Dirichlet Units and Critical Points of Closed 1 – Forms

S. P. Novikov developed an analog of the Morse theory for closed 1-forms. In this paper we suggest an analog of the Lusternik Schnirelman theory for closed 1-forms. For any cohomology class ξ ∈ H1(X,R) we define an integer cl(ξ) (the cuplength associated with ξ); we prove that any closed 1-form representing ξ has at least cl(ξ)− 1 critical points. The number cl(ξ) is defined using cup-products in cohomology of some flat line bundles, such that their monodromy is described by complex numbers, which are not Dirichlet units. 1. Let X be a closed manifold and let ξ ∈ H(X; R) be a nonzero cohomology class. The Novikov inequalities [N] estimate the numbers of critical points ci(ω) of different indices of any closed 1-form ω with Morse singularities on X lying in the class ξ. Novikov type inequalities were constructed in [BF1] for closed 1-forms with slightly more general singularities (non-degenerate in the sense of Bott). In [BF2] an equivariant generalization of the Novikov inequalities was found. In this paper we will consider the problem of estimating the number of critical points of closed 1-forms ω with no non-degeneracy assumption. We suggest here a version of the Lusternik Schnirelman theory for closed 1-forms. We will define (cf. below) a nonnegative integer cl(ξ), which we will call the cuplength associated with ξ. It is defined in terms of cup-products of some local systems constructed using ξ. 2. The cup-length cl(ξ). First we will define the cup-length cl(ξ) assuming that the class ξ ∈ H(X; Z) is integral. For any nonzero complex number a ∈ C∗ denote by Ea → X the complex flat line bundle determined by the following condition: the monodromy along any loop γ ∈ π1(X) is the multiplication by a〈ξ,γ〉 ∈ C. If a, b ∈ C∗ we have the canonical isomorphism of flat line bundles Ea ⊗ Eb ' Eab. Therefore we have the cup-product ∪ : H(X; Ea)⊗H(X; Eb)→ H(X; Eab). The research was supported by a grant from the Israel Academy of Sciences and Humanities and by the Herman Minkowski Center for Geometry