Loop products and closed geodesics

The critical points of the length function on the free loop space (M) of a compact Riemannian manifold M are the closed geodesics on M. The length function gives a filtration of the homology of (M), and we show that the Chas-Sullivan product Hi( ) × Hj ( ) ∗ Hi+j−n( ) is compatible with this filtration. We obtain a very simple expression for the associated graded homology ring GrH∗( (M)) when all geodesics are closed, or when all geodesics are nondegenerate. We also interpret Sullivan’s coproduct ∨ (see [Su1], [Su2]) on C∗( ) as a product in cohomology H( , 0) × H ( , 0) Hi+j+n−1( , 0) (where 0 = M is the constant loop). We show that is also compatible with the length filtration, and we obtain a similar expression for the ring GrH ∗( , 0). The nonvanishing of products σ ∗n and τ n is shown to be determined by the rate at which the Morse index grows when a geodesic is iterated. We determine the full ring structure (H ∗( , 0), ) for spheres M = S, n ≥ 3.