Commentarii Mathematici Helvetici


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Volume 81, Issue 2, 2006, pp. 287–304
DOI: 10.4171/CMH/53

Published online: 2006-06-30

Zero entropy and bounded topology

Gabriel P. Paternain[1] and Jimmy Petean[2]

(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