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manuscripta mathematica

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

On the Lusternik-Schnirelman theory of a real cohomology class

  • D. SchützEmail author
Article

Abstract.

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.

Keywords

Vector Field Smooth Manifold Cohomology Class Close Smooth Manifold Real Cohomology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Mathematisches InstitutWestfälische Wilhelms-Universität MünsterMünsterGermany

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