|
Poincaré-Hopf inequalities
Author(s):
M.
A.
Bertolim;
M.
P.
Mello;
K.
A.
de Rezende
Journal:
Trans. Amer. Math. Soc.
357
(2005),
4091-4129.
MSC (2000):
Primary 37B30, 37B35, 37B25;
Secondary 54H20
Posted:
October 28, 2004
Retrieve article in:
PDF DVI PostScript
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.
References:
-
- 1.
- M. A. Bertolim, M. P. Mello and K. A. de Rezende
Lyapunov graph continuation, Ergod. Th. & Dynam. Sys. 23 (2003), 1-58. MR 2004b:37042 - 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.
- C. Conley.
Isolated invariant sets and the Morse index,
CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, RI, 1978. MR 80c:58009 - 4.
- R. N. Cruz and K. A. de Rezende.
Gradient-like flows on high-dimensional manifolds,
Ergod. Th. & Dynam. Sys. 19 (1999), no. 2, 339-362.MR 2001a:37026 - 5.
- D. R. Fulkerson and O. A. Gross.
Incidence matrices and interval graphs,
Pacific Journal of Mathematics 15 (1965), 835-855. MR 32:3881 - 6.
- A. J. Hoffman and J. B. Kruskal.
Integral boundary points of convex polyhedra.
In Linear Inequalities and Related Systems (H. W. Kuhn and A. W. Tucker, eds.), 223-246, Annals of Mathematics Studies, No. 38, Princeton University Press, Princeton, NJ, 1956. MR 18:980b - 7.
- M. Morse.
Relations between the critical points of a real functions of  independent variables,
Trans. Am. Math. Soc. 27 (1925), 345-396.
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
37B30, 37B35, 37B25,
54H20
Retrieve articles in all Journals with MSC
(2000):
37B30, 37B35, 37B25,
54H20
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:
10.1090/S0002-9947-04-03641-4
PII:
S 0002-9947(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
Posted:
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
Copyright of article:
Copyright
2004,
American Mathematical Society
|
Comments: webmaster@ams.org