Loop products and closed geodesics

Abstract

The critical points of the length function on the free loop space $\Lambda (M)$ of a compact Riemannian manifold $M$ are the closed geodesics on $M$. The length function gives a filtration of the homology of $\Lambda (M)$, and we show that the Chas-Sullivan product ${\displaystyle H_i(\Lambda )\times H_j(\Lambda )\stackrel{*}{\to }{H}_{i+j-n}(\Lambda )}$ is compatible with this filtration. We obtain a very simple expression for the associated graded homology ring $\mathrm{Gr}{H}_{*}(\Lambda (M))$ when all geodesics are closed, or when all geodesics are nondegenerate. We also interpret Sullivan's coproduct $\vee$ (see [Su1], [Su2])) on $C_{*}(\Lambda )$ as a product in cohomology ${\displaystyle H^i(\Lambda ,{\Lambda }_0)\times H^j(\Lambda ,{\Lambda }_0)\stackrel{⊛}{\to }{H}^{i+j+n-1}(\Lambda ,{\Lambda }_0)}$ (where ${\Lambda }_0=M$ is the constant loop). We show that $⊛$ is also compatible with the length filtration, and we obtain a similar expression for the ring $\mathrm{Gr}{H}^{*}(\Lambda ,{\Lambda }_0).$ The nonvanishing of products ${\sigma }^{*n}$ and ${\tau }^{⊛n}$ 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 $(H^{*}(\Lambda ,{\Lambda }_0),⊛)$ for spheres $M=S^n$, $n\ge 3$