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 $\xi\in H^1(M,\R)$ 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.