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String homology, and closed geodesics on manifolds which are elliptic spaces

John Jones and John McCleary

Algebraic & Geometric Topology 16 (2016) 2677–2690

Let M be a closed, simply connected, smooth manifold. Let Fp be the finite field with p elements, where p > 0 is a prime integer. Suppose that M is an Fp–elliptic space in the sense of Félix, Halperin and Thomas (1991). We prove that if the cohomology algebra H(M, Fp) cannot be generated (as an algebra) by one element, then any Riemannian metric on M has an infinite number of geometrically distinct closed geodesics. The starting point is a classical theorem of Gromoll and Meyer (1969). The proof uses string homology, in particular the spectral sequence of Cohen, Jones and Yan (2004), the main theorem of McCleary (1987), and the structure theorem for elliptic Hopf algebras over Fp from Félix, Halperin and Thomas (1991).

string homology, closed geodesics
Mathematical Subject Classification 2010
Primary: 55P50
Secondary: 55P35, 55T05, 58E10
Received: 11 November 2014
Revised: 17 March 2016
Accepted: 26 March 2016
Published: 7 November 2016
John Jones
Department of Mathematics
University of Warwick
United Kingdom
John McCleary
Mathematics Department
Vassar College
Poughkeepsie, NY 12604
United States