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Transactions of the American Mathematical Society

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Poincaré-Hopf inequalities


Authors: M. A. Bertolim, M. P. Mello and K. A. de Rezende
Journal: Trans. Amer. Math. Soc. 357 (2005), 4091-4129
MSC (2000): Primary 37B30, 37B35, 37B25; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9947-04-03641-4
Published electronically: October 28, 2004
MathSciNet review: 2159701
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Abstract | References | Similar Articles | Additional Information

Abstract: In this article the main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse inequalities to hold. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.


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Additional Information

M. A. Bertolim
Affiliation: Department of Mathematics, Institute of Mathematics, Statistics and Scientific Computation, Unicamp, Campinas, São Paulo, Brazil
Email: bertolim@ime.unicamp.br

M. P. Mello
Affiliation: Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific Computation, Unicamp, Campinas, São Paulo, Brazil
Email: margarid@ime.unicamp.br

K. A. de Rezende
Affiliation: Department of Mathematics, Institute of Mathematics, Statistics and Scientific Computation, Unicamp, Campinas, São Paulo, Brazil
Email: ketty@ime.unicamp.br

DOI: https://doi.org/10.1090/S0002-9947-04-03641-4
Keywords: Conley index, Morse inequalities, Morse polytope, integral polytope, network-flow theory
Received by editor(s): February 6, 2003
Received by editor(s) in revised form: December 2, 2003
Published electronically: October 28, 2004
Additional Notes: The first author was supported by FAPESP under grant 02/08400-3
The second author was supported by CNPq-PRONEX Optimization and by FAPESP under grant 01/04597-4
The third author was partially supported by FAPESP under grants 00/05385-8 and 02/102462, and by CNPq under grant 300072
Article copyright: © Copyright 2004 American Mathematical Society