The deficiency of a cohomology class

Abstract

We define the deficiency of a cohomology class u 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 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.