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

Abstract

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