On the Lusternik-Schnirelman theory of a real cohomology class
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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  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.
KeywordsVector Field Smooth Manifold Cohomology Class Close Smooth Manifold Real Cohomology
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