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.