Lusternik—Schnirelman theory for 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 x ? H¹(M,\R) \xi\in H^1(M,\R) we define an integer \cl(x) \cl(\xi) (the cup-length associated with x \xi ); we prove that any closed 1-form representing x \xi has at least \cl(x)-1 \cl(\xi)-1 critical points. The number \cl(x) \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.