COUNTING ZEROS OF CLOSED 1-FORMS

Michael Farber

To the memory of V. A. Rokhlin

Abstract. This paper suggests new topological lower bounds for the number of zeros of closed 1-forms within a given cohomology class. The main new technical tool is the deformation complex, which allows to pass to a singular limit and reduce the original problem with closed 1-form to a traditional problem with a Morse function. We show by examples that the present approach may provide stronger estimates than the Novikov inequalities. The technique of the paper also applies to study topology of the set of zeros of closed 1-forms under Bott non-degeneracy assumptions

§1. Line bundles, Dirichlet units, and zeros of closed 1-forms

In this section we describe some new topological estimates on the number of zeros of closed 1-forms. They use homology with coefficients in finite dimensional flat vector bundles (compare [N3], [P]), replacing the local systems over the Novikov ring in the well-known Novikov’s theory [N1, N2]. We will compute examples (cf. §5) showing that the method of this paper may produce stronger inequalities (and thus predicts existence of a larger number of zeros) than the classical Novikov’s approach.

Estimates on the numbers of zeros of closed 1-forms found interesting applications in symplectic topology. They were initiated by J.-C. Sikorav [S1, S2]. We refer also to the work of H. Hofer and D. Salamon [HS].

1.1. Let M be a smooth manifold, and let ω be a closed 1-form on M, dω = 0. A point p ∈ M is a zero of ω if ω vanishes at this point, i.e. ω_{p} = 0. A zero p ∈ M is called nondegenerate if ω, viewed as a map M → T^{∗}M, is transversal to the zero section M ⊂ T^{∗}M of the cotangent bundle. As is well-known, this condition is equivalent to the requirement that in a neighborhood U of p we may write ω = df, where f : U → R is a smooth function and p is a non-degenerated (Morse) critical point of f. The Morse index of p is well defined (as the Morse index of p as the critical point of f). A closed 1-form ω is called Morse if all its zeros are non-degenerate.

1.2. Recall that a complex number a ∈ C^{∗} is called a Dirichlet unit if it is a unit of the ring of integers of an algebraic number field. Equivalently, a Dirichlet unit is an

Typeset by AMS-TEX

The research was partially supported by a grant from the Israel Academy of Sciences and Humanities and by the Herman Minkowski Center for Geometry; a part of this work was done while the author was visiting Max-Planck Institut für Mathematik in Bonn

algebraic integer such that its inverse a^{−1} is also an algebraic integer. Dirichlet unit a ∈ C is a root of a monic polynomial

a^{k} + β_{1}a^{k−1} + β_{2}a^{k−2} + ⋯ + β_{k} = 0  (1-1)

with β_{1}, β_{2}, …, β_{k−1} ∈ Z and β_{k} = ±1; this property clearly characterizes the Dirichlet units.

In [F2, F3] I used Dirichlet units to construct an analog of the Lusternik - Schnirelman theory for closed 1-forms (i.e. without assuming that the zeros are all nondegenerate).

Consider a flat complex vector bundle E over M. It is determined by its monodromy, a linear representation of the fundamental group π_{1}(M, x_{0}) on the fiber E_{0} over the base point x_{0}, given by the parallel transport along loops. For example, a flat line bundle is determined by a homomorphism H_{1}(X; Z) → C^{∗}, where C^{∗} is considered as a multiplicative abelian group.

Let ξ ∈ H^{1}(X; Z) be an integral cohomology class. Given a complex number a ∈ C^{∗}, we will consider the complex flat line bundle over M with the following property: the monodromy along any loop γ ∈ π_{1}(M) is the multiplication by a^{⟨ξ,γ⟩}. We will denote this bundle by a^{ξ}.

A lattice L ⊂ V in a finite dimensional vector space V is a finitely generated subgroup with rank L = dim_{C} V.

We will say that a complex flat bundle E → X of rank m admits an integral lattice if its monodromy representation π_{1}(X, x_{0}) → GL_{C}(E_{0}) is conjugate to a homomorphism π_{1}(X, x_{0}) → GL_{Z}(L_{0}), where L_{0} ⊂ E_{0} is a lattice in the fiber. This condition is equivalent to the assumption that E is obtained from a local system Ẽ of finitely generated free abelian groups over X by tensoring on C.

1.3. Theorem. Let M be a closed smooth manifold and let ξ ∈ H^{1}(M; Z) be an integral cohomology class. Let E → M be a complex flat bundle admitting an integral lattice. Let a ∈ C^{∗} be a complex number, which is not a Dirichlet unit. Then for any closed 1-form ω on M having Morse type zeros and lying in class ξ, the number c_{j}(ω) of zeros of ω having index j satisfies

c_{j}(ω) ≥ \frac{dim_{C} H_{j}(M; a^{ξ} ⊗ E)}{dim E},  j = 0, 1, 2, ….  (1-2)

Moreover,

∑_{i=0}^{j}(−1)^{i}c_{j−i}(ω) ≥ ∑_{i=0}^{j}(−1)^{i} \frac{dim_{C} H_{j−i}(M; a^{ξ} ⊗ E)}{dim E},  j = 0, 1, 2, …. (1-3)

A proof of Theorem 1.3 is given in §3.

Remark 1. Theorem 1.3 gives interesting estimates already in the simplest case when E is taken to be the trivial flat line bundle. In this case, however, inequality (1-3) follows from the Novikov inequalities, written in the form

∑_{i=0}^{j}(−1)^{i}c_{j−i}(ω) ≥ q_{j}(ξ) + ∑_{i=0}^{j}(−1)^{i}b_{j−i}(ξ),  j = 0, 1, 2, …,  (1-4)

cf. [F1], page 48.

We will show by examples (cf. §5) that by applying Theorem 1.3 with different flat bundles E one obtains stronger estimates, than provided by the Novikov inequalities.

Remark 2. It is easy to show that for transcendental a ∈ C the Betti number dim_{C} H_{i}(M; a^{ξ}) equals the Novikov number b_{i}(ξ) (and in, particular, it is the same for all transcendental a.

Remark 3. Consider the function

a ∈ C^{∗} ↦ dim_{C} H_{j}(M; a^{ξ} ⊗ E).

Then there exist only finitely many numbers a_{1}, a_{2}, …, a_{k} ∈ C^{∗} (they are called jump points) so that the corresponding Betti number dim_{C} H_{j}(M; a^{ξ} ⊗ E) is the same for any a ∈ C^{∗} which is not one of the jump points. Following [BF1], let us denote by b_{j}(ξ; E) the value of dim_{C} H_{j}(M; a^{ξ} ⊗ E) for a not a jump point. The number b_{j}(ξ; E) is a generalization of the Novikov number b_{j}(ξ). For any of the jump points a_{s} actually holds

dim_{C} H_{j}(M; a_{s}^{ξ} ⊗ E) > b_{j}(ξ; E),

i.e. the jumps are always positive.

Suppose that the flat bundle E admits an integral lattice. Then the jump points a_{1}, a_{2}, …, a_{k} are algebraic numbers (not necessarily algebraic integers). If a jump happens at a number, which is not a Dirichlet unit, then Theorem 1.3 applies and we obtain estimate (3-2), which is stronger, than the inequality

c_{j}(ω) ≥ \frac{b_{j}(ξ; E)}{dim E}.

Remark 4. Theorem 1.3 becomes false if we allow a ∈ C^{∗} to be a Dirichlet unit. To explain this, note that any Dirichlet unit a ∈ C is an eigenvalue of an integral square matrix B = (b_{ij}) with det(B) = 1. We may find a diffeomorphism of a compact smooth manifold h : F → F so that h induces the matrix B on homology of some dimension k. Consider the mapping torus M, which obtained from F × [0, 1] by identifying each point (x, 0) with (h(x), 1). The manifold M is a smooth fiber bundle over the circle and so it admits a closed 1-form ω with no critical points, c_{j}(ω) = 0 for all j. The homology H_{∗}(M; a^{ξ}) is nontrivial if and only if the number a is an eigenvalue of the monodromy h_{∗} : H_{∗}(F; C) → H_{∗}(F; C). Here ξ denotes the cohomology class of ω. Hence, if a is a Dirichlet unit, we may construct M so that H_{∗}(M; a^{ξ}) ≠ 0 and class ξ may be realized by a closed 1-form with no critical points.

§2. Morse inequalities for prime ideals

In this section we describe Morse type inequalities for prime ideals in commutative rings, which will be used in the proof of Theorem 1.3. The results of this section are known to experts in commutative algebra, although I am unable to make a reference.

2.1. Let R be a commutative Noetherian ring. Each prime ideal 𝔭 ⊂ R gives a way of assigning Betti numbers to chain complexes over R. Indeed, let C be a chain complex over R

0 → C_{m} → C_{m−1} → ⋯ → C_{0} → 0

with finitely generated free R-modules C_{i}. Given a prime ideal 𝔭 ∈ R, we will denote by b_{i}(C, 𝔭) the 𝔭-Betti number of C, i.e.

b_{i}(C, 𝔭) = dim_{Q(R/𝔭)} H_{i}(C ⊗_{R} Q(R/𝔭)),  (2-1)

where Q(R/𝔭) denotes the field of fractions of the factor ring R/𝔭. Define also the Poincaré polynomial corresponding to 𝔭 as

P_{C,𝔭}(λ) = ∑_{i=0}^{m} λ^{i}b_{i}(C, 𝔭).  (2-2)

Our purpose is to compare the Poincaré polynomials corresponding to two different prime ideals 𝔭 ⊂ 𝔮 ⊂ R.

Given two polynomials

P(λ) = p_{0} + p_{1}λ + ⋯ + p_{m}λ^{m}, Q(λ) = q_{0} + q_{1}λ + ⋯ + q_{m′}λ^{m′}

with p_{i}, q_{i} ∈ Z, we will write

P(λ) ⪰ Q(λ),  (2-3)

to indicate that the differentce P(λ) − Q(λ) may be represented in the form

P(λ) − Q(λ) = (1 + λ)T (λ),

where T (λ) is a polynomial with non-negative integral coefficients. The relation ⪰ defines a partial order on the set of all integral polynomials in λ. It is well known that (2-3) is equivalent to the following sequence of Morse inequalities:

∑_{j=0}^{r}(−1)^{j}p_{r−j} ≥ ∑_{j=0}^{r}(−1)^{j}q_{r−j}, r = 0, 1, ….

2.2. Theorem. Let C be a free finitely generated chain complex over R and let 𝔭 ⊂ 𝔮 ⊂ R be two prime ideals. Then for the corresponding Poincaré polynomials holds

P_{C,𝔮}(λ) ⪰ P_{C,𝔭}(λ).  (2-4)

Proof. Applying twice the Euler - Poincaré theorem to the truncated complex

0 → C_{r} → C_{r−1} → ⋯ → C_{0} → 0,

tensored by Q(R/𝔭) and by Q(R/𝔮), we obtain the following identity

∑_{j=0}^{r}(−1)^{j}b_{r−j}(C, 𝔭) + B_{r}(𝔭) =

∑_{j=0}^{r}(−1)^{j} rk C_{r−j} =

∑_{j=0}^{r}(−1)^{j}b_{r−j}(C, 𝔮) + B_{r}(𝔮)

where B_{r}(𝔭) denotes the dimension over the field Q(R/𝔭) of the image of the map

d : C_{r+1} ⊗ Q(R/𝔭) → C_{r} ⊗ Q(R/𝔭);  (2-5)

the number B_{r}(𝔮) is defined similarly.

Our statement now is equivalent to the inequality B_{r}(𝔮) ≤ B_{r}(𝔭) for all r.

Suppose that the homomorphism d : C_{r+1} → C_{r} is represented by a matrix with entries in R. Then B_{r}(𝔮) equals to the maximal size of the minors of this matrix which do not lie in the ideal 𝔮. The number B_{r}(𝔭) has the similar descritpion in terms of this matrix. Since 𝔭 ⊂ 𝔮, we obtain B_{r}(𝔮) ≤ B_{r}(𝔭). □

§3. Proof of Theorem 1.3

In this section we will describe a proof of Theorem 1.3, which will be based on the Morse inequalities for prime ideals (cf. §2) and on the existence of a deformation complex. While proving Theorem 1.3, we will not use the particular construction of the deformation complex C_{∗}; we will only use its properties (i) - (iv), cf. 3.2 below.

3.1. Suppose that we are in conditions of Theorem 1.3. Since the closed 1-form ω has an integral cohomology class, there exists a map f : M → S^{1}, so that ω = f^{∗}(dθ), where dθ is the standard angular form on the circle. The zeros of ω are precisely the critical points of f. We assume that ω has only nondegenerated zeros, and hence f is a Morse circle valued function on M. Choose a regular value b ∈ S^{1} and let N be the result of cutting of M along the codimension one submanifold V = f^{−1}(b). We have a canonical identification map Π : N → M. Note that the boundary of N contains two copies of V. We will denote them by ∂_{+}N and ∂_{−}N. The notations ∂_{±}N are chosen so that for the normal vector field X on ∂_{+}N, pointing inside N, holds Π^{∗}ω(X) > 0; also, for the normal vector field Y on ∂_{−}N, pointing inside N, holds Π^{∗}ω(Y) < 0.

3.2. In §4 we will construct a chain complex C_{∗} (which we call deformation complex) with the following properties:

(i) C_{∗} is a free finitely generated chain complex over the polynomial ring P = Z[τ], where τ is an indeterminate.

(ii) For any nonzero complex number a ∈ C^{∗} the homology H_{i}(C_{a} ⊗_{P} C_{∗}), as a vector space, is isomorphic to H_{i}(M; a^{−ξ} ⊗ E), where i = 0, 1, 2, . . .. Here C_{a} denotes C with the P -module structure, so that τ acts by multiplication on a ∈ C.

(iii) Let C_{0} denote C with the trivial P -module structure, i.e. τ acts as zero on C. Then the homology H_{i}(C_{0} ⊗_{P} C_{∗}) is isomorphic to H_{i}(N, ∂_{+}N; Π^{∗}E), where i = 0, 1, 2, . . ..

(iv) Let p be a prime number and let Z_{p} denote Z/pZ, considered as a P -module with the trivial (i.e. τ = 0) action of τ. Then the homology H_{i}(Z_{p} ⊗_{P} C_{∗}), as an abelian group, is isomorphic to H_{i}(N, ∂_{+}N; Z_{p} ⊗ Π^{∗}Ẽ), where Ẽ denotes a local system of free abelian groups on M with Ẽ ⊗ C = E.

Intuitively, we may view the deformation complex C_{∗} as a polynomial family of complexes C_{∗}(M; a^{−ξ} ⊗ E), where a ∈ C^{∗} is a parameter, and such that it has a ”singular limit” as a → 0, which is described in (iii) and (iv).

The deformation complex C_{∗} depends on the data M, ξ, E and is not unique. We will use only existence of C_{∗}.

3.3. Let a ∈ C^{∗} be a nonzero complex number which is not an algebraic integer. Consider the prime ideal 𝔭_{a} ⊂ P consisting of integral polynomials

q(τ) = q_{0} + q_{1}τ + ⋯ + q_{m}τ^{m}, q_{j} ∈ Z

with q(a^{−1}) = 0. If C_{∗} denotes the deformation complex, then, using the property (ii) and the notations, introduced in §2, we obtain

dim_{Q(P/𝔭_{a})} H_{i}(Q(P/𝔭_{a}) ⊗_{P} C_{∗}) = dim_{C} H_{i}(C_{a^{−1}} ⊗_{P} C_{∗}) = dim_{C} H_{i}(M; a^{ξ} ⊗ E) and therefore

P_{C_{∗},𝔭_{a}}(λ) = ∑_{i=0}^{dim M} λ^{i} dim_{C} H_{i}(M; a^{ξ} ⊗ E).  (3-1)

Since we assume that a is not an algebraic integer, the ideal 𝔭_{a} contains no polynomials with the free term q_{0} equals 1. Hence, we obtain that there exists a prime number p, so that the free terms of all polynomials q(λ) lying in 𝔭_{a} are divisible by p. In other words,

𝔭_{a} ⊂ 𝔮, where 𝔮 = (p) + (τ) ⊂ P;  (3-2)

in other words, the ideal 𝔮 is generated by p and τ. Using property (iv) in 3.2, we obtain

dim_{Q(P/𝔮)} H_{i}(Q(P/𝔮) ⊗_{P} C_{∗}) = dim_{Z_{p}} H_{i}(N, ∂_{+}N; Z_{p} ⊗ Ẽ),

and thus

P_{C_{∗},𝔮}(λ) = ∑_{i=0}^{dim M} λ^{i} dim_{Z_{p}} H_{i}(N, ∂_{+}N; Z_{p} ⊗ Ẽ).  (3-3)

By Theorem 2.2 we have

P_{C_{∗},𝔮}(λ) ⪰ P_{C_{∗},𝔭_{a}}(λ).  (3-4)

The form Π^{∗}ω is differential of a Morse function g : N → R, dg = Π^{∗}ω. Function g is constant on each connected component of ∂_{+}N. The critical points of g lie in the interior of N and they are in 1-1 correspondence with the zeros of ω. Hence, the traditional Morse theory, applied to the cobordism N and Morse function g, gives

∑_{i=0}^{dim M} c_{i}(ω)λ^{i} ⪰ (dim E)^{−1} · ∑_{i=0}^{dim M} λ^{i} dim_{Z_{p}} H_{i}(N, ∂_{+}N; Z_{p} ⊗ Ẽ).  (3-5)

Now (3-5), combined with (3-3), (3-4), (3-1), gives

∑_{i=0}^{dim M} c_{i}(ω)λ^{i} ⪰ (dim E)^{−1} · ∑_{i=0}^{dim M} λ^{i} dim_{C} H_{i}(M; a^{ξ} ⊗ E),  (3-6)

which is equivalent to (1-3).

We are left to prove Theorem 1.3 assuming that a^{−1} ∈ C^{∗} is not an algebraic integer. We will apply the result proven above, where we replace the cohomology class ξ by −ξ and the flat vector bundle E by the flat vector bundle E^{∗} ⊗ 𝔬_{M}, where 𝔬_{M} is the orientation bundle of M (i.e. a flat line bunlde such that the monodromy along any loop equals ±1 depending on whether the orientation of M is preserved or reversed along the loop). Note that E^{∗} as well as 𝔬_{M} admit intergral lattices.

This gives

∑_{i=0}^{dim M} c_{i}(−ω)λ^{i} ⪰ (dim E)^{−1} · ∑_{i=0}^{dim M} λ^{i} dim_{C} H_{i}(M; a^{−ξ} ⊗ E^{∗} ⊗ 𝔬_{M}).  (3-7)

It is clear that c_{i}(−ω) = c_{n−i}(ω), where n = dim M. Also, from the Poincaré duality we obtain dim_{C} H_{i}(M; a^{−ξ} ⊗E^{∗} ⊗𝔬_{M}) = dim_{C} H_{n−i}(M; a^{ξ} ⊗E). Substitutimg these equalities into (3-7), dividing both sides by λ^{n} and then replacing the indeterminate λ^{−1} by λ, gives (3-6), which is equivalent to (1-3).

This completes the proof. □

§4. Construction of the deformation complex

In this section we will construct the deformation complex and prove its properties (i) - (iv), which were used in §3. The construction here is slightly different from [F3], [F4], although the obtained complex is essentially the same.

4.1. The cell decomposition. In this subsection we will describe a cell decomposition of M, related to the given closed 1-form ω. We will repeat some of the constructions of subsection 3.2.

We assume that ω has an integral indivisible cohomology class. Then there exists a smooth map f : M → S^{1}, so that ω = f^{∗}(dθ), where dθ is the angular form on the circle S^{1}. Clearly f is a Morse map and the zeros of ω are precisely the critical points of f. Choose a regular value b ∈ S^{1}, and let N be the result of cutting of M along the codimension one submanifold V = f^{−1}(b). We have a canonical identification map Π : N → M. The boundary of N contains two copies of V. We denote them by ∂_{+}N and ∂_{−}N. The normal vector field X on ∂_{+}N, pointing inside N, satisfies Π^{∗}ω(X) > 0. Also, for the normal vector field Y on ∂_{−}N, pointing into N, holds Π^{∗}ω(Y) < 0.

The form Π^{∗}ω is exact, i.e. it is differential of a Morse function g : N → R, dg = Π^{∗}ω. Each connected component of ∂_{+}N consists of points of local minimum of g. The critical points of g, which lie in the interior of N, are in 1-1 correspondence with the zeros of ω and have the same indices. Hence c_{i}(g) = c_{i}(ω) for any i.

Fix a cell decomposition of N so that V = ∂_{+}N and ∂_{−}N be subcomplexes and so that the natural homeomorphism J : ∂_{+}N → ∂_{−}N be a cellular isomorphism.

The manifold M is homeomorphic to the factor-space N/ ∼, where we identify each pair of points v ∈ ∂_{+}N and J(v) ∈ ∂_{−}N. In order to give the factor-space N/ ∼ a CW-structure, consider a cylinder V ×[0, 1] having the standard cell-decomposition: for each i-dimensional cell e of V we have two i-dimensional cells e × 0 and e × 1 and one (i + 1)-dimensional cell e × I of the cylinder V × I. Glue each point (v, 1) of the top face of the cylinder with the point v ∈ V ⊂ N; also, glue each point (v, 0) of bottom face of the cylinder with the point J(v) ∈ N. As the result we obtain a CW-decomposition

M = (V × I ∪ N)/ ∼ .  (4-1)

4.2. The chain complex. Here we will calculate the cellular chain complex C_{∗}(M̃) of the universal covering M̃ of the CW-complex (4-1). It is a complex of free left Zπ-modules, where π is the fundamental group of M.

In general the complement M −V may be disconnected. Let us choose base points v_{1}, v_{2}, …, v_{l}, one for each connected component of M − V. For each j = 1, 2, …, l we may find a smooth path μ_{j} : [0, 1] → M, such that μ_{j}(0) = v_{1}, μ_{j}(1) = v_{j} and μ_{j} has intersection number zero with V, i.e. μ_{j} · V = 0. Note that V has a fixed orientation of its normal bundle. To construct μ_{j}, first connect the points v_{1} and v_{j} by an arbitrary path \overline{μ}_{j} in M, and then set μ_{j} = \overline{μ}_{j} − (\overline{μ}_{j} · V)δ, where δ is a closed loop in M such that δ · V = 1. Such loop δ exists since we assume that the cohomology class ξ = [ω] ∈ H^{1}(M; Z) is indivisible.

Each cell of X can be lifted into the covering M̃, and all possible lifts are parameterized by the elements of the fundamental group π = π_{1}(M, v_{1}). In order to fix the lifts of the cells of M, we will describe for each cell e ⊂ M a path γ_{e} in M, starting from the base point v_{1} and leading to an internal point of e. We will call γ_{e} the tail of e. After an arbitrary choice of a lift of the base point v_{1}, we will obtain lifts of all the cells in the covering M̃, determined (in an obvious way) by the choice of the tails.

For the cells e ⊂ N we will choose their tails as follows. Assume that e lies in the component of N containing v_{j}. Then we set γ_{e} = μ_{j}σ_{e}, where σ_{e} is a path in N connecting v_{j} with an internal point of e.

The tail of cells of the form e × I ⊂ V, where e ⊂ V, are constructed as follows. First travel along the existing tail γ_{e} of cell e, which leads from the base point v_{1} to an internal point of e ⊂ V = V × 1 ⊂ N, and then drop slightly down to an internal point of the cell e × I.

After the choice of tails as above we obtain a free basis of C_{∗}(M̃) (over the group ring Zπ) formed by the lifts of the cells of M.

The boundary homomorphism of C_{∗}(M̃), applied to a cylindrical generator e × I is given by the formula

d(e × I) = d(e) × I + (−1)^{i}[e × 1 − e × 0], i = dim e.  (4-2)

Here e × 1 can be identified with e. The generator e × 0 ∈ C_{∗}(M̃) is supported by the cell J(e) ⊂ ∂_{−}N ⊂ N, however it has a different tail. Indeed, it is clear that the tail of the cell e × 0 has intersection number −1 with V, because it first travels along the tail of e × I, and then arrives (staying inside e × I) at the face e × 0 ≃ J(e) of e × I. Hence we may rewrite (4-2) as follows

d(e × I) = d(e) × I + (−1)^{i}[e − g · J(e)],  (4-3)

where

g ∈ π = π_{1}(M, v_{1}), ξ(g) = −1.  (4-4)

Recall that ξ ∈ H^{1}(M; Z) denotes the cohomology class of ω.

4.3. The deformation complex. Let Zπ_{−} ⊂ Zπ denote the subring of the group ring generated by the group elements g ∈ π with ξ(g) ≤ 0. From the description of the chain complex C_{∗}(M̃) (and in particular, from (4-3), (4-4)) we see that the cells of M with their tails specified above, generate a free chain complex C′ over Zπ_{−} such that

Zπ ⊗_{Zπ_{−}} C′ = C_{∗}(M̃).

Let Ẽ → M be a local system of free abelian groups so that Ẽ ⊗ C = E; it exists because we assume that flat bundle E admits an integral lattice. We have the monodromy representation

MonẼ : π → GL(m; Z), m = rank(Ẽ).  (4-5)

Here for g, g′ ∈ π holds MonẼ(gg′) = MonẼ(g′)MonẼ(g), i.e. (4-5) is an antiisomorphism. We obtain a ring anti-homomorphism Zπ_{−} → Mat(m; P), P = Z[τ], where for g ∈ π with ξ(g) ≤ 0 we set g ↦ τ^{−ξ(g)}MonẼ(g). This defines a structure of right Zπ_{−}-module on P^{m}, commuting with the obvious left P -action. Hence, P^{m} becomes a (P, Zπ_{−})-bimodule. We may fanally define the deformation complex C_{∗} as

C_{∗} = P^{m} ⊗_{Zπ_{−}} C′.  (4-6)

4.4. Now we check that the deformation complex (4-6) satisfies the conditions (i) - (iv) of subsection 3.2. Indeed, (i) is obvious. To show (ii), note that for a ∈ C^{∗} we have an isomorphism

C_{a} ⊗_{P} C_{∗} = C_{a} ⊗_{P} (P^{m} ⊗_{Zπ_{−}} C′) ≃ (C_{a} ⊗_{Z} Z^{m}) ⊗_{Zπ} C_{∗}(M̃)  (4-7)

and C_{a} ⊗_{Z} Z^{m} ≃ C^{m} is a right Zπ-module with respect to the action

g ↦ a^{−ξ(g)} Mon_{E}(g) ∈ GL(m; C).

This clearly implies (ii).

Let us now prove (iv). We have

Z_{p} ⊗_{P} C_{∗} ≃ Z_{p} ⊗_{P} (P^{m} ⊗_{Zπ_{−}} C′) ≃ Z_{p}^{m} ⊗_{Zπ_{−}} C′,  (4-8)

where the right action of Zπ_{−} on Z_{p}^{m} is given by

g ↦ \{ \begin{matrix}MonẼ(g), if ξ(g) = 0 \\ 0 if ξ(g) < 0, \end{matrix}  (4-9)

where g ∈ π. From (4-3), (4-4) we see that complex (4-8) will have as its basis the cells of N and cells of the form e × I, one for each cell e of V. The boundary map of (4-8) acts on the cylindrical generators e × I as follows

d(e × I) = ∂e × I + (−1)^{dim e} · e  (4-10)

(the last term in (4-3) disappeares). This implies that all the cells e ⊂ V and e × I ⊂ V × I generate a subcomplex 𝔑_{p} of the chain complex (4-8). Moreover, the map e ↦ (−1)^{dim e}e × I is a chain contraction of 𝔑_{p} (because of (4-10)).

Hence we obtain an isomorphism H_{i}(Z_{p} ⊗_{P} C_{∗}) ≃ H_{i}((Z_{p} ⊗_{P} C_{∗})/𝔑_{p}) and the homology of the factor-complex (Z_{p} ⊗_{P} C_{∗})/𝔑_{p} is clearly H_{i}(N, V; Z_{p} ⊗ Π^{∗}Ẽ). This proves (iv).

The proof of (iii) is similar. We will not use (iii) in this paper.

§5. An example

In this section we show that inequalities (1-2), (1-3) produce stronger estimates than the Novikov inequalities. The example, which is described here, is a modification of example 1.7 in [BF].

5.1. Let k_{N} ⊂ S^{3} be the connected sum of N copies of the trefoil knot. Let X be closed 3-dimensional manifold obtained by 0-surgery on k_{N}. Then H^{1}(X; Z) ≃ Z; we will denote by η ∈ H^{1}(X; Z) a generator.

Recall that the Alexander polynomial of the trefoil is Δ(τ) = τ^{2} − τ + 1. Its two roots we denote by b, b^{−1} ∈ C; they are Dirichlet units.

Consider the rank 2 flat vector bundle F → X, where F ≃ b^{η} ⊕ b^{−η}.

From the knot theory [R] we know that

dim_{C} H_{1}(X; F) = 2N, dim_{C} H_{1}(X; C) = 0.  (5-1)

Let us show that F admits an integral lattice. The monodromy representation of F is a 2-dimensional vector space with a basis e_{1}, e_{2}, on which the meridian γ ∈ π_{1}(X) of k_{N} acts as follows

e_{1} ↦ be_{1}, e_{2} ↦ b^{−1}e_{2}.

Hence the vectors e_{1} + e_{2} and be_{1} + b^{−1}e_{2} form a basis, and the action of γ in this

basis is represented by an integral matrix [ \begin{matrix}0 & −1 \\ 1 & 1 \end{matrix} ]. Content-Length: 9764 X-Lines:

242 Status: RO

5.2. Consider the 3-dimensional manifold given as the connected sum

M = X#(S^{1} × S^{2}).

Thus M = M_{+} ∪ M_{−}, where M_{+} ∩ M_{−} = S^{2} and

M_{+} = X − {disk}, M_{−} = (S^{1} × S^{2}) − {disk}.

Consider a flat vector bundle E → M such that E|_{M_{+}} ≃ F |_{M_{+}} and E|_{M_{−}} is trivial. Let ξ ∈ H^{1}(M; Z) be a class such that ξ|_{M_{+}} = 0 and ξ|_{M_{−}} is a generator.

We want to compare dim_{C} H_{1}(M; a^{ξ}) with dim_{C} H_{1}(M; a^{ξ} ⊗ E) for different a ∈ C^{∗}. Using the Mayer-Vietoris sequence, we obtain

dim_{C} H_{1}(M; a^{ξ}) = dim_{C} H_{1}(M_{+}; C) + dim_{C} H_{1}(M_{−}; a^{ξ}),

and hence using (5-1) we obtain that dim_{C} H_{1}(M; a^{ξ}) = 0 for a ≠ 1. All Novikov numbers b_{i}(ξ) = 0 = q_{i}(ξ) vanish. However, the same argument as above, using (5-1) and the Mayer-Vietoris sequence, yields

dim_{C} H_{1}(M; a^{ξ} ⊗ E) = 2N.

This example shows that an appropriate choice of flat bundle E, appearing in inequalities (1-2) and (1-3), may provide stronger estimates, than when taking E to be the trivial bundle.

§6. Closed 1-forms with non-isolated zeros

In this section we study the topology of the set of zeros of closed 1-forms under Bott type [B] nondegeneracy condition. First inequalities of this type were obtained in [BF].

6.1. Let ω be a smooth closed real valued 1-form on M, dω = 0, which is assumed to have zeros non-degenerate in the sense of R.Bott [B]. This means that the set of points of C ⊂ M, where the form ω vanishes, form a submanifold of M, and the Hessian of ω is non-degenerate on the normal bundle to C.

If N is a small tubular neighborhood of C in M, then the integral ∫_{γ} ω vanishes along any loop γ ⊂ N (since γ is homologous to a curve in C). Thus there exists a unique real valued smooth function f on N such that df = ω_{|_{N}} and f_{|_{C}} = 0. The Hessian of ω is then defined as the Hessian of f.

Let ν(C) denote the normal bundle of C in M. Note that ν(C) may have different dimension over different connected components of C. Since Hessian of ω is nondegenerate, the bundle ν(C) splits into the Whitney sum of two subbundles

ν(C) = ν^{+}(C) ⊕ ν^{−}(C),

such that the Hessian is strictly positive on ν^{+}(C) and strictly negative on ν^{−}(C). Here, the dimension of the bundles ν^{+}(C) and ν^{−}(C) over different connected components of C may be different.

For every connected component Z of set C, the dimension of the bundle ν^{−}(C) over Z is called the index of Z and is denoted by ind(Z). Let o(Z) denote the orientation bundle of ν^{−}(C)_{|_{Z}}, considered as a local system with fiber Z.

The following is a generalization of Theorem 1.3 for closed 1-forms with nonisolated zeros.

6.2. Theorem. Let ω be a closed 1-form on M having Bott type zeros and representing an integral cohomology class ξ = [ω] ∈ H^{1}(M; Z). Let Ẽ → M be a local system of free abelian groups. Let a ∈ C^{∗} be a complex number, not an algebraic integer; let p be a prime number with the property that any integral polynomial q(τ) with q(a) = 0 has top coefficient divisible by p. Then

∑_{Z} ∑_{i=0}^{n} λ^{ind(Z)+i} dim_{Z_{p}} H_{i}(Z; Z_{p} ⊗ Ẽ_{|_{Z}} ⊗ o(Z)) ⪰

⪰ ∑_{i=0}^{n} λ^{i} dim_{C} H_{i}(M; a^{ξ} ⊗ Ẽ).

(6-1)

In the first sum Z runs over all connected components of the set of zeros of ω.

Note that for transcendental a the prime number p may be taken arbitrarily. In the case of transcendental a ∈ C^{∗} inequalities (6-1) turn into the inequalities of [BF], i.e. with C replacing Z_{p} in the first sum (6-1). In fact, the method of the proof, suggested in the present paper, also proves the main result of [BF] (by using property (iii) in subsection 3.2 of the deformation complex). In [BF] the proof used the method of Witten deformation and analytical tools.

Proof. We simply repeat all the arguments of the proof of Theorem 1.3 given in §3, using the Morse inequalities for prime ideals and the deformation complex. On the last stage instead of inequality (3-5) we use now the following inequality

∑_{Z} ∑_{i=0}^{n} λ^{ind(Z)+i} dim_{Z_{p}} H_{i}(Z; Z_{p} ⊗ Ẽ_{|_{Z}} ⊗ o(Z)) ⪰

⪰ ∑_{i=0}^{n} λ^{i} dim_{Z_{p}} H_{i}(N, ∂_{+}N; Z_{p} ⊗ Ẽ),

(6-2)

which is just a slight generalization of the well-known inequality of Bott [B] with Z_{p} coefficients, applied to the manifold with boundary N and to Bott function g : N → R, cf. §3. □

§7. Classes of higher rank

In this section we announce a generalization of Theorem 1.3 for cohomology classes ξ of higher rank.

7.1. Let M be a manifold. We will denote by H the first homology group H_{1}(M; Z). Let ξ ∈ H^{1}(M, R) be a real cohomology class. It can be viewed as a homomorphism ξ : H_{1}(M; Z) = H → R; we denote by ker(ξ) the kernel. Given a polynomial p ∈ Z[H], one defines two numbers d_{ξ}(p) (the ξ-degree of p) and v_{ξ}(p) (the ξ-top coefficient) as follows. Let p = ∑_{j=1}^{n} β_{j}h_{j}, where β_{j} ∈ Z and h_{j} ∈ H. Then d_{ξ}(p) is defined as the maximal number d = d_{ξ}(p) ∈ R such that the sum v_{ξ}(p) = ∑ β_{j}, taken over all j with ⟨ξ, h_{j}⟩ = d, is nonzero.

Let L → M be a complex flat line bundle. We will assume that the monodromy of L is trivial along any loop in M representing a homology class in ker(ξ). L determines a monodromy homomorphism

Mon_{L} : Z[H] → C,  (7-1)

assigning to any h ∈ H the monodromy of L along h. We will denote by I_{L} ⊂ Z[H] the kernel of the homomorphism Mon_{L}.

7.2. Definition. (A) We will say that a flat complex line bundle L → M is a ξalgebraic integer if (i) the monodromy of L is trivial along any loop in M representing a homology class in ker(ξ); and (ii) the ideal I_{L} contains a polynomial p ∈ I_{L} with v_{ξ}(p) = ±1.

(B) We will say that a complex flat line bundle L → M is a ξ-Dirichlet unit if L and the dual flat line bundle L^{∗} are ξ-algebraic integers.

The following statement generalizes Theorem 1.3 on forms with arbitrary cohomology classes.

7.3. Theorem. Let M be a closed smooth manifold and let ξ ∈ H^{1}(M; R) be a real cohomology class. Let E → M be a flat complex vector bundle admitting an integral lattice. Let L → M be a flat complex line bundle, which is not a ξ-Dirichlet unit. Then for any closed 1-form ω on M having Morse zeros and lying in the class ξ, the number c_{j}(ω) of zeros of ω having index j satisfies

c_{j}(ω) ≥ \frac{dim_{C} H_{j}(M; L ⊗ E)}{dim E},  j = 0, 1, 2, ….  (7-2)

The proof will be published elsewhere. It uses the Morse inequalities for prime ideals and an analog of the deformation complex for forms of higher rank > 1.

References

[BF] M. Braverman and M. Farber, Novikov type inequalities for differential forms with non-isolated zeros, Math. Proc. Camb. Phil. Soc. 122 (1997), 357 - 375.

[B] R. Bott, Non degenerate crititcal manifolds, Annals of Math 60 (1954), 248 - 261.

[F1] M. Farber, Exactness of the Novikov inequalities, Functional Anal. Appl. 19 (1985), 40–48.

[F2] M. Farber, Dirichlet units and critical points of closed 1-forms., To appear in ”C. R. Acad. Sci. Paris”.

[F3] M. Farber, Lusternik - Schnirelman theory for closed 1-forms, Preprint math.DG/9811113 (1998).

[F4] M. Farber, Topology of closed 1-forms and their critical points, Preprint math.DG/9811173.

[HS] H. Hofer and D. Salamon, Floer homology and Novikov rings, Floer memorial volume, Birkhauser, 1995, 483 - 524.

[M] J. Milnor, Morse theory, Annals of Mathematics Studies, vol. 51, Princeton Univ. Press, 1963.

[N1] S.P. Novikov, Multi-valued functions and functionals. An analogue of Morse theory., Soviet Math. Doklady 24 (1981), 222–226.

[N2] S.P. Novikov, The Hamiltonian formalism and a multi-valued analogue of Morse theory, Russian Math. Surveys 37:5 (1982), 1–56.

[N3] S.P. Novikov, Bloch homology, critical points of functions and closed 1-forms, Soviet Math. Dokl. 33 (1986), 551–555.

[P] A. Pazhitnov, Morse theory of closed 1-forms, Lecture Notes in Math 1474 (1991).

[R] D. Rolfsen, Knots and links, 1976.

[Sh] V.V. Sharko, Functions on Manifolds, Translations of Mathematical Monographs, vol. 131, American Mathematical Society, 1993.

[S1] J.-C. Sikorav, Thèse.

[S2] J.-C. Sikorav, Un problem de disjonction par isotopic symplectique dans un fibré cotangent, Ann. Scient. Ecole Norm. Sup. 19 (1986), 543-552.

School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel

   E-mail address: farber@math.tau.ac.il