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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
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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  • 1. Andrade, P., On homology directions for flows, Japan. J. Math. (N.S.) 17 (1991), no. 1, 1-9. MR 1115590 (92e:58183)
  • 2. Basener, W., Every nonsingular flow on a closed manifold of dimension greater than two has a global transverse disk, Topology Appl. 135 (2004), no. 1-3, 131-148. MR 2024952 (2004m:57047)
  • 3. Basener, W., Transverse disks, symbolic dynamics, homology direction vectors, and Thurston-Nielson theory, Topology Appl. 153 (2006), no. 14, 2760-2764. MR 2243748 (2007j:37031)
  • 4. Collier, D., Sharp, R., Directions and equidistribution in homology for periodic orbits, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 405-415. MR 2308138 (2008b:37041)
  • 5. Fried, D., The geometry of cross sections to flows, Topology 21 (1982), no. 4, 353-371. MR 670741 (84d:58068)
  • 6. Fried, D., Flow equivalence, hyperbolic systems and a new zeta function for flows, Comment. Math. Helv. 57 (1982), no. 2, 237-259. MR 684116 (84g:58083)
  • 7. Hirsch, M., Pugh, C., Shub, M., Invariant manifolds. Lec. Notes in Math., 583. Springer-Verlag, Berlin-New York, 1977. MR 0501173 (58:18595)
  • 8. Morales, C. A., Axiom A flows with a transverse torus, Trans. Amer. Math. Soc. 355 (2003), no. 2, 735-745. MR 1932723 (2003g:37049)
  • 9. Mosher, L., Equivariant spectral decomposition for flows with a $ \mathbb{Z}$-action, Ergodic Theory Dynam. Systems 9 (1989), no. 2, 329-378. MR 1007414 (90j:58117)
  • 10. Mosher, L., Correction to: ``Equivariant spectral decomposition for flows with a $ Z$-action'' [Ergodic Theory Dynamical Systems 9 (1989), no. 2, 329-378; MR 1007414 (90j:58117)], Ergodic Theory Dynam. Systems 10 (1990), no. 4, 787-791. MR 1091427 (94h:58155)
  • 11. Mosher, L., Dynamical systems and the homology norm of a $ 3$-manifold. I. Efficient intersection of surfaces and flows, Duke Math. J. 65 (1992), no. 3, 449-500. MR 1154179 (93g:57018a)
  • 12. Mosher, L., Dynamical systems and the homology norm of a $ 3$-manifold. II, Invent. Math. 107 (1992), no. 2, 243-281. MR 1144424 (93g:57018b)
  • 13. Palis, J., Jr., de Melo, W., Geometric theory of dynamical systems. An introduction. Translated from the Portuguese by A. K. Manning. Springer-Verlag, New York-Berlin, 1982. MR 669541 (84a:58004)
  • 14. Rolfsen, D., Knots and links. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, 1990. MR 1277811 (95c:57018)
  • 15. Rotman, J., An introduction to algebraic topology. Graduate Texts in Mathematics, 119. Springer-Verlag, New York, 1988. MR 957919 (90e:55001)

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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

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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