Lusternik--Schnirelman theory for closed 1-forms | Zendy

M. Farber | Zendy

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Lusternik--Schnirelman theory for closed 1-forms

Author(s) -

M. Farber

Publication year - 2000

Publication title -

commentarii mathematici helvetici

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.603

H-Index - 46

eISSN - 1420-8946

pISSN - 0010-2571

DOI - 10.1007/s000140050117

Subject(s) - mathematics , cohomology , monodromy , morse theory , pure mathematics , class (philosophy) , novikov self consistency principle , integer (computer science) , algebra over a field , computer science , artificial intelligence , programming language

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.

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