Wednesday, July 31, 2019 - 11:00am
The College of New Jersey
A metric on a compact manifold M gives rise to a length function on the space of maps of the circle into M, 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, relating iteration of closed geodesics and the algebraic structure given by the Chas-Sullivan product and loop coproduct on the homology of LM. We have simplified, chain-level definitions for the “loop” product and coproduct. The new definitions make possible new links between geometry and loop products. For example, If the k-fold coproduct of a homology class X on LM is nontrivial, then every representative of X contains a loop with a (k+1)-fold self-intersection. The talk will emphasize geometric motivation and examples. No knowledge of loop products or string topology will be assumed. Joint work with Nathalie Wahl.