Cycle rank of Lyapunov graphs and the genera of manifolds

Cruz, R.; de Rezende, K.

doi:10.1090/S0002-9939-98-04957-0

Cycle rank of Lyapunov graphs and the genera of manifolds
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by R. N. Cruz and K. A. de Rezende PDF

Proc. Amer. Math. Soc. 126 (1998), 3715-3720 Request permission

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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