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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Cycle rank of Lyapunov graphs
and the genera of manifolds


Authors: R. N. Cruz and K. A. de Rezende
Journal: Proc. Amer. Math. Soc. 126 (1998), 3715-3720
MSC (1991): Primary 58F09, 58F25; Secondary 57R65
MathSciNet review: 1618654
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that the cycle-rank $r(L)$ of a Lyapunov graph $L$ on a manifold $M$ satisfies: $r(L) \leq g(M)$, where $g(M)$ is the genus of $M$. This generalizes a theorem of Franks. We also show that given any integer $r$ with $0 \leq r \leq g(M)$, $r = r(L)$ for some Lyapunov graph $L$ on $M, \dim M > 2$.


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Additional Information

R. N. Cruz
Affiliation: Departamento de Matemática Universidade Estadual de Campinas\ 13083-970 Campinas, São Paulo, Brazil
Email: cruz@turing.unicamp.br

K. A. de Rezende
Affiliation: Departamento de Matemática Universidade Estadual de Campinas\ 13083-970 Campinas, São Paulo, Brazil
Email: ketty@ime.unicamp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04957-0
PII: S 0002-9939(98)04957-0
Received by editor(s): January 22, 1997
Additional Notes: The second author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico under Grant 300072/90.2 and Fundação de Amparo à Pesquisa do Estado de São Paulo.
Communicated by: Linda Keen
Article copyright: © Copyright 1998 American Mathematical Society