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 we define an integer \( \cl(\xi) \) (the cup-length associated with ); we prove that any closed 1-form representing 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.
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M. Farber, Lusternik--Schnirelman theory for closed 1-forms. Comment. Math. Helv. 75 (2000), no. 1, pp. 156–170DOI 10.1007/S000140050117