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The deficiency of a cohomology class


Author: C. A. Morales
Journal: Proc. Amer. Math. Soc. 138 (2010), 4321-4329
MSC (2000): Primary 37C10; Secondary 37C50
DOI: https://doi.org/10.1090/S0002-9939-2010-10433-1
Published electronically: May 24, 2010
MathSciNet review: 2680058
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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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Additional Information

C. A. Morales
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil
Email: morales@impa.br

DOI: https://doi.org/10.1090/S0002-9939-2010-10433-1
Keywords: Cohomology class, deficiency, vector field
Received by editor(s): October 28, 2009
Received by editor(s) in revised form: February 1, 2010
Published electronically: May 24, 2010
Additional Notes: This research was partially supported by CNPq, FAPERJ and PRONEX-Brazil
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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