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

Jones, John; McCleary, John

doi:10.2140/agt.2016.16.2677

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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

DOI: 10.2140/agt.2016.16.2677

Abstract

Let $M$ be a closed, simply connected, smooth manifold. Let $F_{p}$ be the finite field with $p$ elements, where $p > 0$ is a prime integer. Suppose that $M$ is an $F_{p}$ –elliptic space in the sense of Félix, Halperin and Thomas (1991). We prove that if the cohomology algebra $H^{*} (M, F_{p})$ 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 $F_{p}$ from Félix, Halperin and Thomas (1991).

Keywords

string homology, closed geodesics

Mathematical Subject Classification 2010

Primary: 55P50

Secondary: 55P35, 55T05, 58E10

References

Publication

Received: 11 November 2014

Revised: 17 March 2016

Accepted: 26 March 2016

Published: 7 November 2016

Authors

John Jones
Department of Mathematics University of Warwick Coventry CV4 7AL United Kingdom

John McCleary
Mathematics Department Vassar College Poughkeepsie, NY 12604 United States

Volume 16, issue 5 (2016)