Lusternik--Schnirelman theory for closed 1-forms

Abstract

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 $\xi \in H^1(M,\mathbb{R})$ we define an integer \( \cl(\xi) \) (the cup-length associated with $\xi$ ); we prove that any closed 1-form representing $\xi$ has at least \( \cl(\xi)-1 \) critical points. The number \( \cl(\xi) \) 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.