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 | References | Similar Articles | Additional Information

Abstract: We define the *deficiency* of a cohomology class 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 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.

**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 -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 -action'' [Ergodic Theory Dynamical Systems 9 (1989), no. 2, 329-378; MR*, Ergodic Theory Dynam. Systems 10 (1990), no. 4, 787-791. MR**1007414 (90j:58117)**]**1091427 (94h:58155)****11.**Mosher, L.,*Dynamical systems and the homology norm of a -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 -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

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.