JournalscmhVol. 81 , No. 2DOI 10.4171/cmh/53

Zero entropy and bounded topology

  • Gabriel P. Paternain

    University of Cambridge, United Kingdom
  • Jimmy Petean

    Guanajuato, Mexico
Zero entropy and bounded topology cover

Abstract

We study the existence of Riemannian metrics with zero topological entropy on a closed manifold MM 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 MM 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 MM 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.