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