CLOSED 1-FORMS WITH AT MOST ONE ZERO

M. FARBER, D. SCHÜTZ

Abstract. We prove that in any nonzero cohomology class ξ ∈ H^{1}(M; R) there always exists a closed 1-form having at most one zero.

1. Statement of the result

Let M be a closed connected smooth manifold. By Hopf’s theorem, there exists a nowhere zero tangent vector field on M if and only if χ(M) = 0. If χ(M) ≠ 0 one may find a tangent vector field on M vanishing at a single point p ∈ M. A Riemannian metric on M determines a one-to-one correspondence between vectors and covectors; therefore on any closed connected manifold M there exists a smooth 1-form ω vanishing at most at one point p ∈ M. The question we address in this note is whether the 1-form ω which is nonzero on M − {p} can be chosen to be closed, dω = 0?

The Novikov theory [8] gives bounds from below on the number of distinct zeros which have closed 1-forms ω lying in a prescribed cohomology class ξ ∈ H^{1}(M; R). However the Novikov theory imposes an additional requirement that all zeros of ω are non-degenerate in the sense of Morse. The number of zeros is then at least the sum ∑_{j} b_{j}(ξ) of the Novikov numbers b_{j}(ξ).

If ω is a closed 1-form representing the zero cohomology class then ω = df where f : M → R is a smooth function; in this case ω must have at least cat(M) geometrically distinct zeros, according to the classical Lusternik-Schnirelman theory [1].

Our goal in this paper is to show that in general, with the exception of two situations mentioned above, there are no obstructions for constructing closed 1-forms possessing a single zero. We prove the following statement:

Theorem 1. Let M be a closed connected n-dimensional smooth manifold, and let ξ ∈ H^{1}(M; R) be a nonzero real cohomology class. Then there exists a smooth closed 1-form ω in the class ξ having at most one zero.

This result suggests that “the Lusternik-Schnirelman theory for closed 1forms” (see [3, 4] and Chapter 10 of [5]) has a new character which is quite distinct from both the classical Lusternik-Schnirelman theory of functions and the Novikov theory of closed 1-forms.

Date: August 31, 2004.

1991 Mathematics Subject Classification. 58E05, 57R70.

Key words and phrases. Closed 1-form, Lusternik-Schnirelman theory, Novikov theory.

The authors thank the Max-Planck Institute for Mathematics in Bonn for hospitality.

Theorem 1 was proven in [3] under an additional assumption that the class ξ is integral, ξ ∈ H^{1}(M; Z). See also [5], Theorem 10.1. This essentially covers all rank 1 cohomology classes ξ ∈ H^{1}(M; R) since any such class is a multiple of an integral class.

Theorem 1 has interesting implications in the theory of symplectic intersections, compare [9], [2]. Y. Eliashberg and M. Gromov mention in [2] that a statement in the spirit of Theorem 1 was made by Yu. Chekanov at a seminar talk in 1996. No written account of his work is available.

Let us mention briefly a similar question. We know that if χ(M) = 0 then there exists a nowhere zero 1-form ω on M. Given χ(M) = 0, one may ask if it is possible to find a nowhere zero 1-form ω on M which is closed dω = 0? The answer is negative in general. For example, vanishing of the Novikov numbers b_{j}(ξ) = 0 is a necessary condition for the class ξ to be representable by a closed 1-form without zeros. The full list of necessary and sufficient conditions (in the case dim M > 5) is given by the theorem of Latour [6].

2. Preliminaries

Here we recall some basic terminology. We refer to [5] for more detail. A smooth 1-form ω is a smooth section x ↦ ω_{x}, x ∈ M of the cotangent

bundle T^{∗}(M) → M. A zero of ω is a point p ∈ M such that ω_{p} = 0.

If ω is a closed 1-form on M, i.e. dω = 0, then in any simply connected domain U ⊂ M there exists a smooth function f : U → R such that ω|_{U} = df. Zeros of ω are precisely the critical points of f. A zero p ∈ M, ω_{p} = 0 is said to be Morse type iff p is a Morse type critical point for f.

The homomorphism of periods

Per_{ξ} : H_{1}(M) → R (1)

is defined by

Per_{ξ}([γ]) = ∫_{γ} ω ∈ R. (2)

Here ξ = [ω] ∈ H^{1}(M; R) is the de Rham cohomology class of ω and γ is a closed loop in M; the symbol [γ] ∈ H_{1}(M) denotes the homology class of γ.

The image of the homomorphism of periods (1) is a finitely generated free abelian subgroup of R; it is called the group of periods. Its rank is denoted rk(ξ) – the rank of the cohomology class ξ ∈ H^{1}(M; R).

A closed 1-form ω with Morse zeros determines a singular foliation ω = 0 on M. It is a decomposition of M into leaves: two points p, q ∈ M belong to the same leaf if there exists a path γ : [0, 1] → M with γ(0) = p, γ(1) = q and ω(γ̇(t)) = 0 for all t. Locally, in a simply connected domain U ⊂ M, we have ω|_{U} = df, where f : U → R; each connected component of the level set f^{−1}(c) lies in a single leaf. If U is small enough and does not contain the zeros of ω, one may find coordinates x_{1}, …, x_{n} in U such that f ≡ x_{1}; hence the leaves in U are the sets x_{1} = c. Near such points the singular foliation ω = 0 is a usual foliation. On the contrary, if U is a small neighborhood of a zero p ∈ M of ω having Morse index 0 ≤ k ≤ n, then there are coordinates x_{1}, …, x_{n} in U such that x_{i}(p) = 0 and the leaves of ω = 0 in U are the level sets −x_{1}^{2} − ⋯ − x_{k}^{2} + x_{k+1}^{2} + ⋯ + x_{n}^{2} = c. The leaf L with c = 0 contains the zero p. It has a singularity at p: a neighborhood of p in L is homeomorphic to a cone over the product S^{k−1} × S^{n−k−1}. There are finitely many singular leaves, i.e. the leaves containing the zeros of ω.

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

The singular foliation ω = 0 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 [7]. A closed 1-form ω is called weakly complete if it has Morse type zeros and for any smooth path σ : [0, 1] → M^{∗} with ∫_{σ} ω = 0 the endpoints σ(0) and σ(1) lie in the same leaf of the foliation ω = 0 on M^{∗}. Here M^{∗} denotes M − {p_{1}, …, p_{m}} where p_{j} are the zeros of ω.

A weakly complete closed 1-form with ξ = [ω] ≠ 0 has no zeros with Morse indices 0 and n. According to Levitt [7], any nonzero real cohomology class ξ ∈ H^{1}(M; R) can be represented by a weakly complete closed 1-form.

The plan of our proof of Theorem 1 is as follows. We start with a weakly complete closed 1-form ω lying in the prescribed cohomology class ξ ∈ H^{1}(M; R), ξ ≠ 0. We show that assuming rk(ξ) > 1 all leaves of the singular foliation ω = 0 are dense (see §3). We perturb ω such that the resulting closed 1-form ω′ has a single singular leaf (see §4). After that we apply the technique of Takens [10] allowing us to collide the zeros in a single (highly degenerate) zero. We first prove Theorem 1 assuming that n = dim M > 2; the special case n = 2 is treated separately later.

3. Density of the leaves

In this section we show that if ω is weakly complete and rk(ξ) > 1 then the leaves of ω = 0 are dense.

Note that in general the assumption rk(ξ) > 1 alone does not imply that the leaves are dense, see the examples in §9.3 of [5].

Let ω be a weakly complete closed 1-form in class ξ. Consider the covering π : M̃ → M corresponding to the kernel of the homomorphism of periods Per_{ξ} : H_{1}(M) → R, where ξ = [ω] ∈ H^{1}(M; R). Let H ⊂ R be the group of periods. The rank of H equals rk(ξ); since we assume that rk(ξ) > 1, the group H is dense in R. The group of periods H acts on the covering space M̃ as the group of covering transformations. We have π^{∗}ω = dF where F : M̃ → R is a smooth function. The leaves of the singular foliation ω = 0 are the images under the projection π of the level sets F^{−1}(c); this property follows from the weak completeness of ω, see [7], Proposition II.1. For any g ∈ H and x ∈ M̃ one has

F (gx) − F (x) = g ∈ R. (3)

Let L = π(F^{−1}(c)) be a leaf and let x ∈ M be an arbitrary point. Our goal is to show that x lies in the closure L̄ of L. Let U ⊂ M be a neighborhood of x. We want to show that U intersects L. We shall assume that U is “small” in the following sense: ξ|_{U} = 0.

Consider a lift x̃ ∈ M̃, π(x̃) = x. Let Ũ be a neighborhood of x̃ which is mapped by π homeomorphically onto U. We claim that the set of values F (Ũ) ⊂ R contains an interval (a − ϵ, a + ϵ) where a = F (x̃) and ϵ > 0.

This claim is obvious if x̃ is not a critical point of F since in this case one may choose the coordinates x_{1}, …, x_{n} around x̃ such that F (x) = a + x_{1}. In the case when x̃ is a critical point of F, one may choose the coordinates x_{1}, …, x_{n} near the point x̃ ∈ M̃ such that F (x) is given by a±x_{1}^{2}±x_{2}^{2}+⋯+ ±x_{n}^{2} and our claim follows since we know that the Morse index is distinct from 0 and n.

Because of the density of the group of translations H ⊂ R one may find g ∈ H such that the real number F (gx̃) = F (x̃) + g = a + g lies in the interval (c − ϵ, c + ϵ). Then we obtain

c ∈ (a + g − ϵ, a + g + ϵ) ⊂ g + F (Ũ) = F (gŨ). (4)

Hence we see that the sets F^{−1}(c) and gŨ have a nonempty intersection. Therefore the neighborhood U = π(gŨ) intersects the leaf L = π(F^{−1}(c)) as claimed.

An obvious modification of the above argument proves a slightly more precise statement:

Given a point x ∈ M and a leaf L ⊂ M of the singular foliation ω = 0, there exist two sequences of points x_{k} ∈ L and y_{k} ∈ L such that

x_{k} → x and y_{k} → x, (5)

and, moreover,

∫_{x}^{x_{k}} ω > 0, while ∫_{x}^{y_{k}} ω < 0. (6)

The integrals in (6) are calculated along an arbitrary path lying in a small neighborhood of x.

This can also be expressed by saying that the leaf L approaches x from both the positive and the negative sides.

4. Modification

Our next goal is to replace ω by a Morse closed 1-form ω′ which has the property that all its zeros lie on the same singular leaf of the singular foliation ω′ = 0. In this section we assume that n = dim M > 2.

Let ω be a weakly complete Morse closed 1-form in class ξ where rk(ξ) > 1. Let p_{1}, …, p_{m} ∈ M be the zeros of ω. For each p_{j} choose a small neighborhood U_{j} ∋ p_{j} and local coordinates x_{1}, …, x_{n} in U_{j} such that x_{i}(p_{j}) = 0 for i = 1, …, n and

ω|_{U_{j}} = df_{j}, where f_{j} = −x_{1}^{2} − ⋯ − x_{m_{j}}^{2} + x_{m_{j}+1}^{2} + ⋯ + x_{n}^{2}. (7)

Here m_{j} denotes the Morse index of p_{j}. We assume that the ball ∑_{i=1}^{n} x_{i}^{2} ≤ 1 is contained in U_{j} and that U_{j} ∩ U_{j′} = ∅ for j ≠ j′. Denote by W_{j} the open ball ∑_{i=1}^{n} x_{i}^{2} < 1.

Let ϕ : [0, 1] → [0, 1] be a smooth function with the following properties:

(a) ϕ ≡ 0 on [3/4, 1]; (b) ϕ ≡ ϵ > 0 on [0, 1/2]; (c) −1 < ϕ′ ≤ 0. Such a

function exists assuming that ϵ > 0 is small enough. (a), (b), (c) imply that

ϕ′(r) > −2r, for r > 0. (8)

We replace the closed 1-form ω by

ω′ = ω − ∑_{j=1}^{m} μ_{j} · dg_{j} (9)

where g_{j} : M → R is a smooth function with support in U_{j}. In the coordinates x_{1}, …, x_{n} of U_{j} (see above) the function g_{j} is given by g_{j}(x) = ϕ(||x||). The parameters μ_{j} ∈ [−1, 1] appearing in (9) are specified later.

One has ω ≡ ω′ on M −∪_{j}U_{j} and near the zeros of ω. Let us show that ω′ has no additional zeros. We have ω′|_{U_{j}} = d(f_{j} − μ_{j}g_{j}) (where f_{j} is defined in (7)) and

\frac{∂}{∂x_{i}} (f_{j} − μ_{j}g_{j}) = ±2x_{i} − μ_{j}ϕ′(||x||)\frac{x_{i}}{||x||} (10)

If this partial derivative vanishes and x_{i} ≠ 0 then ϕ′(r) = ±2rμ_{j}^{−1} which may happen only for r = ||x|| = 0 according to (8).

We now show how to choose the parameters μ_{j} so that the closed 1-form ω′ given by (9) has a unique singular leaf. Let L be a fixed nonsingular leaf of ω = 0. Since L is dense in M (see §3) for any j = 1, …, m the intersection L ∩ U_{j} contains infinitely many connected components approaching p_{j} from below and from above and the function f_{j} is constant on each of them. We

say that a subset T_{c} ⊂ L ∩ W_{j} is a level set if T_{c} = f_{j}^{−1}(c) ∩ W_{j} for some c ∈ R. Note that f_{j}(p_{j}) = 0. The level set c = 0 contains the zero p_{j}; it is homeomorphic to the cone over the product S^{m_{j}−1} × S^{n−m_{j}−1}. Each level set T_{c} with c < 0 is diffeomorphic to S^{m_{j}−1} × D^{n−m_{j}} and each level set T_{c} with c > 0 is diffeomorphic to D^{m_{j}} × S^{n−m_{j}−1}. Recall that m_{j} denotes the Morse index of p_{j}.

Let V_{j} = f_{j}(L ∩ W_{j}) ⊂ R denote the set of values of f_{j} on different level sets belonging to the leaf L. The zero 0 does not lie in V_{j} since we assume that the leaf L is nonsingular. However, according to the result proven in §3, the zero 0 ∈ R is a limit point of V_{j} and, moreover, the closure of either of the sets V_{j} ∩ (0, ∞) and V_{j} ∩ (−∞, 0) contains 0 ∈ R.

For the modification ω′ (given by (9)) one has ω′|_{U_{j}} = dh_{j} where h_{j} = f_{j} − μ_{j}g_{j}. The level sets T′_{c} for h_{j} are defined as h_{j}^{−1}(c) ∩ W_{j}. Clearly T′_{c} is given by the equation

f_{j}(x) = μ_{j}ϕ(||x||) + c, x ∈ W_{j}.

Hence for ||x|| ≥ 3/4 this is the same as T_{c}; for ||x|| ≤ 1/2 the level set T′_{c} coincides with T_{c+μ_{j}ϵ}. In the ring 1/2 ≤ ||x|| ≤ 3/4 the level set T′_{c} is homeomorphic to a cylinder.

The following figure illustrates the distinction between the level sets T_{c} and T′_{c} in the case μ_{j} > 0.

Examine the changes which undergoes the leaf L when we replace ω by ω′. Here we view L with the leaf topology; it is the topology induced on L from the covering M̃ using an arbitrary lift L → M̃. First, let us assume that: (1) the Morse index m_{j} satisfies m_{j} < n − 1; (2) the coefficient μ_{j} > 0 is positive; (3) the number −ϵμ_{j} lies in the set V_{j}. Then the complement

L −_{⋃_{c∈V_{j}} ; −ϵμ_{j}<c<0}T_{c}

is connected and it lies in a single leaf L′ of the singular foliation ω′ = 0. We see that the new leaf L′ is obtained from L by infinitely many surgeries. Namely, each level set T_{c} ⊂ L, where c ∈ V_{j} satisfies −ϵμ_{j} < c < 0, is removed and replaced by a copy of D^{m_{j}} × S^{n−m_{j}−1}; besides, the set T_{c} ⊂ L where c = −ϵμ_{j}, is removed and gets replaces by a cone over the product S^{m_{j}−1} × S^{n−m_{j}−1}. Hence the new leaf L′ contains the zero p_{j}.

Let us now show how one may modify the above construction in the case m_{j} = n − 1. Since n > 2 we have in this case n − m_{j} − 1 < n − 2; hence removing the sphere S^{n−m_{j}−1} from the leaf L does not disconnect L. We shall assume that the coefficient μ_{j} is negative and that the number −ϵμ_{j} lies in V_{j} ⊂ R. The complement

L −_{⋃_{c∈V_{j}} ; 0<c<−ϵμ_{j}}T_{c}

is connected and it lies in a single leaf L′ of the singular foliation ω′ = 0. Clearly, L′ is obtained from L by removing the level sets T_{c} where c ∈ V_{j} satisfies 0 < c < −ϵμ_{j} (each such T_{c} is diffeomorphic to D^{m_{j}} × S^{n−m_{j}−1}) and by replacing them by copies of S^{m_{j}−1} × D^{n−m_{j}}. In addition, the set T_{c} ⊂ L where c = −ϵμ_{j}, is removed and is replaces by a cone over the product S^{m_{j}−1} × S^{n−m_{j}−1}.

We see that L′ is a leaf of the singular foliation ω′ = 0 containing all the zeros p_{1}, …, p_{m}.

5. Proof of Theorem 1

Below we assume that rk(ξ) > 1. The case rk(ξ) = 1 is covered by Theorem 2.1 from [3].

The results of the preceding sections allow to complete the proof of Theorem 1 in the case n = dim M > 2. Indeed, we showed in §4 how to construct a Morse closed 1-form ω′ lying in the prescribed cohomology class ξ such that all zeros of ω′ are Morse and belong to the same singular leaf L′ of the singular foliation ω′ = 0. Now we may apply the colliding technique of F. Takens [10], pages 203–206. Namely, we may find a piecewise smooth tree Γ ⊂ L′ containing all the zeros of ω′. Let U ⊂ M be a small neighborhood of Γ which is diffeomorphic to R^{n}. We may find a continuous map Ψ : M → M with the following properties:

Ψ(Γ) is a single point p ∈ Γ;

Ψ|_{M−Γ} is a diffeomorphism onto M − p;

Ψ(U) = U;

Ψ is the identity map on the complement of a small neighborhood V ⊂ M of Γ where the closure \overline{V} is contained in U.

Consider a smooth function f : U → R such that df = ω′|_{U}; it exists and is unique up to a constant. The function g = f ∘ Ψ^{−1} : U → R is welldefined (since f|_{Γ} is constant). g is continuous by the universal property of the quotient topology. Moreover, g is smooth on M − p. Applying Theorem 2.7 from [10], we see that we can replace g by a smooth function h : U → R having a single critical point at p and such that h = f on U − \overline{V}.

Let ω′′ be a closed 1-form on M given by

ω′′|_{M−\overline{V}} = ω′|_{M−\overline{V}} and ω′′|_{U} = dh. (11)

Clearly ω′′ is a smooth closed 1-form on M having no zeros in M − {p}. Moreover, ω′′ lies in the cohomology class ξ = [ω′] (since any loop in M is homologous to a loop in M − \overline{V}).

Now we prove Theorem 1 the case n = 2. We shall replace the construction of §4 (which requires n > 2) by a direct construction. The final argument using the Takens’ technique [10] remains the same.

Let M be a closed surface and let ξ ∈ H^{1}(M; R) be a nonzero cohomology class. We can split M into a connected sum

M = M_{1}♯M_{2}♯ … ♯M_{k}

where each M_{j} is a torus or a Klein bottle and such that the cohomology class ξ_{j} = ξ|_{M_{j}} ∈ H^{1}(M_{j}; R) is nonzero. Let ω_{j} be a closed 1-form on M_{j} lying in the class ξ_{j} and having no zeros; obviously such a form exists. §9.3.2 of [5] 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 [5]. 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 ω on M having 2k − 2 zeros which all lie on the same singular leaf of the singular foliation ω = 0. The colliding argument based on the technique of Takens [10] applies as in the case n > 2 and produces a closed 1-form with at most one zero lying in class ξ.

References

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[5] M. Farber, Topology of closed one-forms, Mathematical Surveys and Monographs, Vol 108, American Mathematical Society, 2004.

[6] F. Latour, Existence de 1-formes fermées non singulières dans une classe de cohomologie de De Rham, Publ. IHES, 80, 1994.

[7] G. Levitt, 1-formes fermées singulières et groupe fondamental, Invent. Math., 88(1987), 635 - 667.

[8] S. P. Novikov, The Hamiltonian formalism and a multi-valued analogue of Morse theory, Russian Math. Surveys, 37:5(1982), pp. 1–56.

[9] J.-Cl. Sikorav, Un problem de disjonction par isotopie symplectique dans un fibré cotangent, Ann. Sci. École Norm. Sup. (4)19(1986), 543-552.

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Department of Mathematical Sciences, University of Durham, Durham DH1 2LE, UK

E-mail address: Michael.Farber@durham.ac.uk

Fachbereich Mathematik und Informatik, Westfälische Wilhelms - Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany

E-mail address: schuetz@math.uni-muenster.de