Commentarii Mathematici Helvetici
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Published online: 2006-06-30
Zero entropy and bounded topologyGabriel P. Paternain and Jimmy Petean (1) University of Cambridge, United Kingdom
(2) Guanajuato, Mexico
We study the existence of Riemannian metrics with zero topological entropy on a closed manifold $M$ with infinite fundamental group. We show that such a metric does not exist if there is a finite simply connected CW complex which maps to $M$ in such a way that the rank of the map induced in the pointed loop space homology grows exponentially. This result allows us to prove in dimensions four and five, that if $M$ admits a metric with zero entropy then its universal covering has the rational homotopy type of a finite elliptic CW complex. We conjecture that this is the case in every dimension.
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Paternain Gabriel, Petean Jimmy: Zero entropy and bounded topology. Comment. Math. Helv. 81 (2006), 287-304. doi: 10.4171/CMH/53