Loop Products and Closed Geodesics

The critical points of the length function on the free loop space Λ(M) of a compact Riemannian manifold M are the closed geodesics on M. The length function gives a filtration of the homology of Λ(M) and we show that the Chas-Sullivan product Hi(Λ)×Hj(Λ) ∗ Hi+j−n(Λ) is compatible with this filtration. We obtain a very simple expression for the associated graded homology ring GrH∗(Λ(M)) when all geodesics are closed, or when all geodesics are nondegenerate. We also interpret Sullivan’s coproduct ∨ [Su1, Su2] on C∗(Λ) as a product in cohomology H(Λ,Λ0)×H(Λ,Λ0) H(Λ,Λ0) (where Λ0 = M is the constant loops). We show that ⊛ is also compatible with the length filtration, and we obtain a similar expression for the ring GrH(Λ,Λ0). The non-vanishing of products σ and τ is shown to be determined by the rate at which the Morse index grows when a geodesic is iterated.