JournalscmhVol. 75 , No. 1DOI 10.1007/s000140050117

Lusternik--Schnirelman theory for closed 1-forms

  • M. Farber

    Tel-Aviv University, Ramat-Aviv, Israel
Lusternik--Schnirelman theory for closed 1-forms cover


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 ξH1(M,R)\xi\in H^1(M,\R) we define an integer \cl(ξ)\cl(\xi) (the cup-length associated with ξ\xi ); we prove that any closed 1-form representing ξ\xi has at least \cl(ξ)1\cl(\xi)-1 critical points. The number \cl(ξ)\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.