Presence of minimal components

in a Morse form foliation

I. Gelbukh

Article accepted to: Differential Geometry and Applications, 2005, to appear

Email address: gelbukh * member . ams . org (I. Gelbukh).

Present address: CIC, IPN, Col. Zacatenco, 07738, DF, Mexico.

For a PDF or PS versionof this paper, refer to http://www.I.Gelbukh.com.

Abstract

Conditions and a criterion for the presence of minimal components in the foliation of

a Morse form on a smooth closed oriented manifold M are given in terms of (1) the

maximum rank of a subgroup in H1(M,Z) with trivial cup-product, (2) ker[], and

(3) rk

def = rkim[], where [] is the integration map.

Key words: Morse form foliation, minimal components, form rank, cup-product

2000 MSC: 57R30, 58K65

1 Introduction

Let M be a connected smooth closed oriented n-dimensional manifold and

a Morse form on M, i.e. a closed 1-form with Morse singularities (locally

the di.erential of a Morse function). This form de.nes a foliation F on M /

Sing , where Sing are the forms singularities.

The problem of studying the topology of such foliations was set up by S.Novikov

[9] as far back as in early 80s in connection with their numerous applications

in physics [10,11], which have been recently impulsed by the new

advances in the mathematical theory [2,3].

The topology of a Morse form foliation can be described as follows. Its leaves

are either compact, non-compact compacti.able, or non-compacti.able. A leaf

is called compacti.able if Sing is compact. There is a .nite number of

non-compact compacti.able leaves; thus their union together with Sing has

zero measure. The rest of M consists of a .nite number of open areas covered

by compact leaves (called maximal components) or non-compacti.able leaves

(called minimal components).

Compact leaves have neat properties [8]. All leaves in a maximal component

are di.eomorphic. A maximal component is an open cylinder over any its leaf.

The forms integral by any cycle lying in a maximal component is zero.

Non-compacti.able leaves, on the contrary, have very complex behaviour [1].

Each such leaf is dense in its minimal component. A minimal component can

cover a rather complex set in M; for any M with Betti number 1(M) 2

there exists a foliation whose only minimal component covers the whole M /

Sing . A minimal component contains at least two homologically independent

cycles with non-commensurable integrals [8].

In this paper we consider conditions for a foliation to have minimal components.

The forms singularities give little information on the foliation topology. F is

compact (i.e., all its leaves are compact) if and only if all singularities of are

spherical. Otherwise there always exists a form with the same singularities of

the same indices but with the foliation without minimal components [12].

A more useful characteristic of the form is its rank rk def = rkim[], where

[](z) = _z R, i.e. the rank of its group of periods; it is a cohomologous invariant.

If rk 1, the foliation has no minimal components [9]. For rk 2,

the foliation of a non-singular form is minimal and uniquely ergodic; however,

for forms with singularities the situation is much more complicated.

In any cohomology class with rk 2 there is a form with a minimal foliation

[1]. If the cohomology class of , rk 2, contains a non-singular form,

then F has a minimal component, thoughunlike non-singular caseit is not

necessarily minimal [4]. Existence of non-singular form in a given cohomology

class was studied in [5]; however, the only manifolds allowing non-singular

closed forms are bundles over S1 [13].

We show that for large enough rk any foliation has a minimal component

namely, for rk > h(M), where h(M) is the maximum rank of an isotropic

(i.e., with trivial cup-product) subgroup in H1(M, Z) (Theorem 13). In particular,

the foliation of a Morse form in general position on a manifold with

non-trivial cup-product has a minimal component (Theorem 18).

The mentioned Theorem 13 gives a simple yet powerful practical su.cient

condition for the presence of minimal components. Methods of calculating

h(M) for many important manifolds can be found in [7]; the most useful of

them are listed in Remark 14. For example, F on M2

g with rk > g = h(M2

g )

has a minimal component (Example 16), so does F on Tn (torus) with rk >

1 = h(Tn) (Example 15).

Yet the group ker[] gives more .ne-grained information on the foliation struc-

2

ture than the mere rk = rkim[]. We call a subgroup G H1(M) parallel

if there exists an isotropic subgroup H H1(M, Z) such that any homomorphism

. : G Z is realized by some element of H. If any of the following

equivalent conditions holds then F has a minimal component (Theorem 11):

(i) For any parallel subgroup G it holds rkG - rk(G ker[]) < rk (note

that non-strict inequality here holds for any group).

(ii) The same holds for any parallel subgroup G such that G ker[] = 0.

(iii) The same holds for any maximal parallel subgroup G.

Finally, the foliation F has a minimal component if and only if there exists

z H1(M) /ker[] such that z .[i] = 0 (intersection index) for all (compact)

leaves 1, . . . , M(), one from each maximal component (Theorem 7).

Note that cohomologous invariants of alone do not give much information

on the presence of minimal components, especially when it comes to necessary

conditions (for any form with rk 2 there is a cohomologous form with minimal

foliation [1]). So we had to bring into consideration some characteristics

of the manifold (h(M), parallel subgroups) and the foliation (i).

The paper is organized as follows. Section 2 introduces some de.nitions and

facts connected with Morse form foliation. Auxiliary Section 3 is devoted to

expressing H1(M) in terms of the foliation structure. In Section 4 we give

a criterion (Theorem 7) and a necessary condition for a foliation to have a

minimal component in terms of ker[]. Finally, in Section 5 we give su.cient

conditions for a foliation to have a minimal component in terms of ker[]

(Theorem 11), h(M) (Theorem 13), and cup-product (Theorem 18).

2 A Morse form foliation

In this section we introduce, for future reference, some useful notions and facts

about Morse forms and their foliations.

Recall that M is a connected smooth closed oriented n-dimensional manifold;

n 2. A closed 1-form on M is called a Morse form if it is locally the

di.erential of a Morse function. Sing = {p M | (p) = 0} denotes the set

of its singularities; this set is .nite since the singularities are isolated and M

is compact. On M / Sing the form de.nes a foliation F.

De.nition 1 A leaf F is called compacti.able if Sing is compact;

otherwise it is called non-compacti.able.

3

Note that a compact leaf is compacti.able. The number K() of non-compact

compacti.able leaves 0

i is .nite and can be estimated in terms of the number

of singularities of [8].

De.nition 2 A connected component C of the union of compact leaves is

called maximal component of the foliation.

A maximal component is open; the number M() of maximal components is

.nite and can be estimated in terms of homological characteristics of M and

the number of singularities of [8].

Consider the following decomposition into mutually disjoint sets holds:

M = _

M()

_i=1 Ci _ ., (1)

where Ci are all maximal components and

. = _

m()

_i=1

Cmin

i _ _

K()

_i=1

0

i _ Sing , (2)

Cmin

i being all minimal components of F and m() being their number. The

closed set . has a .nite number of connected components .j .

If Sing = . then F is either minimal or compact. In the latter case it has

exactly one maximal component C = M, which is a bundle over S1 with .ber

F [13].

In the rest of this paper we suppose Sing = .. In this case each maximal

component Ci is a cylinder over a compact leaf:

Ci = i (0, 1), (3)

where the di.eomorphism maps i to leaves of F; this map can be continuously

extended to i[0, 1] [8]. Since Ci . consists of one or two connected

components, each Ci adjoints one or two of .j . Therefore the decomposition (1)

allows representing M as the foliation graph ×a connected pseudograph (a

graph admitting multiple loops and edges) with edges Ci and vertices .j; an

edge Ci is incident to a vertex .j if Ci .j = .; see Figure 1.

De.nition 3 The group H generated by the homology classes of all compact

leaves is called the homology group of the foliation.

Since M is closed and oriented, the group Hn-1(M) is .nitely generated and

free; therefore so is H Hn-1(M).

4

.

1

C

1

.

1

.

4

.

4

C

4

C

4

C

1

C

3

C

3

C

2

C

2

.

2

.

2

.

3

.

3

Fig. 1. Decomposition (1) and the corresponding foliation graph.

A set of elements generating a free group might not contain its basis, e.g.,

Z = _2, 3. However:

Theorem 4 In H there exists a basis e consisting of homology classes of

leaves: e = {[1], . . . , [m]}, i F.

PROOF. Consider a spanning tree T of and the corresponding chords

h1, . . . , hm. We will show that e = {[1], . . . , [m]} is the desired basis, where

i is any leaf in the maximal component hi = i(0, 1) (all leaves in a maximal

component are homologous).

(i) The system e is independent. Indeed, let z be a cycle in the foliation graph

:

z = (p1, x1, . . . , ps, xs, ps+1), ps+1 = p1,

where xi = xj are edges connecting vertices pi, pi+1. For z, a closed curve

in M can be (non-uniquely) constructed from the elements of the cylinders

xi = i(0, 1) connected by segments lying in pi = .i; obviously [].[i] = 1.

For the chords h1, . . . , hm a system of cycles z1, . . . , zm in can be constructed

such that each hi belongs to exactly one cycle zi; denote 1, . . . , m the corresponding

closed curves in M. Then given _i ni[i] = 0, for any j it holds

0 = [j ] . _i ni[i] = nj .

(ii) _e = H. Indeed, consider a leaf such that its maximal component x /

{hi}. Then x T is a bridge connecting two di.erent (non-empty) connected

components: T - x = T_ T__, i.e. - (x {hi}) = T_ T__. The latter

means that {i} separate the two corresponding submanifolds in M, i.e.

[] + _iI

[i] = 0. _

In fact from the proof it follows that for every compact leaf , the coordinates

of [] in the basis e belong to {

1, 0}.

5

3 The manifolds homologies and the foliation

Recall that Ck = k (0, 1), k = 1, ...,M(), are all maximal components and

. = M / (_k Ck ). We will study the relationship between H1(M) and the

decomposition (1).

Theorem 5 Let z H1(M). If z . [k] = 0 for all k = 1, . . .,M() then

z i.H1(.), where i : . _ M.

PROOF. Let .k : kI M, I = (-1, 1) be the di.eomorphisms from (3),

with k = .k(k, 0) M.

Below we will show that z is realized by a closed curve that does not intersect

with any k. Given this, consider M_ = M / (_k k ); z j.H1(M_), j : M_ _

M. By (1),

M_ = . __k

.k _k (-1, 0)_ .k _k (0, 1)__.

Thus . is the deformation retract of M_, the corresponding homotopy on

M_ /. being rs _.k(x t)_ = .k _x (s + (1

s)t)_; recall that .k can be

continuously extended to k [-1, 1] with k {

1} .. This proves the

theorem.

It remains to show that z can be realized by a curve that does not intersect

with any k. Denote = k and . = .k. Let the orientation of be such that

.(x, t) goes along its normal vector as t increases.

Consider a closed curve realizing z, see Figure 2. Without loss of generality

we can assume that is transverse to = k and even that in a small enough

neighborhood U() it goes along the element I of the cylinder im..

Pi

+

Pi

Pi +1

+

Pi +1

+1

1

I

'

'

' '

' '

Pi Pi +1 0

Fig. 2. Removing intersection points of and .

Since [].[] = 0, it holds ၿ = 2p

i=1Pi, where _sgn Pi = 0. Suppose p = 0.

Consider Pi, Pi+1 such that sgn Pi = sgnPi+1 and let P-

i , P-

i+1; P+

i , P+

i+1

6

U(), where Pt

j = .(Pj, t). Since is connected, there is a curve PiPi+1 .

Obviously, [] = [_] + [__], where

_ = _ / (P-

i P+

i P+

i+1P-

i+1)_ P+

i P+

i+1 P-

i+1P-

i

and

__ = P-

i P+

i P+

i+1P-

i+1;

here P+

i P+

i+1 = .(PiPi+1,+) and P-

i+1P-

i = -.(PiPi+1,-). However,

[__] = 0 since __ is homotopy-equivalent to PiPi+1.

The new curve _ has 2p - 2 intersection points with = k. Induction by p

and then by k .nishes the proof. _

Theorem 6 Let e = {[1], . . . , [m]}, i F, be a basis of H Hn-1(M),

De = {D[1], . . .,D[m]} H1(M) a system of dual cycles, i.e. [i] .D[j] =

ij, and DH = _De. Then

H1(M) = _DH, i.H1(.).

Existence of e follows from Theorem 4.

PROOF. Let z H1(M) and ni = z . [i]. Consider the cycle z_ = z -

_niD[i]. Then z_ . [i] = 0 for any i = 1, . . .,m and therefore for any

i = 1, . . .,M(). By Theorem 5, z_ i.H1(.). _

4 Criterion and a necessary condition

Consider the map [] : H1(M) R, [](z) = _z . De.ne rk def = rkim[];

obviously, rk ker[] + rk = 1(M), the Betti number.

For a subgroup H Hn-1(M), denote H H1(M) the subgroup H =

{z H1(M) | z . H = 0}. Note that H1 H2 implies H 2 H 1 .

Theorem 7 F has a minimal component i. H ker[].

PROOF. Suppose F has no minimal components, so that (2) is reduced to

. = _

K()

_i=1

0

i _ Sing .

7

By Theorem 5, H = i.H1(.). Since _z = 0 for any z i.H1(.), we have

H ker[].

Suppose now F has a minimal component A. Consider p A and the leaf

p _ p. Through this point, in some its neighborhood Vp A a (local) integral

curve . A of the vector .eld , () = 1, can be drawn. Since . is transverse

to the leaves and the leaf p is dense in A, there exists a point q p .,

q = p. Let I Vp A be the segment of the integral curve between the points

p and q. The leaf p is connected, therefore there exists a curve J p joining

the points p and q. Then c = I J A is a closed curve and _c = _I = 0.

Since [c] . H = 0, we have H ker[]. _

This implies a necessary condition for F to have a minimal component:

Theorem 8 If F has a minimal component then for any set of compact

leaves 1, . . . , s F it holds

_[1], . . . , [s] ker[].

Example 9 ([6]) If a Morse form foliation on M2

g has g homologically independent

compact leaves then it has no minimal components. Indeed, choose

[1], . . . , [g],D[1], . . .,D[g] (dual 1-cycles) as a basis of H1(M2

g ). Let H =

_[1], . . . , [g]. Since [i] . D[j] = ij , H = H. Obviously, H ker[]. By

Theorem 8 the foliation has no minimal components.

5 Su.cient conditions

We call a subgroup H H1(M, Z) isotropic if u _ u_ = 0 (cup-product) for

any u, u_ H.

De.nition 10 A subgroup G H1(M) is called parallel if there exists an

isotropic subgroup H H1(M, Z) such that any homomorphism . : G Z is

realized by an element of H, i.e. there exists u H such that u|G = ..

Theorem 11 If any of the following equivalent conditions holds then F has

a minimal component:

(i) For any parallel subgroup G it holds

rkG - rk(G ker[]) < rk ; (4)

(ii) Inequality (4) holds for any parallel subgroup G such that Gker[] = 0;

(iii) Inequality (4) holds for any maximal parallel subgroup G.

8

Note that non-strict inequality in (4) holds for any subgroup G and any map

[] out of general group-theoretic considerations.

PROOF. Condition (i) implies existence of a minimal component. Indeed,

suppose F has no minimal components. Consider a group G = DH =

_D[1], . . .,D[m], where [1], . . . , [m] is a basis in H. By Theorem 6, rk =

rkG - rk(G ker[]). However, G = DH is parallel. Indeed, associate with

Hom(DH, Z) the subgroup H H1(M, Z), H = _u1, . . . , um, where ui(z) =

[i].z. Let D : H1(M, Z) Hn-1(M) be Poincare duality map. Then D(ui _

uj) = Dui .Duj = [i] . [j] = [i _j] = 0 since i _j = . for i = j; thus H

is isotropic.

(ii) (i). Let G be a parallel subgroup; G = G_ . (G ker[]) for some

(parallel) G_; then rkG - rk(G ker[]) = rkG_ < rk by (ii).

(iii) (ii). Let G be a parallel subgroup, G ker[] = 0. For a maximal

parallel subgroup H G, choose H_ G such that H = H_ . (H ker[]).

Then rkG rkH_ = rkH - rk(H ker[]) < rk by (iii). _

Example 12 Let M = T3

1 # T3

2 (3-dimensional tori), rk = 2, and ker[] H1(T3

2 ). For any parallel subgroup G such that Gker[] = 0 it holds rkG = 1.

By Theorem 11 (ii), F has a minimal component.

The following Theorem 13 gives a su.cient condition simpler and more practical,

though rougher, than Theorem 11.

Theorem 13 Let h(M) be the maximum rank of an isotropic subgroup in

H1(M, Z). If rk > h(M) then F has a minimal component.

PROOF. Since for any parallel subgroup H it holds rkH h(M), the theorem

follows from Theorem 11 (i). _

Remark 14 Some methods of calculating h(M) in terms of Betti numbers 1

and 2 can be found in [7], for instance:

(i) For r = rkker _ (cup-product H1(M, Z) H1(M, Z) H2(M, Z)),

1 + 2r

2 + 1 h(M)

12 + r

2 + 1

.

In particular, if 2 = 1 then h(M) = 1

2 (1+r); if r = 1 then h(M) = 1;

(ii) If _ is surjective, then

h(M) r +

1

2

+ _ 1 - r -

1

22

- 22;

9

(iii) For the product,

h(M1 M2) =max{h(M1), h(M2)};

(iv) For the connected sum with dimMi 2,

h(M1 # M2) = h(M1) + h(M2).

Example 15 For a torus Tn it holds h(Tn) = 1 and rk n. The foliation

has a minimal component if (Theorem 13) and only if [9] rk > 1.

On a torus, rk characterizes the topology of the foliation. This is, though,

not always the case:

Example 16 For M2

g it holds h(M2

g) = g and rk 2g. The foliation has

no minimal components if rk 1 [9] and has a minimal component if

g < rk 2g (Theorem 13). However, if 2 rk g, the topology of

the foliation may be quite di.erent even in the same cohomology class. For

instance, while in any cohomology class with rk 2 there exists a form with

minimal foliation [1], for any 1 rk g there exists F without minimal

components.

Indeed, consider g tori Ti = M_ i S1, M_ i = S1, with a form i = i dt on

Ti, where t is the coordinate along the S1; Fi is compact. This form can be

locally transformed into a form _i

with some spherical singularities. Using

small spheres around these singularities, a connected sum M2

g = #g

i=1 Ti can

be constructed with i smoothly pasted together into a form on M2

g ; 1

rk = rk{i} g and F has no minimal components.

Consider a Morse form in general position, i.e., with all periods being incommensurable;

rk = 1(M). The foliation of such a form can have no minimal

components: for example, if 1(M) = 0 then all closed forms on M are exact.

What is more, for any given n 3 and k 0 there exists a manifold M,

dimM = n and 1(M) = k, with a form in general position such that F

has no minimal components:

Example 17 The manifold M = #k

i=1Mi and constructed as in Example 16

(Mi standing for Ti and M for M2

g) with M_ i = Sn-1 and rk{i} = k have the

desired properties. Note that here 2(M) = 0; however, by appropriate choice

of M_1

, 1(M_1

) = 0, a similar example can be constructed for any given set of

Betti numbers.

Theorem 18 Let be a Morse form in general position. If _: H1(M, Z)

H1(M, Z) H2(M, Z) is non-trivial then F has a minimal component.

10

PROOF. If _ is non-trivial then h(M) < 1(M) = rk. By Theorem 13,

F has a minimal component. _

In addition, on M2

g all compact leaves of F with in general position are

homologically trivial. Indeed, consider [] = _nizi, where {zi} is the basis of

cycles. Since _ = _ni _zi = 0 and _zi are incommensurable, all ni = 0.

Acknowledgements

The author thanks the anonymous reviewer for constructive criticism and a

number of useful suggestions.

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