Closed 1-forms in topology and dynamics

Closed 1-forms in topology and dynamics

Michael Farber and Dirk Schütz Department of Mathematics, University of Durham, Durham DH1 3LE, UK Michael.Farber@durham.ac.uk Department of Mathematics, University of Durham, Durham DH1 3LE, UK dirk.schuetz@durham.ac.uk
July 18, 2019
Abstract.

This article surveys recent progress of results in topology and dynamics based on techniques of closed one-forms. Our approach allows us to draw conclusions about properties of flows by studying homotopical and cohomological features of manifolds. More specifically we describe a Lusternik - Schnirelmann type theory for closed one-forms, the focusing effect for flows and the theory of Lyapunov one-forms. We also discuss recent results about cohomological treatment of the invariants and and their explicit computation in certain examples.

To S.P. Novikov on the occasion of his 70-th birthday


In 1981 S.P. Novikov [37, 40] initiated a generalization of Morse theory which gives topological estimates on the numbers of zeros of closed 1-forms, see also [42, 43]. Novikov was motivated by a variety of important problems of mathematical physics, leading, in one way or another, to the problem of finding relations between the topology of the underlying manifold and the number of zeros which closed 1-forms in a specific one-dimensional real cohomology class possess. Recall that a closed 1-form can be viewed as a multi-valued function or functional whose branching behavior is fully characterized by the cohomology class.

In [40] Novikov studies various problems of physics where the motion can be reduced to the principle of an extremal action , which is a multi-valued functional on the space of curves, with the variation being a well-defined closed 1-form, cf. [39, 38, 41]. Among problems of this kind are the Kirchhoff equations for the motion of a rigid body in ideal fluid, the Leggett equation for the magnetic momentum, and others.

His fundamental idea in [40] was based on a plan to construct a chain complex, now called the Novikov complex, which uses dynamics of the gradient flow in the abelian covering associated with the cohomology class. The dynamics of gradient flows appears traditionally in Morse theory providing a bridge between the critical set of a function and the global ambient topology.

At present the Novikov theory of closed one-forms is a rapidly developing area of topology which interacts with various mathematical theories. J.-Cl. Sikorav [56, 57] was the first who applied Novikov theory in symplectic topology. Hofer and Salamon [29] pioneered exploiting the ideas of Novikov theory in Floer theory. More recent applications of Novikov theory in symplectic topology and Hamiltonian dynamics can be found, for example, in the work of Oh [44], Usher [63] and references mentioned there. Combinatorial group theory is another area where Novikov theory plays an important role; this connection was also discovered by J.-Cl. Sikorav who realized that the Bieri - Neumann - Strebel invariant can be expressed in terms of (a generalized noncommutative) Novikov homology.

Many problems of Novikov theory are still being actively developed in current research. The list of such topics includes (a) constructions of chain complexes (more general than the Novikov complex) which are able to capture the link between topology of the manifold and topology of the set of zeros, (b) various types of inequalities for closed 1-forms (with Morse or Bott type nondegeneracy assumptions), (c) equivariant inequalities and (d) problems about sharpness of these inequalities.

Two recent monographs [19], [48] give expositions of Novikov theory from quite different angles.

Topology of closed 1-forms is a broader research area which together with the Novikov theory studies a new Lusternik - Schnirelmann type theory for closed 1-forms initiated in 2002 in [16]. The latter theory also aims at finding relations between topology of the zero set of a closed one-form and homotopy information, based mainly on the cohomology class of the form. The words “Lusternik - Schnirelmann type”  intend to emphasize that no assumptions on the character of zeros are made, unlike the ones which appear in the Novikov theory requiring that all zeros are nondegenerate in the sense of Morse.

Although Lusternik - Schnirelmann theory for closed one-forms shares many common features with the Lusternik - Schnirelmann theory of functions and with the Novikov theory of closed one-forms, it is also very different from both these classical theories. The most striking new phenomenon is based on the fact that in any nonzero cohomology class there always exists a closed 1-form having at most one zero (see Theorem 4 below). Hence, the new theory is not merely about the number of zeros but, as we show in this article, about qualitative dynamical properties of smooth flows on manifolds.

Given a smooth flow one wants to “tame” it by a closed 1-form lying in a prescribed cohomology class. More precisely, one wants to find a closed 1-form such that the flow is locally decreasing. This idea leads to the notion of a Lyapunov closed one-form generalizing the classical notion of a Lyapunov function. The key question about the existence of Lyapunov closed 1-forms for flows has been resolved in a series of papers of Farber, Kappeler, Latschev and Zehnder [17, 20, 21, 33, 34]. Note that the classical theorem of C. Conley [5, 6] gives the answer in the special case of the zero cohomology class, i.e. when one deals with Lyapunov functions. When the cohomology class is nontrivial the notion of asymptotic cycle of a flow introduced by Schwartzman [53, 54] plays a crucial role.

A brief account of Lusternik - Schnirelmann theory for closed one-forms can be found in chapter 10 of the book [19] published in 2004. The main idea of this paper is to survey the new results obtained after 2004. To put the material in context we start this survey with a section describing the basic results of the Novikov theory.

In a forthcoming paper [27] we introduce and study the notion of sigma invariants for a finite CW-complex analogously to the sigma invariants of groups of Bieri et al [2, 3]. These invariants carry information about the finiteness properties of infinite abelian covers of . Both the present survey and [27] have a common main theme: they are based on movability properties of subsets of with respect to a closed 1-form which are very close in spirit to the idea of Novikov homology.


Table of contents

1. Fundamentals of the Novikov theory

2. The colliding theorem

3. Closed 1-forms on general topological spaces

4. Lyapunov 1-forms for flows

5. Notions of category with respect to a cohomology class

6. Focussing effect

7. Existence of flow-convex neighbourhoods

8. Proof of Theorem 12

9. Topology of the chain recurrent set

10. Proof of Theorem 14

11. Cohomological estimates for

12. Upper bounds for and relations with the Bieri - Neumann - Strebel invariants

13. Homological category weights, estimates for and calculation of , for products of surfaces

1. Fundamentals of the Novikov theory

S.P. Novikov [37], [40] suggested a generalization of the classical Morse theory which gives lower bounds on the number of zeros of Morse closed 1-forms. In this section we will give a brief account of the Novikov theory. We touch here only the following selected topics: Novikov inequalities, Novikov numbers, and Novikov Principle leading to the notion of the Novikov complex. We refer the reader to the original papers [37, 40] and to the monograph [19] for proofs and more details. Historical information about the development of the subject after [37, 40] as well as bibliographic references can also be found in [19].

1.1. Novikov inequalities

Let be a smooth manifold. A smooth closed 1-form on is defined as a smooth section of the cotangent bundle satisfying . By the Poincaré lemma for any simply connected open set one has where is a smooth function, defined uniquely up to addition of a locally constant function. The zeros of are points such that . If lies in a simply connected domain then if and only if is a critical point of .

Definition 1.

A zero of a smooth closed 1-form is said to be non-degenerate if it is a non-degenerate critical point of the function .

Clearly, this property is independent of the choice of the simply connected domain and the function .

Definition 2.

The Morse index of a non-degenerate zero of is defined as the Morse index of viewed as a critical point of .

We denote the Morse index by . It takes values where .

The main problem of the Novikov theory is as follows. Let be a smooth closed one-form on a closed smooth manifold . Let us additionally assume that is Morse, i.e. all its zeros are nondegenerate in the sense explained above. We denote by the number of zeros of having Morse index , where . One wants to estimate the numbers in terms of information about the topology of and of the cohomology class

(1)

represented by .

In the case of classical Morse theory one has (i.e. where is a smooth function) and the answer is given by the Morse inequalities

(2)

where denotes the -th Betti number of and denotes the minimal number of generators of the torsion subgroup of .

S.P. Novikov [37, 40] introduced generalizations of the numbers and which depend on the cohomology class of (see (1)) and are denoted and correspondingly. We call the Novikov - Betti number; the number is the Novikov torsion number. Their definitions will be given below. The Novikov inequality (in its simplest form) states:

Theorem 1.

Let be a smooth closed 1-form on a smooth closed manifold . Assume that all zeros of are nondegenerate. Then

(3)

where denotes the cohomology class represented by .

Note that for the number coincides with and the number coincides with , i.e. the Novikov inequalities (3) contain the Morse inequalities (2) as a special case.

Our goal in this section is to explain the main ideas of the proof of Theorem 1 without intending to present full details. A complete proof can be found in [19], pages 46 - 48.

Next we introduce the Novikov ring where is a commutative ring. Elements of are formal “power series”  of the form

where is a formal variable, the coefficients are elements of , , and the exponents are arbitrary real numbers satisfying the following condition: for any the set

(4)

is finite. Equivalently, an element can be represented in the form

where , , and tends to . In other words, element of the Novikov ring are “Laurent like”  power series with integral coefficients and with arbitrary real exponents tending to .

Addition in is given by adding the coefficients of powers of with equal exponents. Multiplication in is given by the formula

where

Note that the last sum contains only finitely many nonzero terms and moreover, the set of all the exponents for which is nonzero also satisfies the property that the sets (4) are finite.

For simplicity, we will abbreviate the notation to .

Lemma 1.

The ring is a principal ideal domain. Moreover, for any field the ring is a field.

The second statement is in fact trivial; the proof of the first statement can be found in [19], page 8. The credit for Lemma 1 should be mainly given to S.P. Novikov who discovered and stated it without proof. Full proofs appear in [29], [45] and [19]. Similar statements concerning related rings of rational functions were discovered in [12], [56].

Now we can explain how one associates to a cohomology class the Novikov numbers and . Here we assume that is a smooth closed manifold although the construction is applicable to finite polyhedra. Consider the group ring where denotes the fundamental group . The class determines a ring homomorphism

(5)

defined on a group element by the formula

Here is the evaluation of the cohomology class on the homotopy class . Recall that denote the formal indeterminant of the Novikov ring. Clearly,

i.e. is multiplicative on group elements. It follows that extends by linearity on the whole group ring as a ring homomorphism. By the well-known construction homomorphism (5) defines a local system of left -modules over which we denote by . The homology of this local system is a finitely generated module over the Novikov ring . As we know, is a principal ideal domain, and therefore is a direct sum of a free and a torsion -module.

Definition 3.

The Novikov-Betti number is defined as the rank of the free part of . The Novikov torsion number is defined as the minimal number of generators of the torsion submodule of .

Recall the definition of homology of local system . Consider the universal cover and the singular chain complex . The latter is a chain complex of free left modules over the group ring . One views as a right -module where

Then is the homology of the chain complex

(6)

1.2. Novikov and Universal complexes

The main idea of S.P. Novikov in [37, 40] which finally led him to Theorem 1 was the statement that for any Morse closed 1-form on a smooth closed manifold there exists a chain complex (which is now known as the Novikov complex) having the following two properties:

  1. is a free chain complex of -modules and each module has a canonical free basis which is in one-to-one correspondence with zeros of having Morse index , where .

  2. is chain homotopy equivalent to the complex . In particular, the homology is isomorphic to .

This statement, which we call the the Novikov Principle, trivially implies Theorem 1. It should be compared with the classical Morse Principle which claims that for any Morse function on a closed smooth manifold there exists a chain complex having the following two properties:

  1. is a free chain complex of -modules and each module has a canonical free basis which is in one-to-one correspondence with critical points of having Morse index , where .

  2. is chain homotopy equivalent to the chain complex , where is the universal cover of and .

There are several explicit constructions of Morse theory which lead to the complex . One of them is based on the fact that admits a cell-decomposition with cells in one-to-one correspondence with critical points of . Another well-known construction of is based on the Witten deformation of the de Rham complex.

One is naturally led to ask if there exist other ring homomorphisms

(7)

distinct from (5), for which the analogue of the Novikov Principle holds. To be more specific, we say that the Novikov Principle is valid for a group , a group homomorphism111One has for and a ring homomorphism if for any smooth closed manifold with and for any Morse closed 1-form on representing the cohomology class there exists a chain complex

of free finitely generated -modules having the following two properties:

  1. each module has a canonical free basis which is in one-to-one correspondence with the zeros of having Morse index ;

  2. The complex is chain homotopy equivalent to , where is viewed as a left -module via .

A positive answer to this question was given in [13] in the case of rational cohomology classes and in [15] in the general case. Besides (5) the Novikov principle holds for many other ring homomorphisms (7). Moreover, there exists “the largest”  such homomorphism which we describe below. The chain complex in this case is called the Universal complex. This complex lives over a localization of the group ring .

An element will be called -negative if (finite sum), where and for all . An -matrix over the group ring will be called -negative if all its entries are -negative. Consider the class of square matrices of the form , where is an arbitrary -negative square matrix with entries in .

Note that in the case (when one studies Morse functions) the set of -negative matrices is empty and so the Cohn localization (8) is just the identity map.

We will use the notion of noncommutative localization developed by P.M. Cohn [4]. The universal Cohn localization of the group ring with respect to the class is a ring together with a ring homomorphism

(8)

satisfying the following two properties: first, any matrix , where is -negative, is invertible over , and, secondly, it is a universal homomorphism having this property, i.e., for any ring homomorphism , inverting all matrices of the form , where is -negative, there exists a unique ring homomorphism such that the following diagram commutes

We refer to the book of P.M. Cohn [4], where the existence of localization (8) is proven.

Theorem 2.

Let be any group and be a cohomology class. Then the Novikov Principle is valid for the Cohn localization (8).

Farber and Ranicki in [13] proved this theorem in the case of integral cohomology classes; the general case was treated by Farber [14]. The construction of the Universal complex employs the technique of collapse for chain complexes which is analogous to the well-known operation of combinatorial collapse. This technique was initiated in [13].

One should mention another important closely related ring called the Novikov-Sikorav completion ; it was first introduced by J.-Cl. Sikorav [56] who was inspired by the construction of the Novikov ring . One can view as the noncommutative analogue of the Novikov ring. Elements of are represented by formal sums, possibly infinite,

where and , satisfying the following condition: for any real number the set is finite. Compare this with the construction of the Novikov ring above. Addition and multiplication are given by the usual formulae; for example, the product of and is given by

(9)

The ring homomorphism is the inclusion. If is a -negative square matrix over the ring , then the power series

converges in and hence the matrix is invertible in . From the universal property of the Cohn localization it follows that there exists a canonical ring homomorphism

(10)

extending the inclusion . This implies that the homomorphism (8) is injective.

It is not known whether (10) is always injective. Intuitively, the image of (10) consists of “rational power series”. This is a consequence of the characterization of elements of the Cohn localization as components of solutions of a linear system of equations; cf. [4], Chapter 7.

1.3. Novikov numbers and the fundamental group

It is obvious that one-dimensional Novikov numbers and depend only on the fundamental group and on the homomorphism of periods . Interestingly there is also a strong inverse dependence, i.e. one may sometimes recover information about the fundamental group while studying and . The following result shows that nontriviality of the first Novikov - Betti number happens only if the fundamental group is “large”:

Theorem 3 (Farber-Schütz [22]).

Let be a connected finite complex and let be a nonzero cohomology class. If the first Novikov-Betti number is positive then contains a nonabelian free subgroup.

Theorem 3 has the following Corollaries:

Corollary 2.

Let be a connected finite polyhedron having an amenable fundamental group. Then the first Novikov-Betti number vanishes for any .

Corollary 3.

Assume that is a connected finite two-dimensional polyhedron. If the Euler characteristic of is negative then contains a nonabelian free subgroup.

Corollary 3 follows from Theorem 3 and from the Euler -Poincarè theorem for the Novikov numbers

The proof of Theorem 3 given in [22] is based on the results of R. Bieri, W. Neumann, R. Strebel [2] and J.-Cl. Sikorav [56].

We refer the reader to articles [46], [47], [49] and [50] for additional information.

2. The colliding theorem

As follows from the discussion of the previous section, the Novikov theory gives bounds from below on the number of distinct zeros which have closed Morse type 1-forms lying in a prescribed cohomology class . The total number of zeros is then at least the sum of the Novikov numbers .

If is a closed 1-form representing the zero cohomology class then where is a smooth function; in this case must have at least geometrically distinct zeros (which are exactly the critical points of the function ), according to the classical Lusternik-Schnirelmann theory [8]; this result requires no assumptions on the nature of the zeros.

Encouraged by the success of the Novikov theory one may ask if it is possible to construct an analogue of the Lusternik - Schnirelmann theory for closed 1-forms. It is quite surprising that in general, with the exception of two situations mentioned above, there are no obstructions for constructing closed 1-forms possessing a single zero. Hence, for , the only homotopy invariant (depending on the topology of and on the cohomology class ) such that any closed 1-form on with has at least zeros is or .

Theorem 4 (Farber [16], Farber - Schütz [23]).

Let be a closed connected -dimensional smooth manifold, and let be a nonzero real cohomology class. Then there exists a smooth closed 1-form in the class having at most one zero.

This result suggests that “the Lusternik-Schnirelmann theory for closed 1-forms”  (if it exists) must have a new character, quite distinct from both the classical Lusternik-Schnirelmann theory of functions and the Novikov theory of closed 1-forms.

Theorem 4 was proven in [16] under an additional assumption that the class is integral, ; see also [19], Theorem 10.1. In full generality it was proven in [23].

Let us mention briefly a similar question. We know that if then there exists a nowhere zero 1-form on . Given , one may ask if it is possible to find a nowhere zero 1-form on which is closed, i.e. ? The answer is negative in general. For example, vanishing of the Novikov numbers is a necessary condition for the class to be representable by a closed 1-form without zeros. The full list of necessary and sufficient conditions (in the case ) is given by the theorem of Latour [35].

Theorem 4 was proven in [16] and [19] in the case when the cohomology class is of rank 1. In this section we follow our recent paper [23] and give the proof of Theorem 4 in the case when .

2.1. Singular foliations of closed one-forms

Let be a smooth manifold. A smooth closed 1-form with Morse zeros determines a singular foliation on . It is a decomposition of into leaves: two points belong to the same leaf if there exists a path with , and for all . Locally, in a simply connected domain , we have , where ; each connected component of the level set lies in a single leaf. If is small enough and does not contain the zeros of , one may find coordinates in such that ; hence the leaves in are the sets . Near such points the singular foliation is a usual foliation. On the contrary, if is a small neighborhood of a zero of having Morse index , then there are coordinates in such that and the leaves of in are the level sets The leaf with contains the zero . It has a singularity at : a neighborhood of in is homeomorphic to a cone over the product . There are finitely many singular leaves, i.e. the leaves containing the zeros of .

We are particularly interested in the singular leaves containing the zeros of having Morse indices 1 and . Removing such a zero locally disconnects the leaf . However globally the complement may or may not be connected.

The singular foliation is co-oriented: the normal bundle to any leaf at any nonsingular point has a specified orientation.

We shall use the notion of a weakly complete closed 1-form introduced by G. Levitt [36]. A closed 1-form is called weakly complete if it has Morse type zeros and for any smooth path with the endpoints and lie in the same leaf of the foliation on . Here denotes where are the zeros of .

A weakly complete closed 1-form with has no zeros with Morse indices and . According to Levitt [36], any nonzero real cohomology class can be represented by a weakly complete closed 1-form.

The plan of our proof of Theorem 4 is as follows. We start with a weakly complete closed 1-form lying in the prescribed cohomology class , . We show that assuming all leaves of the singular foliation are dense (see §2.2). We perturb such that the resulting closed 1-form has a single singular leaf (see §2.3). After that we apply the technique of Takens [62] allowing us to collide the zeros in a single (highly degenerate) zero. We first prove Theorem 4 assuming that ; the special case is treated separately later.

2.2. Density of the leaves

In this section we show that if is weakly complete and then the leaves of are dense; moreover, given a point and a leaf of the singular foliation , there exist two sequences of points and such that , and

(11)

The integrals in (11) are calculated along an arbitrary path lying in a small neighborhood of . This can also be expressed by saying that the leaf approaches from both the positive and the negative sides. This statement was also observed by G. Levitt [36], p. 645; we give below a different proof. In general the assumptions that has no centers and do not imply that the leaves of the foliation are dense, see the examples in Chapter 11, §9.3 of [19].

Let be a weakly complete closed 1-form in class . Consider the covering corresponding to the kernel of the homomorphism of periods , where . Let be the group of periods. The rank of equals ; since we assume that , the group is dense in . The group of periods acts on the covering space as the group of covering transformations. We have where is a smooth function. The leaves of the singular foliation are the images under the projection of the level sets ; this property follows from the weak completeness of , see [36], Proposition II.1. For any and one has

(12)

Let be a leaf and let be an arbitrary point. Our goal is to show that lies in the closure of . Let be a path-connected neighborhood of which is assumed to be “small”  in the following sense: . We want to show that contains a point satisfying where the integral is calculated along a curve in .

Consider a lift , . Let be a neighborhood of which is mapped by homeomorphically onto . We claim that the set of values contains an interval where and .

This claim is obvious if is not a critical point of since in this case one may choose the coordinates around such that . In the case when is a critical point of , one may choose the coordinates near the point such that is given by and our claim follows since we know that the Morse index is distinct from and .

Because of the density of the group of translations one may find such that the real number lies in the interval . Then we obtain

(13)

We see that the sets and have a common point . The point has the required property .

2.3. Modification

Our next goal is to replace by a Morse closed 1-form which has the property that all its zeros lie on the same singular leaf of the singular foliation . In this section we assume that .

Let be a weakly complete Morse closed 1-form in class where . Let be the zeros of . For each choose a small neighborhood and local coordinates in such that for and

(14)

Here denotes the Morse index of . We assume that the ball is contained in and that for . Denote by the open ball .

Let be a smooth function with the following properties: (a) on ; (b) on ; (c) .

Such a function exists assuming that is small enough. (a), (b), (c) imply that

(15)

We replace the closed 1-form by

(16)

where is a smooth function with support in . In the coordinates of (see above) the function is given by The parameters appearing in (16) are specified later.

One has on and near the zeros of . Let us show that has no additional zeros. We have (where is defined in (14)) and

(17)

If this partial derivative vanishes and then which may happen only for according to (15).

We now show how to choose the parameters so that the closed 1-form given by (16) has a unique singular leaf. Let be a fixed nonsingular leaf of . Since is dense in (see §2.2) for any the intersection contains infinitely many connected components approaching from below and from above and the function is constant on each of them.

We say that a subset is a level set  if for some . Note that . The level set contains the zero ; it is homeomorphic to the cone over the product . Each level set with is diffeomorphic to and each level set with is diffeomorphic to . Recall that denotes the Morse index of .

Let denote the set of values of on different level sets belonging to the leaf . The zero does not lie in since we assume that the leaf is nonsingular. However, according to the result proven in §2.2, the zero is a limit point of and, moreover, the closure of either of the sets and contains .

For the modification (given by (16)) one has where . The level sets for are defined as . Clearly is given by the equation

Hence for this is the same as ; for the level set coincides with . In the ring the level set is homeomorphic to a cylinder.

The following figure illustrates the distinction between the level sets and in the case .

Examine the changes which the leaf undergoes when we replace by . Here we view with the leaf topology; it is the topology induced on from the covering using an arbitrary lift . First, let us assume that: (1) the Morse index satisfies ; (2) the coefficient is positive; (3) the number lies in the set . Then the complement

is connected and it lies in a single leaf of the singular foliation . We see that the new leaf is obtained from by infinitely many surgeries. Namely, each level set , where satisfies , is removed and replaced by a copy of ; besides, the set where , is removed and gets replaced by a cone over the product . Hence the new leaf contains the zero .

Let us now show how one may modify the above construction in the case . Since we have in this case ; hence removing the sphere from the leaf does not disconnect . We shall assume that the coefficient is negative and that the number lies in . The complement

is connected and it lies in a single leaf of the singular foliation . Clearly, is obtained from by removing the level sets where satisfies (each such is diffeomorphic to ) and by replacing them by copies of . In addition, the set where , is removed and is replaces by a cone over the product .

We see that is a leaf of the singular foliation containing all the zeros .

2.4. Proof of Theorem 4

Below we assume that . The case is covered by Theorem 2.1 from [16].

The results of the preceding sections allow us to complete the proof of Theorem 4 in the case . Indeed, we showed in §2.3 how to construct a Morse closed 1-form lying in the prescribed cohomology class such that all zeros of are Morse and belong to the same singular leaf of the singular foliation . Now we may apply the colliding technique of F. Takens [62], pages 203–206. Namely, we may find a piecewise smooth tree containing all the zeros of . Let be a small neighborhood of which is diffeomorphic to . We may find a continuous map with the following properties:

is a single point ;

is a diffeomorphism onto ;

is the identity map on the complement of a small neighborhood of where the closure is contained in .

Consider a smooth function such that ; it exists and is unique up to a constant. The function is well-defined (since is constant). is continuous by the universal property of the quotient topology. Moreover, is smooth on . Applying Theorem 2.7 from [62], we see that we can replace by a smooth function having a single critical point at and such that on .

Let be a closed 1-form on given by

(18)

Clearly is a smooth closed 1-form on having no zeros in . Moreover, lies in the cohomology class (since any loop in is homologous to a loop in ).

Now we prove Theorem 4 the case . We shall replace the construction of §2.3 (which requires ) by a direct construction. The final argument using the Takens’ technique [62] remains the same.

Let be a closed surface and let be a nonzero cohomology class. We can split into a connected sum

where each is a torus or a Klein bottle and such that the cohomology class is nonzero. Let be a closed 1-form on lying in the class and having no zeros; obviously such a form exists. §9.3.2 of [19] describes the construction of connected sum of closed 1-forms on surfaces. Each connecting tube contributes two zeros. In fact there are three different ways of forming the connected sum, they are denoted by A, B, C on Figure 9.8 in [19]. In the type C connected sum the zeros lie on the same singular leaf. Hence by using the type C connected sum operation we get a closed 1-form on having zeros which all lie on the same singular leaf of the singular foliation . The colliding argument based on the technique of Takens [62] applies as in the case and produces a closed 1-form with at most one zero lying in class .

3. Closed one-forms on general topological spaces

Since closed one-forms are central for our constructions, it will be convenient to operate with the notion of a closed one-form defined on general topological spaces. This notion will allow us to deal with spaces more general than smooth manifolds. The calculus of closed one-forms on topological spaces is very similar to that of smooth closed one-forms on manifolds: one may integrate any closed one-form along a path; the integral depends only on the homotopy class of the path; any closed one-form represents a one-dimensional cohomology class and any continuous function determines an exact closed one-form , the differential of .

In this section we recall the basic definitions referring to the book [19] for proofs and more details.

3.1. Basic definitions

Definition 4.

A continuous closed 1-form on a topological space is defined as a collection of continuous real-valued functions , where is an open cover of , such that for any pair the difference

is a locally constant function. Another such collection (where is another open cover of ) defines an equivalent closed 1-form if the union collection

is a closed 1-form (in the sense of the above definition), i.e., if for any and , the function is locally constant on .

Let be a continuous map and let be a continuous closed 1-form on . Then the induced closed 1-form is defined as follows. Let , where is an open cover of . The family is an open cover of and the functions define a continuous closed 1-form with respect to the cover .

As an example consider an open cover consisting of the whole space . Then any continuous function defines a closed 1-form on , which is denoted by . For two continuous functions , holds if and only if the difference is locally constant.

3.2. Integration

One may integrate continuous closed 1-forms along continuous paths. Let be a continuous closed 1-form on given by a collection of continuous real-valued functions with respect to an open cover of . Let be a continuous path. The line integral is defined as follows. Find a subdivision of the interval such that for any the image is contained in a single open set . Then we define

(19)

The standard argument shows that the integral (19) does not depend on the choice of the subdivision and the open cover .

For any pair of continuous paths with common beginning and common end points , it holds that

provided that and are homotopic relative to the boundary.

3.3. Cohomology class of a closed one-form

Any continuous closed 1-form defines the homomorphism of periods

(20)

given by integration along closed loops with . The image of this homomorphism is a subgroup of whose rank is called the rank of and is denoted .

Recall that a topological space is homologically locally -connected if for every point and any neighborhood of there exists a neighborhood of in such that the induced homomorphism of the reduced integral singular homology is trivial for all . is locally path connected iff it is homologically locally 0-connected.

Lemma 4.

Let be a locally path-connected topological space. A continuous closed 1-form on equals for a continuous function if and only if for any choice of the base point the homomorphism of periods (20) determined by is trivial.

Proof.

If , then holds for any path in , where . Hence if is a closed loop.

Conversely, assume that the homomorphism of periods (20) is trivial. Our assumption about implies that connected components of are open and path connected. In each connected component of choose a base point . One defines a continuous function