The deficiency of a cohomology class

Morales, C.

doi:10.1090/S0002-9939-2010-10433-1

The deficiency of a cohomology class
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by C. A. Morales PDF

Proc. Amer. Math. Soc. 138 (2010), 4321-4329 Request permission

Abstract:

We define the deficiency of a cohomology class $u$ with respect to a vector field as the set of limit points in the ambient manifold of long almost closed orbits representing homology classes on which $u$ is nonpositive. We prove that, up to infinite cyclic coverings, the sole vector fields on closed manifolds exhibiting nonzero cohomology classes with finite deficiency are the gradient-like ones. We also prove that if the manifold is not a sphere, every singularity is hyperbolic and there is a closed transverse submanifold intersecting all regular orbits, then there is also a nonzero cohomology class with finite deficiency.

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