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

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.

**1.**M. A. Bertolim, M. P. Mello, and K. A. de Rezende,*Lyapunov graph continuation*, Ergodic Theory Dynam. Systems**23**(2003), no. 1, 1–58. MR**1971195**, 10.1017/S014338570200086X**2.**M. A. Bertolim, M. P. Mello and K. A. de Rezende*Poincaré-Hopf and Morse inequalities for Lyapunov graphs*. To appear in Ergod. Th. & Dynam. Sys.**3.**Charles Conley,*Isolated invariant sets and the Morse index*, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR**511133****4.**R. N. Cruz and K. A. de Rezende,*Gradient-like flows on high-dimensional manifolds*, Ergodic Theory Dynam. Systems**19**(1999), no. 2, 339–362. MR**1685397**, 10.1017/S0143385799120893**5.**D. R. Fulkerson and O. A. Gross,*Incidence matrices and interval graphs*, Pacific J. Math.**15**(1965), 835–855. MR**0186421****6.**A. J. Hoffman and J. B. Kruskal,*Integral boundary points of convex polyhedra*, Linear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N. J., 1956, pp. 223–246. MR**0085148****7.**M. Morse.*Relations between the critical points of a real functions of independent variables*,

Trans. Am. Math. Soc.**27**(1925), 345-396.

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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:
http://dx.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