String topology and the geometric decomposition of three manifolds
Moira Chas (SUNY Stony Brook)
Abstract: In the late nineties, in joint work with Dennis Sullivan, we
generalized Goldman's Lie algebra structure to a graded Lie algebra on
families of loops (defining the equivariant homology of the free loop
space of a manifold of any dimension). This graded Lie algebra,
together with other operations in spaces of loops is known now as
String Topology.
We will describe the String topology bracket on the free loop space
of three manifolds and how this structure can be used to recognize
hyperbolic and Seifert vertices and the gluing graph in the
geometrization of three manifolds.
Time permits, we will discuss other applications of String Topology to
three manifolds (These are different works of Abbaspour, Basu, Gadgil,
McGibbon, D. Sullivan, M. Sullivan and myself).