A metric on a compact manifold M gives rise to a length function on the

free loop space LM whose critical points are the closed geodesics

on M in the given metric. Morse theory gives a link between Hamiltonian

dynamics and the topology of loop spaces, between iteration of closed

geodesics and the algebraic structure given by the Chas-Sullivan product on

the homology of LM. PoincarA(c) Duality reveals the existence of a

related product on the cohomology of LM.

A number of known results on the existence of closed geodesics are naturally

expressed in terms of nilpotence of products. We use products to prove a

resonance result for the loop homology of spheres. There are interesting

consequences for the length spectrum, and related results

in Floer and contact theory.

Mark Goresky, Alexandru Oancea, Hans-Bert Rademacher, and Nathalie Wahl are collaborators.