on Bn+I (which is not zero

on P^ I i i_ l in general). If we can eliminate 0, i.e. write w\ — OJ2 — d(Y^~' a,dx,),

then we will havew = (J^^"1 dxiAdzi—J^1 dxiAdai+u}2 = E " ~ ' ^ A J ( i i -

where u^ is some are form on P 2 , and 7r* means the lifting.

c) Generating functions F \ . , . , Fn and coordinates I , , . , . , xn_, commute pairwise.

The last statement yields that the foliation by Liouvilie tori is completely deter-

mined by common level sets of Fn and ny coordinates i , , . . . ,x n _i .

Proof. Let x; be the Hamiltonian functions of Hamiltonian vector fiel s £; in

Proposition 3.5, Xi(N) — 0. Remark that the Tn~l- group action in Proposition 3.5

gives rise to a trivial T^'-foliation. Denote its base by B™+1. Let L be a section

of this foliation, and define functions z,{i = 1 , . . . ,n - 1) by putting them equal to

zero on L and setting dz.-ff,-) = {x,-, Zi} = 1. Set Ui = u> — J^"~' ^ i ^ ^-z>- Then

one checks that L^wi = i^.wi = 0. It means that u , is a lift of some closed 2-form

from B" + 1 to i/,vr(iV), which we will also denote by LJ[. Since w is non-degenerate,

it follows that iii\ is non-degenerate on every 2-surface (with boundary) P*t Tn_l =

UM{N) n {xi,... ,xn-\fixed}. Using Moser's argument [Mo], one can construct a

diffeomorphism <j> : B"+1 ->• Dn~l x P2, under which UJ, restricted on P^ I n_ l does

not depend on the choice of Xi,..., xn-t. In other words, there is an are form ui2 on

P 2 such that u>! —w2 vanishes on every P^ I__ i. Since d(ui — w2) = 0, we can write

and the theorem will be proved by putting jf; = z; — a,-. Let us show now how to

eliminate 0. 0 restricted on every /JJll...,Ill_1 is a closed 1-form, hence it represents

a cohomology element \p](xi xn_i) E Hl{P2). If [/?](- , £ n - i ) = 0 then

/3 = dF — bidxi — ... — 6n_idin_i for some functions F, 6 i , . . . , 6n, and we have

Wi — u)j = d(X)(a> "~ b,Xi)dxi). In general, we can achieve [/?](JI, . . . , i n _ i ) = 0

by induction on the number of generators of H^(P^) as follows. Let 7 be a simple

curve in P"1 which represents a non-zero cycle. Set 6(JTI, . . . ,x n _i) = < [/?],7 >

(ari, . . . , x n _i ) . Immerse P2 in an anuulus so that only simple curves homotopic

to 7 go to non-zero cycle there, other simple curves go to vanishing cycles. By

this immersion we have a (non single-valued) system of coordinates (u,u) on P2,

u — modi. Wj has the form u)3 = adu A dv for some positive function a. Change u2

for the following 2-form on UM(N); U; = uij + db A du. It is clear that ujj and u>?

restricted on every P^ I n i are the same. Moreover, ui'2 is closed and of rank 2.

Thus the distribution by its (n-l)-dimensional tangent zero-subspaces is integrable,

and it gives rise again to a diffeomorphism $ : Bn+1 —> D""1 K: / J i . Replacing UJ2 by

wj, we havew, - w j = rf(X)""1 a,rfa:, + (3'), with ^ = 0-bdu, whence < [/?'], 7 > = 0.

•

With the above propositions, one can have the same notion of stability of codi-

mension 1 singularities in higher dimensions. They can be classified (topologically)

again by Fomenko's letter-atoms.

4 Nondegenerate systems

In this section we recall some known topological results about singular points of a

Poisson action, and give a notion of non-degenerate systems. After that we discuss

19 20

how affine structures of orbit spaces behave near singularities.

As before, assume that we are given a Poisson /faction generated by XH,XF

on a symplectic manifold (M4,w). Let 0 € M" be a fixed point of the this action:

dH(O) = dF(O) = 0. Linearizing the action at the point 0, we obtain an linear

Poisson R1 action on symplectic space (R4 = {(pi,Pz,qi,qi)},u>0). Take 2 generators

of this action. They are given by two quadratic Hamiltonians, Ho and Fo say, which

are the quadratic parts of H and F:

H0(x) =< Ax,x > /2,F0(x) =< Bx,x > /2,

where A, B are symmetric operators from R* to (R4)' = R4. The corresponding

Hamiltonian vector fields are linear:

Here / denotes the (operator with) matrix

0010

0001

- 1000

0- 100

The matrices IA and IB belong to the algebra sp(2). Since Ho and Fo commute

these elements are also commuting in sp(2). Note that the Cartan algebra of sp(2)

has rank 2. If IA and IB are non-collinear (i.e. they span a Cartan subalgebra),

then the Poisson action is called non-degenerate at point O. Otherwise it is called

degenerate. We have the following definition.

Definition 4.1 An integrable Hamiltonian system on a compact symplectic j-manifold

(may be with boundary) is called non-degenerate if it has only non-degenerate fixed

points, and all i/it singular one-dimensional orbits are either non-degenerate or

closed simply degenerate.

To our knowledge, all the well-known examples of algebraically integrable sys-

tems are no i-degenerate by this definition. Later we will consider with non-degenerate

systems.

21

From now on we will consider only non-degenerate systems.

Let's look at non-degenerate fixed points. They are also called non-degenerate

singular points (of codimension 2). Up to a symplectic linear change of coordinates

in R4 and of generators of the Poisson action, there are four cases, with the following

linearization (cf. [LU]):

1. center-center, Ho = p\ + q], Fo = p\ + i\

2. center-saddle, Ho = p\ + q\, Fo = P2<jt

3. saddle-saddle, Ho = Pi?j, Fo = p^qj

4. focus-focus, Ho = piqi + pjft, Fo = Pi<ii - Pa?i

Note that "center" is a synonym for elliptic singularity, and saddle is a synonym

for hyperbolic. All the 4 cases above appear in classical mechanics (see e.g. [Fl, Os]

for a lot of examples).

The local theory of these non-degenerate singularities were studied by Lerman

and Umanskii [LU1]. In some cases, better results were obtained later by Eliasson

[El] and others (see e.g. [CDM,DM]) by using fine division techniques. Note that if

one complexifies the system (provided that it is analytic), then the first three types

become the same.

The semi-local study of the first three types was partially carried out in [LU2,

Bo], Here by "semi-local" we mean the study in a neighborhood of a singular leaf in

the Lagrangian foliation containing the singularity. For the center-center case, it is

the same as local. The center-saddle case is more. Under some "stability" conditions

(cf. [Bo]), topologically it is a product of letter-atom A with another (orientable)

letter-atom, but the symplectic form may not be a sum of two area forms.

The saddle-saddle case is much more complicated: they are not a product of

two letter-atoms apriori [LU,Bo]. Also there are many things common for general

dynamical systems: homoclinic and heteroclinic orbits, horseshoe after small per-

turbations, etc. We will call a (semi-local) saddle-saddle singularity (topologically)

simple if the Lagrangian foliation near that singularity is diffeomorphic to a product

of two letter-atoms (i.e. surface singularities). If moreover, the symplectic form

22

ill this product also splits, i.e. is the sum of 2 area forms, then wn say that this

singularity is of product type. The same definition applies for center-saddle points

and also for singularities of higher codimensions.

When there is only one (saddle-saddle) singular point over the saddle-saddle

point in the bifurcation diagram, to[. ologically there are' four different cases (cf.

[LI-2, Bo]). When there are two saddle-saddle points (with heteroclinic orbits form

one point to another one) the number of possible cases is much larger [Bo].

Figure .'i

Here we have a "principle" that many, if not all, semi-local saddle-saddle cases

can be obtained AS a quotient by a finite group action of simple singularities, i.e.

products of two letter-atoms without stars (in the terminology of Foiticnko). Here

we illustrate this idea on all the 4 subcases of the one-saddlp-poini case. One of

these I cases is just a (topologically) simple singularity, that is it can be realized

as a product of two "letter-atoms B". Note that the letter-atoms B,DUC2 (see

Figure 3) admit a foliation-preserving involution (rotate them by ISO0). Denote this

involution by o. Then a acts as Z2 group also on the products H x P , and B x C2.

Note that these actions are free. Divide that two products be the action, we obtain

2 of three other subcases of one-saddle-point case.

Note that Ci admits another involution, which also permutes the two saddle

points, and preserves everything, (it is easy to see if one embeds Ci in S2). Denote

it by b. Note also that a and b commute. On C? x Ci define the following action of

a(x,y) = (ax,ay),b(x, y) = {bx, a(by)).

Then the quotient by Zi ® Zj will coincide with the fourth subcase.

These quotients are perhaps not accident, and many singularities met in me-

chanics are quotients of something more simple. For example, I guess that all the

singularities of geodesic flows on multi-dimensionat ellipsoids (and quadrics?) are

quotients of simple singularities (see e.g. [Z3| for the list of singularities for the

ellipsoid problem, [Au2] for the algebraic treatment).

As another example to illustrate the above idea, we are going to show that if

the saddle-saddle singularities are not simple, then the DelzanTs integral property

may be violated. Recall that, for Delzant polytopes (cf. [De]), every vertex gives a

basis for the integral lattice Z ! of R2 (see more about it, in the following sections).

In our cases, it may give a basis not for Z2 but onty for a 2-dimensional sublattice

of Z2, and this sublattice can be arbitrary, just 2-dimensional. Let n.k be a pair of

relatively simple natural numbers (n > 2), In the letter-atom Vn there is a natural

Zn action (Figure 3). Denote this action by <f>'{i £ Zn). Define the following action

of Zn on Vn xVn:

Then one can check easily that the "local polytope" which contains an orbit 71 x 72

has the following property: its vertex corresponds to the sublattice of Z2 generated

by 2 elements (1,0) and (k,n). In particular, the mark in Fomenko-Zieschang in-

variant of the corresponding edge of the system around the saddle-saddle point will

be k/n. (See §7, [FZi,BMF] for the definition of this invariant). In the previous

example there are already some cases where the mark is 1/2, and the sublattice is

of index 2.

Every non-degenerate integrable system has a natural orbit space associated with

it: it is the space of (maximal closed) leaves of the Lagrangian foliation by Liouville

24

tori. From the definition above, one sees that orbit space of a non-degenerate in-

tegrable system is a locally finite stratified 2-dimensional manifold (CW-complex).

The momentum map often used in physics is actually a map from the orbit space

to an Euclidean space. Inside the 2-domains of this orbit space, there is an integral

affine structure, according to Arnold-Liouville Theorem. We are now interested in

how does this affine structure behave near the singularities.

For the codimension 1 singularities, the answer is rather simple, because of the

canonicai form given by Proposition 3.2. Let's look at hyperbolic singularities first.

In the canonical coordinates given by Proposition 3.2, xs can be viewed as a local

function from the orbit space to the real numbers. Since A'x, = djdx2 has a periodic

flow, the ievel sets {xt = co-ist} are geodesic in the affine structure. In other words,

we have

Proposition 4.2 Let C2 be an orbit space of a non-degerttrate integrabte system.

Then every saddle boundary component of a 2-domain in C"2 is transversal to some

geodesic direction. We witl call it a preferred direction. •

Note that i j is denned on all the 2-domains of the orbit space which are contigu-

ous to our hyperbolic singularity. The 1-stratum, which represents the hyperbolic

codimension 1 singularity, in general is not a geodesic in (the extension of) any

2-domain contiguous to it, as one can easily construct examples by changing F, H

in the canonical form. The relative behavior of this 1-stratum with respect to the

affine structure on every contiguous 2-domain is arbitrary, aside from the fact that

it must be transversal to the preferred direction. There is only one condition how

they must behave in total, which can be stated in terms of cohomology, or simply

in terms of linearity: if we fix on every contiguous 2-domain a second coordinate

y,, which increase? as a point goes out go from the hyperbolic 1-stratum and which

complements Xj to the integral affine system of coordinates, then for points t in the

hyperbolic stratum the sum Ej[;(() depends linearly on ijfi). 1 guess that the arbi-

trary behavior of saddle curves in affine structured orbit spaces is an obstruction for

general smooth integrable Hamiltonian systems to admit any additional structure

(e.g. to be bi-hamiltonian in some sense). See e.g. the papers by Turiel and Brouzet

25

in [La] for some related results.

The case where there is a simply degenerate closed orbit is similar to the above

saddle case, because of the assumption about good systems of coordinates. In par-

ticular, there still exists a transversal geodesic direction.

The case of elliptic singularities is very simple, because elliptic singularity means

that essentially we have a torus action [El]. In particular, near the elliptic singu-

larity the orbit space is affinely equivalent to a local half-plane, and the elliptic

1-dimensional stratum in this orbit space is geodesic (boundary line of the half-

plane). So in elliptic case we have a uniquely determined direction, which is parallel

to the elliptic boundary. Functions n in Proposition 3.2 still gives us a transversal

geodesic direction, which is not unique.

Let us try now to understand the affine structure near non-degenerated fixed

points.

The case center-center is again very simple. Locally, one has a correct (integrally

affine) angle. In the center-saddle case, one can see from the above discussion that,

near a center-saddle point, the preferred direction associated to the hyperbolic strata

going to this center-saddle point is parallel to the geodesic direction given by the

elliptic stratum. We do not know much about the affine structure in the saddle-

saddle case. What we know is only that outside the saddle-saddle points, there

are transversal directions, which don't have to make up a right angle in every 2-

domain. There is one case when we have the right angle, namely the case when our

saddle-saddle singularity has product type.

In the rest of this section we pay our attention to the focus-focus case. Focus-

focus type singularities appear quite often in physics and mechanics, for example,

in Lagrangian, Klebsch tops, four-dimensional rigid body motion (see e.g. [El, Fl,

Os]). It is somewhat surprising that people usually ignored this case (see [El]), but

in fact the geometrical picture of this case is rather simple and beautiful, The affine

structure near the focus-focus points also have a very simple behavior.

The following simple but important fact was observed by Holsinov: the singular

fiber corresponding to a singularity of focus-focus type consists of a chain of n

26

I* !t •> ,« -*!

l.agrangian spheres (n being the number of singular points), each of which intersects

transvivsally with two other When n — 1 it is just one sphere ivith one point of

sell-interspctiou. The reason is that this singular fiber cannot contain singular points

of other i.ypes. This fact follows from the fact that near a focus-focus singular point

in M* KV have two Lagrangiau invariant siibmanifolds intersecting transversally at

this point, an the Poisson R2 action has. an orbit type of ainiulu.s on that Lagrangian

subrnaiiifnlds \\A .

We have the following

P r o p o s i t i o n 4 . 3 Lit n hi llx number of l.nynntgian *iiktrt» for a i/u-in focus-focus

.-angularity in aboci. I'hcn in a nnghborhood of tint* singular poml m tin orbit space,

Ihi njjnif fitructuir- run bt nbtaiurd from the standard ftnt struct tin tti IP = {x,y}

mar tin origin O by cutting nut Ike anglt I {(0. 1), ( - n , I)} ami ijluimj tht eAijts of

tlif n s l t i M j f t h a - 1)1/ t h t l u l i r / n i l l i n m r I r a n s f t i n i i t i h w ) ( . r . i / i ~ i ( . , - + u i / . i / ) . ( F i i / u r t

5ft

Figure 4

Before [ir<nmg l.|»" above Proposition let us now const met KU algebraic model for

'his singularity

Near the origin O in I he l<n id standard symplectic space H1.^: — i/^j Ai/</| +dpi A

tlifi we have two generating functions for a Poisson /fraction with the singularity of

the typi* focus-focus:

/ i = Pi?i + Pl<i2

wmwmw ssi:

Set zi = pi — J 2,2j = <Ji + i<?2 • Here we define a complex structure:

Then f\ and f2 are the real and imaginary part of function Ziz2. In particular, the

level sets of (full) ar e the (complex) level set of z^z-i in C2.

Remark. More generally, for any (germ of) analytic function g : C2 —> C, Reg

and Img are in involution with respect to the symplectic form ui defined above.

Thus we have a local model of a integrable system. The. interesting case is when g

has a singularity at 0. Suppose the Milnor number of that (isolated) singularity is

fi. For the function z^z2 one has p — 1. If ft > 3 then the level surfaces near the

singuiarity have at least 2 handles, so it can not be a local model for an integrable

system with general torus orbit. It is an interesting question to consider the cases

when n — 2 and /i = 3.

Consider first the simplest case, when the preimage of a focus-focus point in the

image of a moment map is a single point. We can construct the model as follows:

Take the conformal map 4> : (z,, z2) -> (z^1, Z\z\). We will see in the next lemma

that this map (when it is well-defined) is aymplectic. Consider a small neighborhood

D X CP1 of the sphere 0 x CP1 {z2 lies in CPl). Gluing the points near (0,0) to

the points near (oo,0) by the map 0 we obtain a complex space M which has a

natural symplectic form ui (because 0 preserves the symplectic form). Furthermore,

it is clear that the analytic map Z\z2 : M —• C is well-defined on M and it is the

moment map for a desired H? Poisson action.

Lemma 4.4 $ is a sympleciic mapping.

Proof. Note that w — Redzx A dz?, and * is a complexifica-tion of a real area-

preserving map. •

Remark. From the "pure" symplectic point of view, the symplectic structure

of M is uniquely determined (near the singular sphere fj=f2 — 0) by Moser's

28

principle (see e.g. [Mo, Br]), and there is a symplectic submersion from a small

neighborhood of the zero section in T'S* to Af, which sends the zero-section to that

singular sphere. But it seems quite hard to define a Poisson action in this way.

The case when the preimage of focus-focus point of the bifurcation diagram

consists of (finitely) many points is similar. Take n samples Ci(t = l,...,n) of

D x CPl and define n local maps 4>i • Ik -> tf,+i((7n+i = f/i), which in local

coordinates have the same form $ as above. Glue U, together by these maps. Again

we obtain a symplectic manifold, and the function z^2 provides us a moment map

for a Poisson R1 action on that manifold, now with u focus-focus points over a

(single-point) singularity in a (local) bifurcation diagram. More geometrically, what

we do is just take an n-covering of the one-point case. Under this covering every

Lagrangian torus also goes to an n-covering of itself.

Topologically, this construction is unique. From the construction it is easy to

see the topological type of a "isoenergy" 3-manifold around the singularity (more

precisely, the manifold {j^j l = ( > 0}. It is a locally flat fibration with torus fiber

over a circle. We will compute the hoionomy mapping of this fiber bundle. Our

compulations in fact do not depend on the above specific model, but only >n the

properties of ail the focus-focus points.

Note that, from the results of [LU, El] and the above observation of Bolsinov, one

sees that semi-locally there is a Hamiltonian vector field A' = A'3, which is tangent

to the singular leaf, and which flow is stationary at focus-focus points and periodic

of period 1 elsewhere. In our model g — / j . Fix a small circle {|*i*j| = e} =

{/(2 + / | = e2} in the bifurcation diagram. Every point in this circle corresponds to

one Lagrangian torus. Fix one point {/, = t, f2 = 0}. On the torus corresponding

to this point fix a basis of generators of the fundamental group, so that the second

generatoi is induced from the symplectic vector field Xj3, an the orientation on T2

given by ihese two generators coincides with that one given by A'/,,X/S. Oenote

these generators by f,5 resp. When the point {/i = t,/g = 0} moves along the

circle in the positive direction (anti-clockwise), 7 and 6 also move homotopicaily,

and in the end come back to some new cycles ~ineui,&new °n the old torus.

29

Proposition 4.5 With the above notations we have:

1) The curves g — const are straight lines in the affine structured orbit space.

2)

Proof. 1) follows from the fact that the flow of X3 is periodic with constant

period. We prove 2) for n = 1. Then one can use n-covering to see it for any n.

Since the assertion is topological, then it is enough to prove it in our model. On

the submanifold {|zi22| = e} in M4 denote by 9 the cycle where z\ = const and arg

Zi decreases, A the circle where z2 = const and arg J[ increases. Then S — 6 + X.

Recall that when we go around by 7, the coordinate system changes by the rule:

(jJ""',^J"u) = (zj1 ,z\,z\). Fix a point A in one representative of 7. Moving A

along A by some angle e means increasing arg Z\ by this angle. Making things go

homotopically around 7, what we get is that argz"'1" increases by e, zj51i remains

const. By the above rule, in the old coordinates arg z\ increases by 2e, and arg 22

decreases by e. That yields that after going around 7, A becomes to move on A + S

with the same angle.

It follows that 7T,,™ = 1 + 6. •

Proof of Proposition 4.3. It follows directly from Proposition 4.5 D

In higher dimensions, it is natural that the singularities of higher codimensions

do occur. Here codimension at some singular point means the minimum corank of

(dFi,... ,dFn) at this point among all then-tuples of first integrals of our Lagrangian

foiiation.

A singular point is called non-degenerate if some condition of transversal non-

degeneracy is satisfied (see e.g [El, DM, LU]). This condition means the following.

Suppose we have a singular point of codimension it. Then the orbit of the Poisson

action through it has dimension n — k. Let us linearize the action at this point. Then

the transversal direction to the orbit is a (n + fc)-dimensional coisotropic subspace.

After symplectic reduction of this space we obtain a Poisson action on a symplectic

2^-space, which is generated by quadratic Hamiltonians, i.e. symmetric matrices,

say A\ An. Then nondegeneracy means that I At, - • •, I Av span a Cartan subal-

30

gebra in sp(k), where / is the k-dimensional analogy of the matrix / in the beginning

of this section. Under this non-degeneracy condition, it is clear the (2fc)-symplectic

space admits a decomposition into direct sum of 2-dimensional and 4-dimensional

subspaces which are invariant with under IA\,..., IAk. Two dimensional subspaces

correspond to codimension 1 singularities, and 4-dimensiona] subspaces correspond

to codimension 2 focus-focus singularities. In other words, the linear part of a

non-degenerate singularity can be decomposed into product of codimension 1 and

focus-focus (codimension 2) singularities (cf. [Ds]). There is an important fact: the

action near to every non-degenerate singular point is (homcomorphically) conjugate

to its linear part (cf. [El, LU, DM]).

Thus locally, things are more or less clear. But semi-locally not. Already in case

of codimension 2, we have seen that the topological structure of the foliation near

the singular leaf can be very complicated. Let us pick up the simplest singularities

by the following

Definition 4.6 A singularity (i.e. a tubular neighborhood of a closed singular leaf)

of codimension k (1 < k < n) is called of product type (or geometrically simple^, if

it is symplectomorphic via a foliation-preserving morphism to some product of the

type Tn~k x Dn~k x Pt y. P? ... X P,, where Pi is either a sympledic surface with

a non- degenerate singular foliation on it, or a symplectic 4-dimensional tubular

neighborhood of a focus-focus singularity.

We don't give a rigorous definition of non-degenerate integrable systems in higher

dimensions here. Anyway, one can have a "rough definition", by requiring all codi-

mension 1 singularities to be non-degenerate or simply-degenerate, all codimension

2 singularities to kc non-degenerate or simply-degenerate, and so on.

31

5 Orbit space and integrable surgery

This section is devoted to a general question: which topological 2-complex (and

n-complex) can be an orbit space for some non-degenerate integrable system? By

the way there will arise the notion of "integrable surgery", which seems to be useful

in symplectic geometry.

Here by a 2-complex we mean a finite 2-dimensional CW-complex with a given

decomposition into a finite number of 2-domains and a the singular part, consisting

of a finite number of 1-dimensional strata and singular points. Given such a 2-

complex, we want it to be an orbit space of some non-degenerate integrable system

with 2 degrees of freedom, so that 1-dimensional strata correspond to non-degenerate

codimension 1 singularities, and singular points correspond to non-degenerate sin-

gular points or simply-degenerate orbits. More generally, suppose we are given an

n-cornplex. We may ask whether it is an orbit space for some non-degenerate inte-

grable Hamiltonian system with n degrees of freedom.

The most important thing in the orbit space, from our point of view, is the

affine structure. Using geodesies in the affine structure allows Condevaux, Dazord

and Molino to give a simple proof of the Atiyah, Guillemin, Sternberg and Kirwan's

convexity theorems. There are obstructions for manifolds to admit integral affine

structures (see e.g. [Suj). It means that, in particular, many n-complexcs can not

be orbit spaces for integrable systems.

For dimension 2, J. Milnor proved that the only closed 2-manifolds admitting an

affine structure are torus and Klein bottle (see e.g. [Su]). In our case 2-domains are

usually not closed: their closer contains smaller dimensional strata. Still we have:

Proposition 5.1 Lei C2 be an orbit space of a non-degenerate integrable Hamillo-

nian system on a compact (may be ivitk boundary) symplectic manifold M*. Fix a

point x € C2 and denote by C2 the connected 2-domain which contains x. Assume

that Ci is relatively compact in C1, and points in the boundary of Ci correspond

to singular leaves of tht Lagrangian foliation (if M* is closed then this condition is

satisfied automatically). Then C? is homeomorphic to either an annulus, a Mobiiis

32

band, a Kitin bottle, a torus, or a disk.

Remark. Only the case when C2 is a disk in Proposition 5.1 really often appears

in mechanics and physics. The other cases are somewhat "artificial".

Proof. We will prove for the case Ci is orientable. Then the non-orientable case

can be treated by taking a double covering. For the moment assume that theit: are

no focus-focus points. Suppose that Ci contains exactly k handles (k > 0). Take a

smooth loop 7 to be a collar of all of these handles.

Provide 7 with a natural orientation (such that the boundary E = dCi is "out-

side" of 7, if it is not empty). Then homotopically we have 7 — ajbjnf 1ij" l...a^16j'.

Fix a positive basis (eo,/i0) at x0 = 7(0) = T(1). Transfer it by the curve 7 with

respect to affine structure of C2 , we get a family of bases (?(t), h(t)). Transfer it by

7 with respect to some metric where 7 is a geodesic, we gel. another family of bases

(et, ht). Consider the angle function Ay(t) = L(c(t),e,) with respect to orthonormal

basis (ei,/st). Of course, the angle function A(i) can be chosen to be a continuous

function.

Lemma 5.2 If k > 0 thtn .4,(1) - A,(0) < 0.

We will prove Lemma 5.2 only in the case of one handle. The case of many

handles is completely analogous.

At every point y(t) we have: (e,, A,) = By,,(e{t),k(t)) for some operator R,.,. In

particular, (e<1,h0) = (c,, Aj) = fl,(e(l), h{l)), where ft, is a "holonomy operator" :

By homotopy relations, we have H, = BaBbB^Bb-,. It is easy to check that

Now Lemma 5.2 follows from the fact that the absolute difference between two

A^,ho(l)- Aa,caM(0) and ^ 1 ) W l , ( 0 ) - ^ . . , ^ 1 | W l ) ( l ) is less than

33

JT, and also the absolute difference between two terms J4»,C.(I),AO(I)(1 ) — -4t,«o(i)./..(i)(°)

a n d >s ' e s s than TT.D

Note that the value in Lemma 5.2 does not depend on the choice of a metric.

Note also that the initial direction of ea can be chosen arbitrarily.

Now assume that the curve 7 is going around some topological component of the

boundary (but is not necessary a collar). Then we have the following:

Lemma 5.3 The initial direction of e0 can be chosen so that A~,(l) — .4,(0) > 0.

Proof of Lemma 5.3. There are two cases: the topological component of bound-

ary is smooth closed, or it has singular points.

Case 1. Smooth closed curve. If this curve corresponds to elliptic singularities,

then an action function (i.e. a local function on an orbit space whirl] Hamiltonian

flow is periodic with period 1), which is 0 on this boundary, is well-defined near it.

Then just take 7 to be a near to zero level set of this action function. If this curve is

of hyperbolic point, then as was shown before, there is a well-defined (every where

near this curve) geodesic direction, which is transversal to this curve. Then take e0

to be parallel to this transversal direction.

Case 2. There are singular points (of saddle-saddle, center-saddle or center-center

types). For simplicity, we wilt assume that all singular points are of saddle-saddle

type. The other cases can be treated similarly. Then near every saddle-saddle point

there are two preferred directions, tvhich are transversal to 2 edges of this vertex.

Note that every transversal direction is a preferred direction for two vertices. Let

e0 be one of the two preferred directions near one vertex. Then simple comparisons

show that A,( l ) -A,(0)>O.D

Now Proposition 5.1 follows from the fact that when A goes from one boundary

component to another one and back, the angle increases for more than it.

Let us consider the focus-focus points. From Proposition 4.3 one can check that

when 7 goes to a focus-focus point, makes a turn to the right and come back, in

total the angle remains the same or increases. Thus focus-focus points can affect

only positively on our computations. O

34

What Proposition 5.1 really asserts is an obstruction to the existence of a suitable

affine structure. We can ask the inverse question. Suppose on a given n-complex Cn

there is given also a suitable affine structure. Is it an orbit space for some integrable

system? Here by "suitable" we mean that near every singularity (strata of smaller

dimension) this orbit space is an orbit space for some local integrable system with

that singularity. In other words, we have some pieces of integrable systems, and

we ask if it is possible to glue them together to get an integrable system on some

symplectic (closed, for example) 2n-dimensional manifold. Thus we come to the

notion of integrable surgery

Surg< y is one of the most useful technics in topology and geometry. Contact

and symplectic geometry is not the exception (see e.g. [Gi, Go, We2]). Here we want

to use integrable surgery to construct symplectic manifolds with integrabio systems

on them. By an integrable surgery we mean a gluing of two sympiectic manifolds

with Lagrangian foliation (with non-degenerate singularities in the sense of §4) on

them along some common boundary so that the symplectic form and the Lagrangian

foliation can be naturally extended from the two manifolds to their union.

The simplest form of integrable surgery is via orbit space. It, is the case when the

Lagrangian foliation is tangent to the boundary along which we are going to glue

2 symploctic manifolds. Later on, we will consider only integrable surgeries which

can be projected to the surgery of orbit spaces.

Thus we are in the following situation: how to do surgery on the level of orbit

spaces? In other words, can we glue two orbit spaces together and lift it to obtain

an integrable surgery? First of all, we need to extend the integral affine structure

from two (pieces of) orbit spaces to get a affine structure on their union (so it is a

problem of "affine surgery"). Obviously, there are obstructions to do affine surgery.

But we will not consider that problem here, and we will always assume that we can

do it. In other words, we are interested only in how to lift an affine surgery to an

integrable surgery.

It turns out that Lpgrangian sections play a very important role in integrable

surgery, since locally for action-angle coordinates (p,, <?;) there are natural Lagrangian

35

sections to the foliation by Liouville tori, which are given by {q, = const}. We will

consider first the case where the orbit space is a (open) manifold without singularity.

Our situation now is similar to that of real polarizations (see e.g. [Wo, Chap. IV])

Let E : (M2 T \LJ) -^v Bn be a I^-bundle over a base manifold B", fibers of

which are Lagrangian tori. Hence on B" there is a natural integral affine structure,

uniquely determined by this bundle. We will assume that B has finite type (e.g. it

can be viewed as an interior part of some compact manifold wil.i smooth boundary).

Proposition 5.4 a) if-Ki(B) = 0 then E is a principal Tn-bundle. Moreover, if

there, exists a Lagrangian section, then E is a trivial bundle and any two Lagrangian

sections are homotopic via Lagrangian sections.

b) If there is a decomposition of B into a union of finite number of contractible

submanifolds BU...,B, such that B, 0 B, n Bk = 0 Vi ^ j ^ k and TT, (/?,- nfl ,) =

0 Vi, j (in particular, Xi(B) = 0), then there always exists a Lagrangian section. The

bundle E in this case is unique, in the sense that if E' ; (A/',u/) —> B is another

bundle by Lagrangian tori, then there is a bundle-preserving symplectomorphism

from M to M', which is identity on the base.

Remark. If ^i[B) / 0, then bundle E is not a principal bundle in general. It

is easy to construct examples with 7ri(B) = 0,7r2(Z?) ^ 0 (e.g. B is a thickened

2-sphere in R?) and the bundle E by Lagrangian tori of a symplectic manifold over

it, which is non-trivial (and therefore there is no Lagrangian section). The above

propositions tell how Lagrangian fiber bundles are far from general 7"1, and principal

T"1, bundles.

Proof, a) On Bn locally there are n action functions pt,...,pn. They can also

be viewed as multi-valued functions on B. If i*i{B) is trivial, then these functions

are actually single-valued. Hence the Hamiltonian flows generated by them give rise

to a T"-action on M. This action makes our bundle to be a principal bundle. If

there is a Lagrangian section, say C, then we can define the functions (<j,), requiring

them to be zero on C and {p,,q,} = 1. It is straightforward to check that q, are

well-defined modulo 1, and they are actually the angle functions, i.e. the symptectic

form has the form w = £ dpi A <fq,. q< give rise to a trivialization of the bundle E. If

36

C\ and £j are two Lagrangian sections, and suppose q; and q[ are angle coordinates

corresponding to them, then we have 0 = 5^ dp< Adq>i — £ dp;AdflJ = ^ dp< Ad(qt-q\).

Note that <j, — ijj are locally well-defined functions depending only on p3, and since

X\(B) = 0, they are globally well-defined. Denoting them by a,- = a^pi,... ,pn),

we have rf(^a;rfp;) = 0, i.e. £<2,c(;»; is a closed 1-form, hence an exact 1-form:

a, = df/dp; for some function / from B to the real numbers. Lagrangian sections

C and C can be joined by Lagrangian sections Ct, which are given by the equations

qi = (a, in the coordinates (pi,<ft).

b) We can use the assertion a) to prove it by induction by the number k in the

decomposition of B. Note that, although we have a Lagrangian section, in general

we don't have globally defined (mod 1) angle coordinates. Anyway, if E' is another

bundle over S, then we can construct a morphism from E to E', by mapping a

Lagrangian section of £ to a Lagrangian section of E' first, and then extend this

map in a. natural and unique way. •

By the way, we also have the following

Proposition 5.5 Over every integral affint structured manifold B there exists a

unique bundle E by Lagrangian tori from some symplcctic manifold M, which admits

a Lagrangian section.

Proof. The uniqueness is clear. The existence can be seen by constructing angle

coordinates (<;;) locally, and let the transformation maps of the fibers be the linear

maps (ql) = A~l(q{) with respect to the afflne transformation maps in the base B

of the type (p.) = AT(p,) + (c,), where AT denotes the transpose. O.

Lagrangian sections are useful because of the following

Abstract Principle. If two orbit spaces can be glued together to be an integral

o^ne structured spaces, and if in every of the symplectic manifolds corresponding

to them there is a given Lagrangian section over a neighborhood of their common

boundary (for gluing), then there is a unique canonical way to do integrable surgery

from the. .. data.

37

Rigorously, at the moment we can have a proof of the above principle only in

case when orbit spaces are manifods without singularities, and the proof is then

trivial, as one glues 2 Lagrangian sections together and extend that gluing in the

most natural way. For applications, however, sometimes we will need to use the

above principle also in cases when there are singularities.

In case the base B has only elliptic singularities, the situation remains almost

the same. Locally, there always exists a lagrangian section, which is a lagrangian

manifold with the same type of boundary and corners as B has. More precisely,

locally we have B = Dn~k x Ck, where Ck is the right angle {pn-t+l > 0, . . . ,pn >

0} C Rk, and JD""* = {-£ < pi < t,...,-i< pi < i}. Coordinates (qt) are still

defined modulu 1, with gn_jt+, undefined over the boundary part {pn-i,+l = 0} of

B. (M,bJ = Yldpi A dqi) is a trivial bundle over B outside the boundary of B. Its

general fiber collapses to a torus of smaller dimension when a base point approaches

the boundary (cf. [El]}. Nevertherless, Lagrangian sections {qi — const) are well-

defined, and they are Lagrangian submanifolds with the same type of boundary

as that of B. Furthermore, locally 2 Lagrangian foliations can be again joined by

a family of Lagrangian foliations. This fact can be obtained easily from general

results of Eliasson [El]. It means that we can deal with base spaces with elliptic

singularities in the same way as with base spaces with no singularities (one can view

the "bundles" over them as a gereralized kind of principal bundles, or "Satakian

bundles").

Example 5.6 Delzanl polytopes

Recall that, in view of convexity theorems and structure of elliptic singularities, for

Hamiltonian n-torus action on a closed symplectic 2n-manifold, all the singularities

are elliptic, and the affine structured orbit space is equal to a polytope in fl" with

the following properties; the set of (integral vectors lying on) edges from any of

its vertices form a basis of the integral lattice Z" in Rn. Such polytopes are called

Delzant polytopes (by suggestion of V. Guillemin). The main theorem of Delzant in

[De] is:

Proposition (Delzant). Every Detzant polytope in R" corresponds to a unique

38

closed symplectic 2n-manifotd with a Hamittonian n-torvs action on it, In other

words, for every Dttzant polytope there is a symplectic closed manifold with a Hamil-

tonian action, the image of the moment map of which is integral affinely equivalent to

the given polytope, and for any two such manifolds there is a toriis action preserving

symptectomorphism between them.

Delzant [De] proved the above theorem by using some Weinstein-Marsden sym-

plectic reduction. Here, using symplectic surgery, one can have another proof as

follows.

Proof. Locally there is a Lagrangian section (and an integrable system). Since the

base space (i.e. Delzant polytope) is obviously contractible, then actually we have

a global Lagrangian section. Now the proof follows immediately from Proposition

5.5. •

Example 5.7 Blowing up

Suppose we have an Poisson /?™-action on a symplectic 2n-dimensional manifoid

M2" with a elliptic fixed point. Then the orbit space near to this point equals to a.

right angle {i, > 0} in the Euclidean space R* = {x;}. To do symplectic bio ving-

up, just cut this angle out by a hyperplane Sii = const > 0 in Rn to get a new

orbit space. This orbit space corresponds to a blown-up symptectic manifold {see

e.g. [Aul]).

For hyperbolic singularities, the situation is different: except for the simplest

codimension 1 singularities, in general we can not have a (singular) connected La-

grangian section homeomorphic to the base. The only thing we can do is to consider

Lagrangian sections outside hyperbolic singularities, and require them to be some-

what compatibie with that singularities. Near every singularities of codimension

1, we have a 7"1"1 action and a corresponding, which gives rise to a trivial prin-

cipal T'^'-bundle. Thus instead of Lagrangian sections, it is natural to lo k for

coisotropic (n+1 (-dimensional sections to the above bundle. Then Lagrangian sec-

tion outside the singularities will be contained in that coisotropic submanifolds when

they approach the singularities.

Definition 5.8 We will call a 2-complex with a given integral affine structure on

39

it suitable, if near every singular point it can be realized as an orbit space for some

integrable non-degenerate system such that these systems coincide topologically on

their overlapping orbit space domains. (In other words, every point in an 1-stratum

corresponds to a unique tetter-atom).

Proposition 5.9 Every suitable integral ayfRne 2-complex is an orbit space for some

non-degenerate integrable system-

Proof. First we prove that a tubular neighborhood of the singular part of

our space corresponds to some integrable system. It can be achieved by gluing

coisotropic sections explained above, and by Propositions 3.2, 4.2. Now we have

to extend the system from that tubular neighborhood into 2-domains to get the

system over the whole space. Since the connected boundary components of this

tubular neighborhood is obviously simple closed curves, over them there exist la-

grangian sections (Proposition 5.4). hence the theorem follows from Proposition 5.5

and Abstract Principle. •

Remark. We don't claim the uniqueness in the above proposition. In general,

there may be many different integrable systems (in different symplectic manifolds)

with the same orbit space, because over every circle there are many non-homotopic

Lagrangian sections. If the orbit space satisfies some strong conditions, then one

can still show the uniqueness. But we will skip that problem here. If somehow, the

Lagrangian section near the singularities is given, then we have a unique natural

system (there may be other "non-natural" systems, which don't accept the given

Lagrangian section). For example, if all the singularities are of product type, then

we are likely to have a canonical Lagrangian section.

Example 5.10 Torus handle

Let us look at the following example of integrable surgery, which will be used in the

next section. Set V — T* x D2 = {(xuyilx2,yi)},xi,yl -modi with the symplectic

form u; = dxi A dy\ + dx? \ dy2 The "central" symplectic torus is xj — 1/2 = 0-

The Lagrangian foliation is given by xi = const, / ( i j , 512) = const where f is a

Morse function. (Clearly, all that foliations are Lagrangi?n and non-degenerate).

40

H\ rli,MINIUM / inside a -rimll di.-c (MI IIIHI / t h o s a m e olll .sicle i t ) w e ca.il

;i-.-nnir iti.ii ni'iii- [mini .,-_, - i/_. = (I f IIH.S the form / = r{ 4- i/^. Then this foliation

li.i- ,1 luiinlc^i-inTiiU' -in^iiliiriiy ni !••> — ij2 = 0 For the operation below we will

ii-siimr / is sum of square- ol'roordiiiiii™. U:e will rail the- const ruded above piece

*>l s\ i I I | >l<-cf ii Mifiin lo l i i cin '"] ciU-^i A I i lc I or i is handle .

]i i \\i • -y 11 i|>J«-'i i. 1- ii i,i r M 11 >3i t> tv i ih La^ rH i i ^M j i fo l i a t ion I I H W H I I rmbc-cidc-i, si an -

l i . m l i i i i i ' u r . i i i h - i i - n i - f i ' i i u l i r [ u l i r T e i l i c L i i j in i i i f t ta i i lol i i - t l ion on t h a t hand le is an

hi I in i'< i lu l i;i i toi l i J Iri'it o i i f * 'i r i i ] ! u i i l i lii's*1 f o i us l i f iml l f s and g l i i f Oir t w o n iH i i -

i f o l r l - i . \ i r i i t u t i i u i i t i i i i i i n l j | - y / ' • ' . l i i < y i i ! [ ) k T t i i ' l i T i i i s . I l i < - c o n s t r u c t i o n i s c l e a r :

i , i k i ' . i i - i i i n n ' i l e d - n i I I i , | 1 \ , , I O I - J I I - u i 1 f a 1 1 > u i l 11 c i n i r r l i i i a l r s ( ' • ; , . i y j , ) i i n d f . r ^ , \ j i l ) .

\ I n - « O T M I I 1 ' I c I - I N n h . i ^ s- m i l ' u i l i n . i l , i \VA l o r n i i n i j n r o d I m m t h e t i v n s m f n i c s . N o w

r i i ^ i 1 i p i - > i l i i < l i l i . i i i ^ j r l i i t 1 . 1 v v i i h / - , , i - i i ' l i [ i - ! i p r o i l u r i L I | M - r u j j l c r t i i m i i i u f < i ] ( l s .

O l i 1 - . i n n - \ \ . I l i i 1 l . ) i . i > r i i n u i f i i i I ' j l i . i i i d i i - . o i i i ' I I n ' I w o H I M n i f o l i Is f i r e " l u e d t o ! > ! - t l i c r i n fl

I M ' ' . ' . : ' . ' i J '1 I V ' i 1 i r f , , 1 ' j 1 j l 1 I-J 3,1 [ I fn I [ | , 1 ! I f j l | 1 1 \ \ I I I I 1 S I I I I ]

I N I ! n - . i l i c M 1 i \ i I | I - I: I I : d l v IVi1 I I J I M ' l i v n o r l i i l - | i , i i c - u l t h e H | j r o f rtll ; i m i l l I n s

. i !i< J . I I-H i .1 I J L! I ri I I L^i-I I I -* 'H I H >li I H i J^1 T' I i j l l l - 1 l lOMl ! - A\\ o l IVJOUS | | ] a t I CI .

l-'i«urc

5.11 l-nru.--l-'ririi.- imml.-

I mu K "i is ii 11 c\iiii 11• 11- uf ,i I uu-i|i>iiian] 1 II an orbil sjjdcc. ]|M Ins e\'Hni|jle, wi1 Jiavo a

I1 i i i i m l c « 1111 I l o r ' i i s - l o c i i s p o i i i l s i i i i l . S o t h e r e i s a s i m p l e ' " i n a r l i i n a r y ' " t o p r o d u c e

.! I.if <>f I I M I I S f u r i i s >i n ^ u l i i ! il i ' 1 - . l ! "<-X|)[fli t i s " w h y f o c u s - f o c u s s i l i g u l t f n t i e s a . |>p*?ar

- i i . i t t . ' i i i n i i r n c i i r c .

II

It seems to us that integrable surgery is easier to deal with than general symplec-

tic surgery. Further, it is not much more restricted than symplectic surgery, since

locally we can always have a Lagrangian foliation.

6 Gompf's construction

Gompf [Go] has constructed a closed symplectic 4-manifold with any given funda-

mental group, using symplectic surgery. It is a very interesting result, since it shows

that symplectic geometry is quite far from Kahler geometry. The first example of

a non-Kahler closed symplectic manifold was constructed by Thurston [Th]. Actu-

ally, Thurston constructed a symplectic 4-manifold with fundamental group Z-\ and

used the fact that the first cohomology group of a Kahler manifold must have even

dimension. More subtle topologica! invariants allow Gompf [Go] to show that there

are simply-connected closed symplectic 4-manifolds which are not Kahler.

In this section, using a construction of Gompf [Go] and integrable surgery, we

are going to prove the foilowing:

Proposition 6.1 Any finitdy-gentrated group is a fundamental group of some com-

pact symplectic 4-manifold with a non-degenerate integrable system on it.

The are two reasons why I am interested in such a proposition: the first one is

that to test how useful is integrable surgery. The second one is to try to answer

the following question, posed sometime ago by Fomenko: is it true that every closed

sympiectic 4-manifold admit an integrable Hamiltonian system non-degenerate in

some sense? In general, this question seems to be very difficult. Due to the work

of McDurT, now one try can check it for symplectic rational and ruled manifolds

(analogues of rational and ruled complex algebraic surfaces [McDl]).

Let G be an arbitrary given finitely-generated group. Then as in [Go] there is a

42

closed surface P,a, finite number of cycles (7i,..,7i) in P and a closed one-form p

on P, such that

2. p restricted to every curve y, is a volume form (i.e. positive nondegenerate).

Furthermore we can assume that:

3. p is an integral form (i.e. [p] £ II'(P,Z)).

4. p is a Morse one-form.

Then e have a map pi : P —• S1 denned as follows:

pAx) = / P.

where x0 is a fixed point in P.

Take A/i — P x 7'2, where TJ is a torus with coordinates 8, X(mod]). Construct

the following symplectic form on A o:

= u,

Here Wi is some area form on P.

Jp A <i0

Define the following Lagrangian foliation on Mo : A = {pi = const, A = const}

Then A is a non-degenerate system (in the sense of §2).

Following Gompf, We define some special symplectic tori in Afo.

Take T; = A; X Q,(T = 1,..., it), where ft C f is defined by A = const,. On T, the

restriction of u> is p A d9. Thus T, are symplectic. Note that T< are not embedded,

but only immersed. So we have to perturb them to obtain embedded symplectic

tori. To do this, let r be a coordinate of A,, and substitute f, by T, = IJr A,(r) x QT,

where QT = {A = g{r)\. Here g(r) is a function of r, which is close to const, in

C°°, such that g{l) ^ ir(u) if A,(i) = A,(u). Then T, will be symplectic embedded

disjoint subtori of Mo. Take also To = x0 X T2 C Mo. Here we assume that x0 does

not belong to the curves A;.

The next step in the Gompf's construction is to glue, along T,, k+1 samples of

C P 3 # 9 C P 2 (equipped with a natural symplectic form) to Mo, to kill ^i{T2) and

43

the geufi-atoi'.s A, ol ~ i ( / ' ) , thus to obtain a rompnrl. sy inp l rd i r •1-iiifinifold wi th

the given fuii(Umcnta] fimup G. Hen- we " i l l maks- tl i is process by an iiilegrablo

surgery. Also we will rcplncr CP2#!)f7J J bv another manifold which is easier for us

t it deal wil h .

Our gual now is to li i id a neighborhood of tori 7', win h arc intcgral)le torus

Im ndlps.

1'ix a nimibci- i = l , . . . k . and io\ siinpticity below we wil l drop the index i. Define

a small i intnei MVI cy l im ln in I' aniund the curve A,. On this cylinder the (induced)

a r e a l o v m c a t ; l>e w r i t t e n . is -.•] = ( / / / A dr. ^ l i t ' i e ,r j ; ; c o o i < l i t t a t e m i 1 l i e c y l i n d e r ,

. r ( A , t -.-. I I .

W ; i r / \vr i , u i ] r u T I t f t 1 ic >\ M I ple< [ i< U,i 111 _* 11^ l o l l u w ^-i

_ •••• ,l,i, ' .l.i •• ,10 : ,l.\ r ,l,i, A ,11)

- . / I , . , - \ i .'• ,1\.<- + I)) + ,l,\ A ,l.r

Sine i ' r/i , j , I i - e l iK - i - 1,i I he r u i i - ; . i l i i i \v'I Hi 11 w e - e t = I] J. 1 l i e e i p n i l i o n A — i/(p, J is

I ' f l l l i v H l c i i t t n t i l l - e [ | l | , | t u rn A — / , l ; l | -f \ I l u l s u l l i e - . M i o i ' l l l I'l l l n i i l l l l /) w l l i l ' l l is a l s o

D e l i n i \i,'W i L I U I i11n i l i L ' - :

/ ' I -• I- • - \

I - 1 •-•• . 1

A i — A + In p , )II, - I) - ., ( I - 1,'t/i, M

O n e s ( . ( ' S i l i i M t h e s e a r e r e a | ] \ ' i ~ v ^ t e t n u l , . H i r d i i i a t e s f o r a n e i j f h b o r l i o o d o f T.

I n l l i e s . ' n e w . n , , i [ | i i i a t e - . / i - ; i , i \ e n I n i , = A | - ( I . , n n l i h e L a g r a n g i a n f o l i a t i o n

i s j ^ i v i ' i i L \ \ , - < i i n - i . , j | - i - n i i s i i n n l i e i i i K , n - . t he s \ i n p l e i 1 ic f o r m n o w l i a s t h e

form:_• •-• I / / J , •'• < / [ « , i - r l / i / i f i ) t i / | A | - h \ i n I ) / , d . r

= ,/ ;<, ,'-.,10, -i- , / A , / . , / , • ,

T l m . s w e i . b i a m i n l e . m a l i l i - i u r n - , l i a n d l , ^ n e a r t h e l o i i I',. \ - { k. T h e c a s e o f

t o r us I,, is s! n i i In K a n d sj n i f i l e r .

l.el in now iudii ate what we wil l use in place of ( 7 ) J#<)( " / J J Look at the Figure

(). Lei s explain whiii dues llns picture means; ii i . an miaj;e of a orbit spa.ee in R2

^ i ri - i » n ii 1 1 1 0 1 1 n ! i l i i i r i \ i . I h i ! D i M i [ H T >>{ s i n - | > i > i n 1 ^ i n I ! i r - | i i c h i M ^ f f o r c < - i < h l i < l o m < ) i n

H i ; ;-.r I >i i i ] I 1 : , i i i o ] M l i i ^ r ; i i n s - > J u n v n i n I l i e j j i r l n i v . A K o I h e 1 V f x ' S o f S l l i g n l a n l [ ^ s a r c

- r - i i x " i " 1 1 • ) i n r . I ' if L Li I f h r - i d u i i i ^ i v I h r | ) i c i i i f c s h o w 1 * i v h < T f ; i u - h o w m a n y c o u t i g u o n s

H I L J I I M M I - i n ! I n - L i r l i i t s | i , n r \i> I h o c | < - l f ( M - ; i t o i n s . 5 j i o m N i n ! \ \ r " r r n U 1 ] 1 " n w a u f o c u s -

i ^ ' U v | i , t i i i K . | u l j i i l y w e J N I V I ' .! • 1 - [ 1 l i > i n - ^ - f u M i s | J O I J ] 1 S i n | [ i c u r l i i l s j > ; u ' c \ T h o

"• c e• r e i 3 -.11 j i . i 1 1 o l i I i f p i r T i n c i n i i t a t o i ! i , i i o f < ' \ a i i i | ) l c . '.. '. \ \ ^ i m p l f f h ' g t M K M a < i c s ,

] i '•• | ; ( • ! - • I HI .11 H r i r h I I ' l l ! 111 ]< • L j | - l }\\ - " k i l l i ^ n h c M I n 1 r I t i p i - o i [ i n c r i n o i l - s i n f i l i a l L n ^ r a i ^ i H n

In i ln- " l u w c r p u r l " of l- ' ijiwv (i. ivr li . ivc s-omr [ j o i n l s on t h e s i n g u l a r l eve l of

' iiiuiiKvii i n , i | i . i'ur whi i ' l i I lie p n ' i n i t i ^ i ' i< a n o n - s i n g u l a r Lagr i i i ig i i i i i t o n i s in t h e

I")

f o l i a t i o n . T h e l o w e r p a r t o f t h e o r b i t s p a c e in t h i s p i c t u r e r e p r e s e n t s a n i n t e g r a b l e

torus handle.

It is straightforward to check that the orbit space itself is corvtractible, and

the corresponding symplectic manifold, even after cutting out a torus handle, is

simply-connected. Thus we can take it as our model. Gluing k+1 samples of this

model to Mo via the constructed above integrable torus handles, we obtain a desired

symplectic manifold with a non-degenerate singular Lagrangian torus foliation on

it. D

Proposition 6.1 is also valid in higher dimension. The simplest way to show this

is, as usual, taking multiple products with symplectic spheres S2.

The following question remains unsolved: How to produce an integrable system

on a compact symplectic manifold with non connected boundary of contact type

(see [McD2] for examples of such symplectic manifolds in dimension 4).

7 Fomenko's theory

In this section and the next 2 sections we study the systems restricted on a compact

non-singular isoenergy 3-manifold. In the end of each section we will talk also

about higher-dimensional case. We will assume that our systems only have non-

singular (codimension 1) singularities (see §3). Fomenko and his collaborators have

classified all such systems up to topological equivalence. Here two systems are

called topologically equivalent if their Lagrangian foliations by Liouville tori are

homeomorphic.

The main observation of Fomenko is that tubular neighborly. >(i of every singular-

ity is a Seifert fibered space (see §3). As a consequence, the isoenergy 3-manifold it-

self is a graph-manifold (also called a YVaidhausen manifold [Wa]). Graph-manifolds,

and Seifert fibered spaces in particular, are a very important class of manifolds in

46

3-dimensional topology. There is a very beautiful topological classification of them

by Waldhausen, Orlik, Vogt and Zieschang [Wa, OVZ]. See e.g. [JS] for their role

in general 3-manifolds.

According to [FZi, BMF], for every system (Q3,v) we can associate to it an

invariant called its topological type, or Fomenko-Zieschang invariant, and denoted

by [{v) (actually it was denoted by I"(v) in that papers, but we will drop the

asterisk here), by the following way: !{v) is a marked graph, every vertex of it is

a letter-atom which denotes a codimension 1 singularity (see §3), every edge of it

represents an 1-dimensional family of iagrangian tori going from one singularity to

another singularity, and there are some additional numbers, called "marks", which

represent the "topological torsions": they depend on how one glues Seifert fibered

spaces (which correspond to singularities) together via homeomorphisms of a torus.

It is natural that two integrable systems are topologically equivalent if and only if

they have the same topological type [BMF]. The proof is somewhat straightforward,

once one knows the classification theory of graph-manifolds. Conversely, Fomenko

and Brailov (see [F2]) proved that, given any abstract topological type (i.e. a graph

with marks and letter-atoms) there will be an integrable system with that topological

type. This result is very clear from the point of view of integrable surgery: one can

construct the required system by construct it near the singularities first (e.g. use the

canonical form in Proposition 3.2 and require F to be ^[-invariant for simplicity),

and then glue them to get the required system. But on the level of orbit space, it is

a trivial task.

One application of Fomenko's theory is that sometimes it gives a very simple

way to look at integrable systems. For example, a geodesic flow on a usual ellipsoid

can be represented by the marked graph in Figure 8 (cf. e.g. [Fl, LZ]).

This graph gives much information about the dynamics of the above geodesic

flow. For example, looking at the picture, we can say that there are exactly 4 stable

periodic critical geodesic curves (2 if one does not count the orientation), and 2

unstable ones.

Another application of Fomenko's theory is that it detects some obstructions to

47

-HI m. m m-w

•AFigure 8:

integrability. For example, if a 3-manifold is not a graph-manifold (and it often

happens theoretically) then no system on it can be nondegenerate integrable.

Another example is the following obstruction to integrability on S3. The case of

other graph 3-manifolds is similar, due to the theory of Waldhausen.

A knot 7 in S3 is called a generalized torus knot if it can be obtained from the

trivial knot by iterations of the following two operations:

I. Torus winding-. 7 is an embedded knot in dU(-j,), where U{~n) is a tubular

neighborhood of some generalized torus knot 7!.

II. Connected sum: 7 = 7,+ 72, where 7] and 72 are generalized torus knots.

Theorem 7.1 (cf. [FZu]) If v is a (non-degenerate) integrabk system on S3, then

every periodic orbit of v is a generalized torus knot.

Proof. Using the theory of reductions of graph-manifolds, we have proved it

for critical one-dimensional orbits [FZu]. Now, using the same reduction of Wald-

hausen's graph-manifolds as in [FZu], one verifies that every cycle in any Lagrangian

torus will be a torus winding of some critical periodic knot, after some step of re-

duction. D

Note that the class of all generalized torus knots is quite small in general knot

theory. Therefore Theorem 6.2 is a kind of obstruction to integrability.

In Fomenko's theory there arises the following natural cobordism problem. The

48

question is: when two systems oti two 3-manifolds can be connected by a system on

some compact symplectic 4 IIIHnifolcl ? Equivalently, which system on a 3-manifold

bounds (it1 is a boundary of a system in some compact syinplectic 4-manifold) ?

Here again by ;\ "system on 3-manifold Qy we mean a (germ of) integrable

system on some (Q x I,UJ), I = (—t, +t) with the Haniiltonian which is a projection

// ; Q x / -> / . We will denote tiie system v on Q by (Q, v) or (<5,u.\ v) as in §2. A

system (Qi,uilt rt) is called ( u p ) cobardanl to a system ((^,1^2, v?) if there exits a

compact, symplectic 1-manifohl {A/, u,') with the boundary dM = (Q\.ui\) — (Q2.W2)

and KII integrable Haiiiiltonian vector field A'/j on M, which coincides with us and v?

on Q\ Hud Q2. respectively. In this case [Q^v-i) is called dowti-cobordtint to {Qi,i'i).

If Qi is einp'y then {Q\,Vi) is called cobordnnt to 0.

If we arc given not the s\ .items , but only their topological types (in particular

we are not given the synipk'ctic germs), then we have the following result.

T h e o r e m 7.2 (cf. [HZJ) (lircn 11111/ ~3 topologiait types It on Q, and l2 on Q2

thin t.vi.*t two .-i/.tlfinx (Qi. i'i), (Q-i, r2) with lopalogical types / j and /2 nxpectively

.^iifli ihnt r t ts rohordtud li> i' v\a mi nittgrnblf syxlf in au fi i-nnurrtcd s

Uiniiii'k. In the case of l l ami l ton iau torus action similar results were obta ined

in [An. C h . l V ] .

I'o prove '['liporem 7.2, we will use technics of "cu t t ing" and "gluing" of edges of

i i i t rgrable sys tems , which are special cases of integrable surgery. Let us s t a r t with

some definitions. As explained in \$, we can assume that all t he s ingular i t ies of our

s\ s t ems are elliptic or orieuUible hype rboh i .

.Suppose we are given the topological type / . which conta ins an edge \V : .4 | —J-

A2 with I lie m a t r i x D on it. Let us recall the meaning of this m a t r i x . Near the ver tex

.1i(i.e s ingular i ty which it corresponds to) fix a basis ( f | , ^ i ) on JTi(7^)(.r £ W) so

that f 1 is a vanishing cycle if . 1 , is elliptic, and h\ is homoto[)ii t o hyperbol ic periodic

orbi ts in . 1 | i f , l | is hyperbolic . Analogously near .42 fix a basis (ti.hi) • Then we

have:

Note that D is defined only up to some equivalence class, and it is an integral

matrix with det — —1 (the negative sign comes from orientation convention).

Now assume that we have

where Dt and D2 are integral matrices with detDi = detD2 = - 1 . Change the edge

W for the following two edges: Wt : Ai-^-A , and Wj : A^-A3 (where, as before,

letter A means an elliptic singularity). Then we obtain a new topological type / ] .

Definition 7.3 The operation I -> /, is called a formal cutting of the systems. The

inverse operation is called a formal gluing.

Topological type of Hopf systems is called a formal Hopf system.

Lemma 7.4 / / / consists of only one edge and two elliptic vertices, then by a finite

number of formal cuttings it can be reduced to a disjoint sum of several samples of

the formal Hopf system.

The proof of Lemma 7.4 is straightforward. It consists just of algebraic compu-

tations of 2 x 2 matrices,

A topological type / is called specialil its graph is a tree, with only one hyperbolic

vertex, and all the topological torsions (mark ;) are trivial.

Lemma 7.5 For every special topological type I there is a up-cobordant to 0 system

v with l{v) = /.

Proof. In this case Q has the form Sl x Efl with some closed orientable surface

Ea. Then Q is a boundary of the symplectic manifold D3 x £3 with some product-

wise symplectic form. There is a function F : S s -f R such that the lift of F on

50

Q gives rise to a foliation equivalent to that given by /. Denote by p2 the square

radius function on the disk D2, lifted on D2 x Es. Then a Hamiltonian system for

almost any Hamiltonian H of the type H = H(F,p1) will give us a desired system.

•

Lemma 7 6 Every topological type I can be reduced by formal cuttings to a disjoint

sum of special types and samples of the formal Hopf system.

The proof of Lemma 7.6 follows from Lemma 7.4.O

Lemma 7 " If l2 is obtained from ^ by formal cutting or gluing, then for every

system vt with /(i>i) = I\ there is a up-cobordant and a doum-cobordanl to V\ systems

v2 and v'2 such that I(v2) = l{v'2) = h-

The proof of the above lemma follows from general facts about integrable surgery

(§5). Note that here we make integrable surgery only near elliptic singularities, and

for elliptic singularities the affine structure is very simple and determines completely

the foliation (54,5). •

Proof of Theorem 7.2. It follows from Lemmas 7.5, 7.6 and 7.7. •

If on Q the system and the symplectic germ is given, but not only the topological

type, then the problem is much harder. In section 9 we will give some partial res alts

about this case.

Observe that, if Vi is up-cobordant to v2, then there is no reason why vs must

be down-cobordant to v2- (And in fact generally it is not the case, see below). That

means that our cobordance relation is not an equivalence relation. If we want to

extend it to get an equivalence relation, then by the arguments similar to that of

the proof of Theorem 1 one can perhaps show that all the integrable systems will be

equivalent. In other words, if we want to make some cobordism group of integrable

systems, then this group will be trivial. (See [F3] for various definitions of cobordism

groups).

For higher dimensional (non-degenerate and stable) integrable systems, Fomenko

[F3] denned a topological invariant, which is called "net", by the following way. For

51

every integrable system, its net is actually a graph, with vertices of 2 kinds: vertices

of the first kind represent domains in the orbit space, vertices of the second kind

represent codimension 1 singularities. Every edge goes from a vertex of the first

kind to a vertex of the second kind, and represents the fact that a codimension 1

singularity is contiguous to a domain in the orbit space. Obviously, it is a natural

generalization of his invariant of systems with 2 degrees of freedom From the defi-

nition, it is clear that codimension 2 and higher singularities are missing there. But

the (semi-local) topological study of codimension 2 and higher singularities is very

complicated in general, unless one imposes some strong conditions to simplify the

problem (see §4).

Since 3 = 2 + 1 = 2 x 2 — 1, one can expect two different generalizations of the

cobordism problem in higher dimensions: cobordism of (2n-l)-dimonsional isoenergy

hypersurfaces, and cobordism of (n + l)-dimensional integrable "slices". Again due

to singularities of higher codimension, it seems too hard to carry out the study of

cobordism in the case of (2n-I)-hypersurfaces. In case of (n + 1) dimensional slices,

we can assume that all the singularities are of codimension 1, and then we can obtain

a result similar to Theorem 7.2

First note that, topologically, integrable (n + 1)-dimensional "slices'1 can be clas-

sified by (a modification of) Fomenko-Zieschang invariant, which we will also denote

by !{v). Of course, the marks of /(t)), which determine how to glue pieces together

via diffeomorphisms of a n-dimensional torus, are more subtle in this case, but the

ideas are the same. We have

Theorem 7.8 Given any 2 topological types lt on Q"+t and !2 on Q%+i there exist

two integrable systems restricted on (n+l)-dimtnsional slices (Qi,ui), (Q2^2^ with

topological types I\ and I2 respectively such that Vi is cobordant to v2 via an integrable

system on a connected (n+2)-dimensional coisotropic slice (with a symplectic germ

Proof. The proof of the above theorem repeats exactly the proof of Theorem 7.2.

Let us just indicate what we mean here by a formal Hopf system (on (n+l)-slices).

From the definition it will be clear that slices in the higher dimensional Hopf system

52

are formal Hopf systems. Recall that for every (codimension 1) elliptic singularity,

the (1-dimensional) vanishing cycle is uniquely determined homotopically, and for

every hyperbolic singularity, the "hyperbolic (n-l)-subtorus" is uniquely determined

homotopically. Using Poincare's duality, we can define the pairing of any two 1-

dimensional cycles in a (oriented) n-torus. The topological type, which has only

one edge and 2 elliptic vertices, such that two corresponding vanishing cycle have

pairing equal to 1 is called the formal Hopf system. This definition is in complete

analogy with the case of 2 degrees of freedom. •

8 "Torsions" of integrable systems.

In this section we define some invariants for the triple {Q3,UJ,V), where, as before,

v is assumed to be integrable non-degenerate. Unlike Fomenko-Zieschang invariant

I(v), these invariants will depend on the symplectic germ. We will also assume

that v is stable, that is it is a restriction of some system on Q3 x (t, —t) which is

non-degenerate and all singularities of which are stable.

Consider now an edge W of I(v). Recall that, by the Arnold-Liouville theorem,

every point t of the edge W corresponds to a Lagrangian torus T,2, and there is a

natural flat structure on this torus. Furthermore, v restricted to the torus T(2 is a

constant vector field.

Let us make some conventions about the orientations. On M* = Q3 x D1

the positive orient?fion is given by u A u. On Q3 the positive orientation is the

orientation for which the gradient of the Hamiltonian generating v is pointing outside

(with respert to the orientation on A/4). Suppose we fix on every edge of I(v) an

orientation. Then on T,1 the positive orientation with respect to the orientation on

W is the orientation for which the product orientation on 7 2 x W is negative in

Q. For exai 'pie, M4 — R2 x T2 = {(pi,P2, 1i><j2),0ii<ft — modi} with the natural

foliation by tori, u = Ldpi A dqitQ = M4 n [pt = 0}. The orientation on Q is

53

given by the basis (92»9i(pa)- Q has one edge W parametrized by p?. Let pa give

the positive orientation on W, then (qi,qj) gives the positive orientation on T2 with

respect to it, since (<?i,<?2,pj) gives the negative orientation on Q.

Take now some basis (71,72) on Tti(Tt). We can assume that the homotopy

classes of 7i(0>7a(0 (in {UT<,( € W}) do not depend on t. (full) gives us a

linear basis in Tt. Let this basis be orthogonal, i.e. provide Tx with a conformal

metric. Then we can talk about the algebraic angle ^(71, i>). Note that for every

t € W, this angle is constant on Tz, so that we obtain an angle function Ang :=

^npiw.-n.Ta) : W -* R/2TTZ, Ang(t) = L(-n(t),v(t)). This function is obviously

continuous (at least outside the singularities of the system v). Clearly, invariants

of these angle functions (for example, their local min-maxs) are invariants of the

system v. We want to derive from these angle functions some numerical invariants,

called "indices". The idea is very simple : these indices measure "total geometric

torsions" of systems on their edges (i.e. one-parameter families of Liouville tori).

To define the index on the edge W, we will need some canonical choice of the

basis (71,72). Recall that there are 2 types of nondegenerate critical periodic orbits

: elliptic and hyperbolic. For shortness, we denote elliptic by "el" and hyperbolic

by "hyp". Every vertex of I(v) must be of one of the two above types. Thus there

are 3 cases for the vertices of the edge W:

I. el elII. el hyp

III. hyp hyp

Consider the first case: e/j el^. (By the way note that in this case Q is a

lens space and l(v) has only one edge). Then near every one of these two elliptic

critical cycles there is a uniquely defined (homotopic) vanishing cycle. Denote them

by ei and e2 respectively. There are two subcases:

la. C] = e2 (up to the orientation). In this case we do as follows: Fix an

orientation on W (It is easy to show that the definition of index does not depend

on the choice of orientation on W). Take 71 = e, = e2, and fix an arbitrary cycle -/2

such that (71,72) is a positive basis on Tt2 (with respect to the orientation of W). In

the cases below we will also require that the orientation given by (71, 7J) be positive

with respect to the orientation of W.

Ib. e i A e j ^ 0.

In the second case eli hyp, near the first end of W there is a uniquely

defined vanishing cycle ei, and near the second end of W there is a uniquely defined

"hyperbolic" cycle hi, which is homotopic to the critical hyperbolic periodic cycles

of the second vertex of W. There are also two subcases:

[la. ej = h2 (up to the orientation).

lib. e, A W O .

In the subcases Ib, Ha, lib, also take 7, to be d.

Similarly, in the third case there are two subcases:

Ilia, ftj = hi (up to the orientation).

Illb. ft, A h2 ^ 0.

In both subcases take 71 to be ht.

72 can be taken arbitrarily, since the definition below will not depend on it.

To be sure that our "indices" will be well-defined integers, we need the following

Proposition 8.1 The angle function Ang(t) (for any fixed pair of cycles (71,72),)

has a limit when x 6 W tends to an end ofW (i.e. to a singularity). More precisely,

this angle function converges to I{ft,hyp)(mod2ir) in the hyperbolic case, and to

some value different from l(iuel)(mod7!) in the elliptic case, where, as above, hyp

denotes the cycle homotopic to the hyperbolic orbit with the orientation given by XH,

and el denotes the vanishing cycle m the elliptic case.

Remark. This proposition doesn't hold if the system is non-stable. Actually, in

non-stable case, limit still exists but it can be anything. Non-stable systems require

a separate treatment, and we will not consider them here.

We postpone the prove of this proposition to the end of this section. Let us

now define the index. Instead of the angle function Ang(t), it is more natural and

55

geometric to consider the "direction map"

Dir{t):~ exp{iAng(t))

According to the above proposition, this map gives a continuous path in Sl C R2,

which is parametrized by [0,1] and smooth outside the end points. In the case Ib we

have to add t.o this direction path a (smallest) arc in the circle from Dir(l) to the

point (-1,0), if Dir(l) and (-1, 0) lie in the same side of the line generated by e2,

or from Dir(l) to (1,0) if otherwise. We will call this new path Dir(t) and forget

about the old path in the case Ib.

Now define the index of W as twice a Maslov index of the direction curve in the

plane with respect to the straight line given by 7J. More precisely, we imitate the

following definition of Maslov index from Duistermaat [Du]. Let (,[t) be a continuous

piecewise smooth curve

Then the Maslov index of the path f with respect to the line {x — 0} is given as

follows: in case both ((0) and ((1) lie outside the line {z = 0} the Maslov index is

the algebraic number of times the curve crosses the line {x = 0}, i.e. points where

it crosses {x — 0} in anti-clockwise direction are counted as positive. Note that by

homotopy theory this number is well-defined. If at least one of the ends of C,{t) lies

in {x = 0}, then perturb C, a little bit in the anti-clockwise direction: (' = exp(it)Q,

f > 0 small. Then the Maslov index is the algebraic number of times £' crosses the

line {ar = 0}, if C(O),C(1) € {x = 0}, this number plus 1/2 if only C(0) € {x = 0}, or

this number minus i/2 if only ((1) € {x — 0}. With the above definition of Maslov

index, we have

Definition 8.2 (cf. [BZ]j Twice the Maslov index of the path Dir(t),t € W, with

respect to the line given by •yl, is called index of the system v on the edge W.

Remark. It follows from the definition that in the cases la ,IIb, Ilia, index is an

even number. In the other cases index is an odd number. Aside from this fact, any

choice of indices is admissible, i.e. there will be a system with the given topological

56

type and given indices. This fact is trivial from the point of view of orbit space and

integrable surgery.

Example 8.3

Hopf systems have index 1. (Recall that they have only one edge). Integrable

geodesic flows on a 2-dimensional Riemantiian torus with a quadratic on momenta

additional first integral have all zero indices. It can be easily derived from [K, ZlJ.

Let us mention the following trivial fact.

Proposition 8.4 (cf. [BZ]) If(Qi,Vi) and (Q2,v2) are orbitally equivalent, then

their corresponding angle functions are (C°) conjugate. In particular, indices are

invariants for the orbital equivalence. •

Remark. More recently Boisinov and Fomenko found a < omplete set of invariants

for the orbital equivalence. Their papers will appear in Math. Sbornik in Russian.

Proof of Proposition 8.1. The existence of the limit does not depend on the choice

of (71,72), so we will choose the basis as we like. Let's recall from Proposition ...

that we have the following canonical form near the elliptic and hyperboiic singularity

for the symplectic germ in thickened Q, i.e. Q x (t, —t):

ui = dx\ A + 7T*(wi)

with respect to decomposition of Uiw(N) into Dl x Sl x P2. Furthermore, the

Hamiltonian H and first integral F are invariant with respect to x2. The vector

field djdx2 is Hamiltonian and periodic with period 1. In the elliptic case, P2 is a

disk with the usual area form on it, and H has the type H = H(x!,(x2 + rj)) with

dH/dzi ^ 0 (cf. [Ei]). The proof in this case is trivial, since the flows of xv and

it(x\ + x\) are periodic of period 1 and give rise to a basis in Liouville tori. Consider

now the hyperbolic case. For simplicity assume that we have only one singular circle

in N, i.e. we have a letter-atom B. Let 0 be the singular point of F restricted to P2.

Because of the stability, we can assume that (dFjdx\)(O) — 0 (see §3). Let us look

at one of the orbits of Xf in an annulus A in the singular level in P2xSl. Every such

57

orbit is homoclinic with respect to the hyperbolic curve. Since (dF/dxi)(O) = 0, one

can verify that this orbit does not wind around the hyperbolic curve. In other words,

the limit set of it in positive time is just one point, say £1, in the hyperbolic curve,

and in negative time is another point, say £3. Now suppose T,2, t ~ 0, is a Liouville

torus in Q3, which is near to the annulus A in the sense of Hausdorff. let 7t be the

(homotopic) hyperbolic cycle in T(s, and y2 be the cycle given by T2 n {x2 = const}.

Then by continuity, on T2 the flow af XF going from a point E\(t) near to E, will

meet the circle SE,(f) = {[x3,x^ = (x3(Ei(t)),xt(E,{t)))} C Tt2 in the first time at

a point Ej{t) near to E2. In the flat structure of Tf, homotopically the orbit of Xf

from £i(() to £?}(<) can be written as c-yi + 72, where c tends to a finite number as

( -* 0. The time for XF to go from £,(/) to Ei(t) will be denoted by d(t). Note that

d(t) -> 00 as ( -*• 0. Now recall that djdx2 = adH + bdF, and a(O) ^ 0 because

(dFjdx{)(0) — 0. It means that, homotopically, the orbits of the time 1 flow of Xfj

on T(2 are equivalent to

1 b 1 , , .1 b . ba ad(t) a ad(t) ad[t)

When t -> 0, it tends to (I/a)-/!. It means that angle 1(XH,~II) tends to 0 with

respect to the basis (71,72). Thus we have proved our statement for one family of

tori. For the other families of tori the proof is the same. D

Remark. If a system (Q,UJ,V) is "embedded" in some system in a compact

symplectic manifold (M4,ui,v), then (Q,UJ,V) can be represented by a graph in the

orbit space of (MA,u>, v), and the angles discussed above in this section are much

similar to that ones in §4.

The notion of angle functions can be generalized naturally for higher dimensions

as follows. If Qin~x is a regular isoenergy hypersurface of some integrable system,

then it corresponds l o a n - 1-dimensional subspace (don't confuse it with (n-1)-

strata) in the orbit space. So, instead of edges of Fomenko-Zieschang invariant, in

this case we have n — 1-dimensional domains. Every such a domain means a n — 1-

dimensional family of Liouville tori, Fix one domain and, for simplicity, assume it

to be homeomorphic to a disk. Fix an positive (under some orientation conventions)

integral basis on one torus in this family and extend it homotopically to obtain a

58

basis on every torus of this family. The Hamiltonian vector field XH is constant and

non-zero on every torus, so it defines a direction (with respect to the chosen basis).

This way we have a Gauss map from our n — 1-dimensional domain to 5"" ' . These

Gauss maps are clearly a natural generalization of angle functions defined before.

One can attempt to define an index from these Gauss maps, roughly as the index

of maps between closed manifolds with the same dimension. But technically, it is a

difficult problem.

9 Compatible contact structures

In this section we discuss how to construct compatible contact structures for a given

non-degenerate system (Q3,ui,v), As in the previous section, we will assume that

our systems are stable.

First let us recall the main known results in the theory of contact 3-manifolds.

For dimension greater than 3, Gray [Gr] proved that there are topological obstruc-

tions (in terms of characteristic classes) to the existence of a contact structure on

a given manifold. In dimension 3 the situation is different. Martinet [Ma] proved

that any orientable 3-manifold admits a contact structure. The idea of Martinet is

to glue together contact handle-bodies.

For the sphere S3 one can construct a contact structure as follows: imbed S3 in

the complex plane C2 in a standard way. Then S3 will be a convex hypersurface (see

e.g. [Ho]). Thus the distribution of tangential complex lines on S3 will be a contact

structure on it. This contact structure is called the standard contact structure on

S3. One can give a simpler proof of Martinet's theorem by using Dehn surgery

and the normal neighborhood theorem for transversal curves to construct contact

structures on any given orientable 3-manifold from the standard contact structure on

.53. Normal neighborhood theorems in symplectic and contact geometry are mainly

due to the Moser's principle [Mo]. In case of transversal curves, it was also proved

59

by Martinet [Ma].

Contact 3-manifolds which are invariant under some free S'-action were classified

by Lutz [Lu]. The next important result is due to Bennequin [Be], who proved that

there exists a contact structure on S3 which is not equivalent to the standard one.

Recently, Eliashberg in his series of papers has made a great progress in the subject.

In particular, he classified all the overtwisted contact structures, all the contact

structures on S3, and proved that overtwisted contact manifolds do not bound. For

more precise statements, see his papers or below. Note also the paper of Giroux, in

which he proves the existence of a so called convex contact structure (cf. [EG]) on

every orientable 3-manifold.

As explained in §3, in our case near to singularities of (Q,^\ v) there are natural

5'-actions. Outside a small neighborhood of singularities we have a natural S1 x

Sl = Traction, which can be thought of as a large set of S'-actions. So every

point of Q lies under at least one natural S'-action. We call the set of all that S1-

actions a local S' -action on Q (with respect to a given choice of small neighborhoods

of singularities). Thus for every triple (Q,^, t')there is a natural local S] action

associated wilh it. Therefore we are interested in 2 questions: how to construct

compatible contact structures, and how to construct compatible contact structures

which are invariant under a local 5L-action?

In R2n, star-shaped hypersurfaces are contact. An analogous situation holds in

our case. We will define the notion of almost-convex system and show that such

systems admit a compatible contact structure.

Fix a orientation on every edge of /(f), and fix a positive basis (7i(*)i72(0) on

every Liouville torus, so that tori over the same edge have "the same basis" (see

§7). Associate to first end and second end (with respect to the orientation) of any

edge W two (arbitrary) algebraic angles *4i(W) and AjfW), respectively, so that

Ai[W) < Ai{W), We say that these angles satisfy Property (P) if the following

conditions hold:

L. Near every elliptic singularity, the angle of the only one end-of-edge

contiguous to it, say /4(W), is equal to /(f1(W/},e) modulo 2TT, where e

60

is a positive vanishing cycle, positive means that (e,Xff) gives rise to a

positive basis.

2. Near every hyperbolic singularity, say N, suppose A ^ ^ , . . , , A(W|;)

are angles of end-of-edges contiguous to it, and Si,.. ., Sk are their "sign":

s, equals +1 if A(W) is At(W) and -1 if A(W) is Ai(W). Denote by e,

the positive boundary cycles of 2-surface P2 in the normal form (+) in §3

associated to this singularity: DP2 = £e; (recall that P1 has a positive

orientation given by the symplectic form). Every angle A(W,) is given

(module 2JT) by a direction in the flat structure on a torus Tt,t £ W.

Denote this direction by j , - in case s, = 1, or the inverse of this direction

by gi in case s, = — 1. We require that j , - can be given by in the flat

structure by e, + a< hyp for some numbers a;, where /typmeansa common

positive hyperbolic cycle for all the contiguous edges. Now our property

(P) says that Ea, = 0.

The meaning of the above property will be clear in the construction of locally

^'-invariant contact structures. With the notations as in the previous section we

have the following definition.

Definition 9.1 A stable system (<3,w,u) is called almost-convex if, for some choice

of orientation of edges in I(v) and positive bases (fi[W), 72(W))> there are angles

Ai(W), A2(W) satisfying the property (P), so that for every edge W, identified with

interval (0,1), the following conditions hold:

t. Ang(t) - Ang(t') > -JT if t > t'

3. min[0,,]Ang(t) > At(W) > A(0) - TT

3- maxp^Ang(t) - tr < A2(W) < A[l)

Definition 9.2 A stable system (Q,UJ,V) is called convex if all the indices tre

positive, and for some choice of orientation of edges in I[v) and positive bases

(7i(H'),72(VV)), angle functions art monotone strictly increasing on every edge.

It is clear that Definition 9-2 does not depend on the choice of bases, i.e. if

angle functions are increasing with respect to some choice of bases, then they are

61

so with respect to any choice. The same is true for Definition 9.1. Just note that

if we change orientation on some edge W and change the basis (71,72) over it to

(12,71)1 then Ang{t) changes to TT/2 - Ang(t), and Ai(W),A2(W) can be changed

to -Ai(W) - JT/2, -A2(W) - IT/2 respectively.

Usually, increasing monotonicity of angle functions implies that index is positive,

except few cases when it can be -1 or 0. We exclude such cases from our definition of

convex systems. If our system is imbedded in some integrable system in a compact

symplectic 4-manifold, then, roughly speaking, convexity means that every curve

given by this system in the orbit space is convex in the affine structure: locally in

every affine chart it is a convex curve.

Proposition 9.3 [f some stable system is convex, then it is also almost-convex.

The proof of Proposition 9.3 is somewhat straightforward, so we will omit it

here. P

Proposition 9.4 If a stable system (<2,t>) is almost convex, then it admits a com-

patible locally Sl-invariant contact structure.

Proof. We will construct a contact 1-form near singularities first and then extend

it along families of nonsingular tori to get a global contact structure.

Near every elliptic singularity N, simply take a canonica! contact 1-form in a

neighborhood of a transversal simple closed curve, which is in our case an elliptic

periodic orbit: a = dxi + ^(xdy — ydx), where i 2 is a coordinate in a canonical

form (+) (see §3), x,y- Euclidean coordinates in a disk D2 = UQ(N) D {T2 = const}.

Note that a in this case can be made Sl x S1 = T2-invariant. y.lso, it is clear that

O(XH) > 0 if Uq(N) is small enough, since at every (fixed) elliptic orbit XH is equal

to &Xi times a positive number

For every hyperbolic singularity Ar, suppose it has k contiguous end-of-edges.

Recall that we have a set of numbers fli,...,ajt, which corresponds to the set of

boundary components £1 , . . . ,& of P ' = UQ(N)C\{XI = const} under some canonical

62

form (+), and Sa< = 0. Set a = t/i2 + /?, where /9 is some one-form on P2 with the

following properties:

a) d0 is a positive nondegenerate (area) form on P2 • Recall that P2 is a symplectic

submanifold, thus it has a predefined positive orientation.

b) For every i = 1, , . . , k, /?(C.) — ai ' s positive and small (as small as we want).

c) Near the boundary of UQ{N) (for some UQ(N)), a is T2-invariant.

d) a is compatible: o(X#) > 0.

Properties a), b), c) can be achieved simply by Poincare's Lemma. Property d)

can be achieved as follows: Take any Qj = dx^-\-fii which satisfies a), b), c). Denote

by g'Xp the flow with time s of vector field Xf of an additional integral F chosen as

in Proposition 3.3. Then set

One can check that, for S big enough, the above form a will satisfy also the property

d).

Now we have to extend the above constructed contact compatible S'-invariant

one-form a near the singularities to the contact one form on the whole Q, making

it 7'2-invariant outside singularities.

For every edge W, identify it with (0,1), and write a T2-invariant 1-form a which

we are looking for in the following way:

a = aidqi 4- a?dq-} -f a$dt,

where qi,(j2 are coordinates with respect to the chosen basis (71,72), t € (0 + e, 1 — ()

and al,a^,a3 - functions, depending only on t (e positive small, since we consider

only outside singularities).

The compatibility yields that a does not vanish when restricted to the tori T2 x

{pt}. In other words, at every point at least one of two functions 01,02 does not

vanish. We can assume that a) / 0 (locally). Then

a A da = dq2 A dq{ A dl(-^-cn - - ^ O J ) .

63

and positively contactness of a means that

da2 da,

It is equivalent to the fact that the angle arctgfa/ai) increases when ( increases.

Denote by ea(t) oc ai(i)djdq\ — oJiJS/Sqj the characteristic direction field given by

contact the form a on T2(t). Then the positively contactness of a is aiso equivalent

to the fact that the angle Z(7i,eo(()) is increasing with (.

The compatibility of or now means that l("n,ea(t)) can be chosen so that

i(-n,ea(t)) < Ang(t) < Z(",,,ea(0) + T.

In other words, it is sufficient and necessary to chose an angle function B(t) =

^(7i,eo(t)) s° t n a t B{t) < Ang(t) < B(i) + ir and R[t) satisfy some "bound-

ary conditions": it can be made to be equal to the value given by a constructed

before 1-form a near the singularities. These boundary conditions mean exactly

that B({ — t) — J43(W) is positive and near to zero, B(d + t) — 4i(W) is negative

and near to zero. In other words, we should be able to extend B to a monotone

increasing angle function on [0,lj such that B(0) = A\(W),B(\) = A[W), and

B{t) < Ang(t) < B(t) + * for all ( e [0,1]. Such an angle function B can be

constructed as follows: for t 6 [0, c), B(i) is a linear function going from ^ 1 (W7) to

min{min\o,i]Ang(s), Ai{W)} — ti for some small positive ti. For t e [(, 1], B(t) has

the form B(t) — min{min[til]Ang(s),Ai(W)} - t?(t), with some small correcting

function t2, c2(l) = 0, such that B(t) is continuous strictly increasing.•

Going in the inverse direction of the above proof, we obtain the following

Proposition 9.5 If a stable system (Q,CJ,V) admits a locally Sl-invariant compat-

ible contact structure, then it is almost convex,

The proof of Proposition 9-5 repeats the proof of Proposition 9.4, except that

one has to be a little bit more careful at singularities. We will omit the details here.

a

If contact compatible structures are not required to be locally 5 '- invariant, then

their behavior can be rather complicated, especially at singularities of the systems.

64

Anyway, some conditions near singularities and along the Lagrangian tori, not much

weaker than almost-convexity, are still necessary. Here we will give a result only

about conditions along Lagrangian tori.

Proposition 9.6 // a system (Q,u,v) admits some compatible contact structure,

then for every its angle Junction Ang(t) on any edge we have that Ang(t) — Ang(t') >

-TTlft> t'

Proof. Suppose we have a compatible contact distribution f with contact 1-form

a over the edge W of (Q,w,u). Identify W with the interval (0,1)- Then for every

t <z (0,1) the tangent pla-ies of the torus T*{t) intersects with f transversally, and

give rise to a characteristic direction distribution ((z) € TTT2(t),z € T2(t). This

direction is, generally speaking, non-constant with respect to the torus action on

T2(t), unlike the locally S'-invariant case. Still we can define an "average direction"

of C(i) over T"*(t) as follows. Note that ((x) is transversal to Xfj(x). So it is also

transversal to some constant vector field X(x), which is near to Xfi(x), and which

has periodic orbits. Define a basis (71,72) on T2(t), such that 72 is parallel to the

above periodic orbits, and denote (qi, qi) the coordinates system corresponding to

this basis. (qi,q2) will be an Euclidean system of coordinates for fl2, which is Z2-

covering of T3(i). In these coordinates, ((x) can be given by the vector field of

the form djdqt -+- a(qi,()2)d/d<l2, where 0( 1,(72) is some real function, periodic with

period 1 in both variables. Suppose g'(qi,q2) = (qi + s,g,(q1,q2)) is a time s map

of the flow of the above vector field. Then take the limit

Vim gs(qi,qj)/s,I—>OO

One can check that this iimit exists and does not depend on the initial value (1)1,92)

(see e.g. [Ar2]). Denote it by o(() and denote by f (i) the direction given by d/dqx +

a(t)djdqi. C(i) changes smoothly with respect to (, so ((() is a continuous function

on t. From the construction it is also clear that £(() is transversal to XH- Denote

by B[t) the angle t("t\,(,(t)) with respect to some "common" basis (71,72) over the

edge W. Then Proposition 9.6 is a consequence of the following Lemma.

Lemma 9.7 B(t) is a monotone increasing function.

65

Proof. As before, we can assume that a — atdqi + a?dq? + a3dt. Then

. do, da2 da3 daz[( ? + (

da2 da,

~ ^ a i ~ ~ & a i ) ] q* qi-

Positively contactness of a means that

Fix any (0 e (0,1). Near T2(t0), define new form a' = a\dqv + a'2dq2 + a'3dt such

that a' A da' = 0 and restrictions of a and a' on T2(t0) coincide. It, implies that

(near to T2(t0)),da2

Consequently, at T2(tg) we have

da'2 , da[

d a'2

dta{'

It follows that, for ( > (Q and near to io, the average angle given by a is greater than

that one given by a'. But since the plane distribution given by a' is integrable (by

Frobenius's Theorem), the evarage angle given by it does not depend on t (because

of the conjugacy). Thus we have B(t) > Ba,(t) = Ba'{t0) = B[t0) for t > t0, t near

to to. Analogously, B(t) < B(t0) for t < tQ, t near to (0- a

Corollary 9,8 // a system (Q,u},v) has a edge with index smaller or equal to -2,

then it does not admit any compatible contact structure. •

Remark. From Proposition 9.5 it is easy to construct an example of a stable

system, which admits a compatible contact structure, and which has an edge of

index -I.

The following definition of overtwisted contact structure appeared first in the

work of Bennequin [BeJ. Suppose [Q3,£) is a contact 3-manifold, and P2 C Q is an

arbitrary embedded surface (may be with boundary) in it. Then the intersection of

f with the tangent planes of P2 gives rise to a line fields (with singularities) on P1.

The foliation given by this line field is called a characteristic foliation on P2. So it

is an unique foliation by Legendrian curves on P2. A contact 3-manifold (Q3,£) is

66

r i i l t f r l on rhri*f(il if it c o n t a i n s nn e m b e d d e d d i s k w i t h I lie <:liai a c t e i i s t i c f o l i a t i o n

i>iw]i in |-i"urr> !la. o r < ' ( | i i i \ a l cn l l y , in F i g u r e 91> ( n o t k o t h a t t h e b o u n d a r y i t s e l f is a

l . rnt - iu l i I rin c n r \ i \ c! [He. I".! ,2]). [t is n o t i l i i l i cu l l t o c o n s t m e t e x p l i c i t e x a m p l e s o f

iivi-i-l\vi<ifil i D i i l a c t s i r u c t u r c o n a n y o r i c u l a b l e 3 - m a n i f o l d . B e n i i e q i i i i i [Be] s h o w s

' 11i-i 1 ii .i < o i i l . n l s l r n c i u r i ' in S'1 is o v c r t w i s t p d . t h e n it is n o n - s t a n d a r d , b y u s i n g

- • I ' l l i r i l i ' h i ; i l r M a r k o v M i r l . u c s 3 e c h l l k l U C s .

Figure !i

| - u l l ( , \ v i i m L l i a s l l l i i - r j i T . l ] . l \ e > H Y ( l u l l a l - O l l i H C l S l l l H 111 l o l l I l l O i m d n - < l / l l l j ) l l f f i r a l i l j

! " i - .1 i i n i i . i ' t U -11 • - I i n n I I I l . i \\ l i > r M > I I I I - s v m p l i ' c l i r < o n i ] > a r i I - n i i i n i f o l d . a n d b o u n d . *

'" < <<ur t i U .. I' < • , , : - - i \ i t i ^ . i 1 1 1 L c i - . i t ( P I I \ i ' \ l ) o n n i h i r \ ' l o r s o m e a l i t u i s l r o u i p h ' X c o i n j i a c t 4 -

11 I . I M d c M I I . . m i l t h e i i m l i u I - I I u . T n i l - i s i j , i v c ' i i l i v c o i i ! [ ) l c x l i n e s . O f c o u r s e , i f s o m e t h i n g

l i i n i i n i - - \ i n h l r i i I I a l l v . I I H ' I I i i ; i l s o I H I I I I H I S i n i u i i i | ) l c s I c r n i s . ' s i n g t < > r h n i ( | u e s o f

h l l i i i L , l ) y l i u l i j i i i d i | ) l m I ' H I M ' S a l l ( j w r d K l i a s l i l x ' i g k i [ i r o v c i h c I D I I O W I I I ^ i n i i > o r t a n t

I I - - I I 1 1 - U I I I . 1 1 t ' o l l l H i l l s t 1 1 1 - l l ' s l l l i o f B l ' l l l H - < | l l ! l l .

T h c o r t ' i n 9 . 9 ( E l i a s h b e r g ) If n cnntiti-l .i-iiiunifulil i* un Hti-i.-tlril. Ihm if f/rir.-

t 'orollary 9.10 / / nu inliiirnhli »J»IM« i

hu> rrlmii / ^ tutt!ifi>hit, thin llti.-- *tfsl(

.^.r) admit* a tompnlilili conflict strue-

/> no! tiji-fy>boiili!tit to 0.

.i t oriM'ij uriiee. we have:

Theorem 9.11 If a system (Q,u,v) is almost convex, and there is a final edge of

I(v), i.e. an edge which is contiguous to at least one elliptic singularity, with index

greater than 2, then (Q,ui) does not bound.

Proof. We will show that the contact structure constructed in Proposition 9.4

in this case is overtwisted, by detecting an "overtwisted disk".

Suppose v is convex, edge W of I{v) has index greater than 2, and the first end

of W is elliptic, i.e. B(0) = AjflV) = i(-/i,el). From our assumptions, it follows

that there exists a (unique) point t0 g (0,1) such that B(to) ~ £?(0) + TT. Consider

the solid torus n = u[=0TJ(*). By changing coordinates in a T2- equivariant way, we

can assume that £0 = T and the contact structure can be associated to the following

contact form:

a — cost dq2 + ( sint dq,.

Here (<ji,q2) are coordinates on tori, {qi — const} C\ T2{t) are vanishing cycles, and

t € [0, ?r]. In these coordinates, the disk D2 = {ft = 0} C tl is an overtwisted disk,

i.e. it has a characteristic foliation as in Figure 8b. •

Example 9,12

Let (M,o;) = (R4,ui0)/Z where Z is the group action generated by y2 —> y2 + 1.

Consider the Hamiltonian H — [x\ + y})2 + x\. Then one can easily see that the

restriction of Xu on every positive energy surface is a up-cobordant-to-0 system

on S1 x S2, topological type of which has only one edge, and with index 2. This

example shows that the assumption "greater than 2" in Theorem 9.H is necessary.

Corollary 9.13 There are non-degenerate integrable systems (Q,ui,v), which are

not a boundary of any integrable system on a compact symplectic ^-manifold.

Actually, there are plenty of such systems, since by integrable surgery, we can

choose our angle functions rather arbitrarily. In particular, we can chose them to

be monotone, with indices as high as we want.

Proposition 9.14 If a system (Q,u>,v) is almost convex, and there is an edge of

I(v) with index greater than 6, then [Q,u),v) does not bound.

68

Proof. Suppose the edge W in I{v) has index greater than 6. The by "cutting"

this edge (see §7) we can show that this system is down-cobordant to another almost-

complex system which has a final edge of index greater than 2.O

Before going to higher dimensions, let us make a small remark. From the results

of Eliashberg and Lutz, one sees that any contact structure on S3 is invariant under

some S'-action. It is interesting to know wether every contact structure on a graph

3-manifold is invariant under some local .S'-action or not.

Let (Q*n~l,Lj, v = XH) be an integrable system on a {2n-l) manifold with a

symplectic germ on it. We may ask if there exists a locally Sl- invariant compatible

contact structure. Here, for simplicity, we will assume that all the singularities have

product type (i.e. they are products of non-degenerate codimension 1 singularities,

and focus-focus codimension 2 singularities). Now "locally .S'-invariance" means

that it is ^""^-invariant near every codimension k singularity (under some natural

Hamiitonian Tn'k-action)(k = l , . . . ,n — 1) and, outside all the singularities, it is

r"-invariant (more precisely, it is invariant under Hamiitonian flows of first inte-

grals, because T"-actions may be not well-defined globally even in the non-singular

domains).

Below we try to construct compatible locally ^'-invariant contact structure, un-

der the conditions that our systems behave "very well": it is convex, its singularities

are of product type, and every stratum of its orbit space is contractible.

Let us =tudy such a contact structure near some non-singular Liouville torus Tj1.

A tubular neighborhood of Tg in Q is diffeomorphic to O""1 x T", and we provide it

with a system of coordinates ti,. • •, tn-\, q\, ..,</„, where q, are angle coordinates,

and ti are first integrals. A T"-invariant 1-form a can be written as

a — aidqi + .. . + andqn + btdti + . . • + &n_i<ftn_i

with da,/uqj = dbi/dq3 = 0. Simple computations show that

a A An~lda - (Det) d<)t A (ft, A . . . A <*<}„., A dta-, A dqn

69

where Det means the following determinant:

Det := Det

\an daj'dtx

The contactness of a means that Det > 0. This condition can be characterized

also as follows. The kernel of a on every n-torus can be viewed as an oriented (n — 1 )-

subspace in /T*. Thus we have a Gauss map from our n — 1-dimensiona! domain to

G+(n,n - 1) = S"'1 given by this 1-form a. Then a is (positively) contact if and

only if this Gauss map is a local (orientation-preserving) diffeomorphism.

The compatibility of the contact 1-form a means that the above (n- l)-subspaces

given by a is (positively) transversal to XH- In case our system is convex, i.e.

the Gauss map given by XH is a local (orientation-preserving) diffeomorphism, the

existence of a compatible contact 1-from a in every non-singtilar contractible domain

is clear.

Under product type conditions, every neighborhood of a codimension k singular-

ity is diffeomorphic to (£)""*"' x Tn~k) x Pu, where P7k is a symplectic manifold

with an exact symplectic form (since f2t is a product of surfaces (codimension 1 sin-

gularities) and symplectic 4-manifolds of focus-focus singularities). Then the locally

contact 1-form can be constructed in such a neighborhood by the following

Proposition 9.15 //V, = V^"1"1 x Vj", and a is a contact 1-form on V2, u; = dO

is an exact symplectic form on V$, then a + 8 is a contact 1-form on Vj.

Proof. The proof is straightforward.•

By the above scheme, one can recover the standard contact structure on 52"~' =

isoenergy hypersurface of a higher dimensional Hopf system.

70

S. SL:

10 Nonstandard symplectic structures in i?4

Consider the Euclidean space R2n with some symplectic form w on it. Because u

is closed in Rin, it is also exact: UJ — da for some one-form a. Such an 1-form a

is called a Liouville form. If L is a Lagrangian submanifold in (/i2n,w) then a is

a closed one-form on L. The Lagrangian submanifold L is called exact if a is an

exact one-form on L. In other words, the symplectic action of any 1-dimensional

cycle in £{:= integration of a over this cycle) is 0. Gromov [Gr] proved that in the

standard symplectic space (R2",wo) there does not exist exact closed Lagrangian

submanifolds. Viterbo [Vilj proved that, in case L admits a Riemannian metric

of non-positive curvature, then for every Lagrangian embedding of L in (R?n ,w0)

there is a cycle in L with positive Maslov index not greater than n + 1 and positive

symplectic action.

A symplectic structure w on R2n is called non-standard if (/?2",u>) can not be

symplectically imbedded into (fi2",t4t0). The above theorems of Gromov and Viterbo

yield that if a symplectic structure on R2n admits a closed Lagrangian submat ibid

which is exact, or which has a "non-standard" Maslov class, then this symplectic

structure is non-standard. Up to now all the known examples of non-standard

sympiectic structures on R2n were proved to be non-standard by these ways (cf.

[Vil, Vi2, Mu, Ba, BP]).

The first explicit example of an exotic syinplectic structure was found by Bates

and Peschke [BaP]. Later Bates [Ba] observed that their construction has some

symmetry, and by Noether's theorem, it should be obtained from some integrabie

system. He found an explicit integrabie system for his case, and raised the question

about the study of general integrabie systems (on S3).

Here, following Bates and using integrabie surgery and topological classification

of integrabie systems on S3 [FZu], we obtain an infinite series of exotic symplectic

structures in Rl. Our series has the property that there are structures with as many

"obvious" exact disjoint Lagrangian tori as one wishes (there may be other exact

tori that we don't know?), and the relative topological structure of these tori in R4

is quite complicated (usually they are not regular homotopic).

A vector field X in a symplectic manifold (M,OJ) is called Liouville if Lx"J = w,

i.e. ixuj is a Liouville one-form. That means that the flow of this vector field

enlarges the symplectic form u>, Liouvitle vector fields play the central role in the

theory of convex symplectic manifolds [EG]. By constructing a explicit Liouville

vector field, in some cases by extending these vector fields out from the manifolds

as far as possible, one can see that our symplectic structures can be made "almost

complete" in the following sense: there exists a Liouville vector field in R* which

defines a group action of R in R* almost everywhere.

Let us start constructing our series of non-standard symplectic structures. Take

any topological type / = I(v) which can be realized as ,JI non-degenerate integrabie

system on S3. Now construct an integrabie system on a symplectic manifold (M* =

S3 x Dl ,u-') with the above topological type (on every isoenergy level set, see §?).

Since M* has trivial second cohomological group, every 1-cycle in it has a well-

defined symplectic action. Now take an arbitrary (non-singular) Liouville torus

To in M. Such torus lies in a 2-parameter family of Liouville tori, one parameter

is bounded by singularities, and another parameter is bounded by the boundary

of M. Near the torus To fix a system of coordinates (p,,q,(modl)) given by the

Arnold-Liouville theorem. Suppose in these coordinates the symplectic action of

the cycles generated by d/dq] and djdqi in our torus is c\ and cj respectively. By

adding constants, we can assume pi(7o) = C[ and p?{Ta) = c2. Now there comes

a simple but important observation: for every Liouville torus in this 2 parameter

family, symplectic actions of the cycles given by d/3qi and djdq? are p, and p2,

respectively. Remember that the coordinates (pi,p2) in the (2-domain of the) orbit

space are the system of coordinates for the integral affine structi.^e associated with

the system. From integrabie surgery, we know that if we change the affine structure,

leaving it unchanged near the singularities, then still we have an orbit space for an

integrabie system. Now we can change (pi,p2) in such a way tliat it satisfies the

above condition, and there is a point in the 2-domain at which we have (pi,p2) =

(0,0). It means that in the new integrabie system (which has the same topological

foliation by tori as the previous system) we have a torus, for which symplectic action

u r t h o l h e c r l e s ^ e u e t T i l e d l>v i)l<)t[\ a n d ()j()>{i a r e 0 . i . e . d t i e x ; i c l t o r n ^ ! T h n s w e

h a v e a s y m p l i ' c l i c m a n i f o l d ( . A / 1 = N ' x / J ' . u ; ) w i t h Hi) e x a c i t o r u s . s i d i n g i n s i d e - .

D c l d i i i n n p o i n t f r o m > ' . w e o U i n i u a s y m p l e r t u - m a n i f o l d [It' — (>••* - pi) x D ' . o . ' )

w i n c h l ias MII ( ' .we) c i n l j c r l d c d f.-Hgrangian t o r u s . T i m s I h e s v m p l c t f if s i r I U R -

< I I I I ^ I r u t I e d A I U A C S n o n - s l a i i r l a i 1 ' 1. N O W l a k e ,\ ' . " o m p l K a l e d t o p o l o g i e s ! t y p / ( ( ' )

m i , s " . i . e . w i i h i n . u i y r t ! i ! , r s ; U K I r o i i i ] ) l i ( H l I ' d i . i r s i o n s . i i n r i ' p t ' c i t t h e a b o v e s c h e m e

w i t h i i n ! o n l y o n e L ' - d m i i i i i r i i n t h e o r b i t . s p a c e s , b u l w i t h m a n y 2 - d o i i i a i n s , T h e n w o

w i l l ] i , i \ i ' j S V I H I J I C C I i r s i r u < i u r e u i i If u i l h a s m a n y d i s j o i n l I . H g r a n g i i ; : i t o r i a s w e

w i s h . ; t n d i I n - i i ) | i i i l < i ^ v i l l ! I n 1 r o i ] i ] i l r i n c i i l t o t h e s c t o r i c a n l i e i i l s o v e r y c o m p l i i a l c d .

N L j | c t h . i t t h r r i l » o v r ( D i l s t i-n j • • I l u t i < ; i n b < = ^ c l l t M ' f i l l / c d l u r j t i n i l l i n c i i s i o n . - I l i s t ^ t ^ r I

I n i n i i n r v i I i l l ' j ; t i d i l c s \ — 1 • • i l l o n > ' - " - ' , I ) ' i h e l i i ^ l i c r i l i i n r r i s i o n a l H o p I s y s t e m f o r

I ' X i i m p l i 1 . i i i i ' l T 1 1 c -11 i l i . i n ^ c I I n 1 o r l i i l s p i i c c .

10:

h i l l i c s i i n f i l r ^ - i r i i " - r . u h r r r o u r > \ s l c i n i s *i f n r n i f i f [ l u p f s y s t c i n . i . e . t l i e g r a p h

u l ' ' I n ' f o p u l u n i i i l l l y p i - i < j n < i f i n i n i e i A r i l i v i l i i i w u v c r l i c e s . a n c \ a n i | i l e < i f a m > n -

- t , i i i i i . n ' i s \ M i j i l r r l If s i i i t e l i ] u - i i ^ i v r n i n l e n s e s o f u i t r ^ i f i ! i l e s v M r i n ^ ] IJ< s h a w i i l i t

j 1 - ^ 1 1 1 ;• I [ | . I n l h i s | > 3 r -1 I I t t t b e i i j l i n e s i i u r I i l l r i s t h e i i i v e i s i - i n i . i ^ r o f t h e s t H l l i l d r d

. i l h i i r - I i i n I n r e u i i l { - l l e i e o n e c ; i i i r e c u v e r I l i e o r i g i n i l l i m i s I n t c i i o n o f H a t e s a n d

I ' l . - e l i k i ' j H I ' l . w i l l l o l l ! , l l l \ r o i n n i l t . l t i o n .

7:1

Acknowledgements.

First of all I would like to express my deep gratitude to Michele Autiin, who

made me write this paper, and without whom this work would not have been done.

The present work is carried out while the author is visiting the Mathematics Section,

ICTP. He would like to thank Prof. Abdus Salam, Prof. James Eells and Prof. M.S.

Narasimhan, the International Atomic Energy Agency and UNESCO for hospitality

at the International Center for Theoretical Physics, Trieste. This work develops

some aspects of the theory by A.T. Fomenko about topoSogy of integrable systems.

Various parts of this work were presented at the Workshop on Dynamical Systems

- Trieste 6/1992, CIMPA School on Symplectic Geometry - Nice 7/1992, Moscow

University 9/1992, Workshop on Symplectic Geometry - MPI Bonn 4/1995, and

University of Strasbourg 5/1993. I would like to thank all of the participants of

the above seminars for their interest, especially Prof. Michele Audin, Prof. Daniel

Bennequin, Prof. Anatoly Fomenko, Dr. Alfred Kuenzle and Prof. Jacob Paiis. The

person who always encouraged me and taught me a lot of mathematics if Alberto

Verjovsky. There are so many other persons, my friends, my colleagues, that I would

like to thank here. In particular, Alexey Bolsinov, Eleonora Ciriza., Le Hong Van

and Kaoru Ono helped me much by mathematical discussions, in correcting my

English and style and so on.

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