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, Volume 113, Issue 1, pp 85–106 | Cite as

On the Lusternik-Schnirelman theory of a real cohomology class

  • D. SchützEmail author


Farber developed a Lusternik-Schnirelman theory for finite CW-complexes X and cohomology classes ξ H 1 (X;ℝ). This theory has similar properties as the classical Lusternik-Schnirelman theory. In particular in [7] Farber defines a homotopy invariant cat(X,ξ) as a generalization of the Lusternik-Schnirelman category. If X is a closed smooth manifold this invariant relates to the number of zeros of a closed 1-form ω representing ξ. Namely, a closed 1-form ω representing ξ which admits a gradient-like vector field with no homoclinic cycles has at least cat(X,ξ) zeros. In this paper we define an invariant F(X,ξ) for closed smooth manifolds X which gives the least number of zeros a closed 1-form representing ξ can have such that it admits a gradient-like vector field without homoclinic cycles and give estimations for this number.


Vector Field Smooth Manifold Cohomology Class Close Smooth Manifold Real Cohomology 
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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Mathematisches InstitutWestfälische Wilhelms-Universität MünsterMünsterGermany

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