TOPOLOGY SEMINAR

4/17/2008

Nancy Hingston
College of New Jersey

Loop products and closed geodesics

The critical points of the energy function on the free loop space L(M) of a compact Riemannian manifold M are the closed geodesics on M. Filtration by the length function gives a link between the geometry of closed geodesics and the algebraic structure given by the Chas-Sullivan product on the homology of L(M). Geometry reveals the existence of a related product on the cohomology of L(M). For manifolds such as spheres and projective spaces for which there is a metric with all geodesics closed, the resulting homology and cohomology rings are nontrivial, and closely linked to the geometry. I will not assume any knowledge of the Chas-Sullivan product. Joint work with Mark Goresky.