We study the existence of Riemannian metrics with zero topological entropy on a closed manifold 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 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 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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Gabriel P. Paternain, Jimmy Petean, Zero entropy and bounded topology. Comment. Math. Helv. 81 (2006), no. 2, pp. 287–304DOI 10.4171/CMH/53