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.
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Mathematics Subject Classification (2000): Primary 37C29; Secondary 58E05
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Schütz, D. On the Lusternik-Schnirelman theory of a real cohomology class. manuscripta math. 113, 85–106 (2004). https://doi.org/10.1007/s00229-003-0423-z
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DOI: https://doi.org/10.1007/s00229-003-0423-z