The primary aim of the course is to explain some recent homotopy theoretical results about the free loop space of a manifold. We hope to suggest new relations between these and certain other branches of mathematics.
String topology was introduced by Sullivan and Chas. Building on geometrical ideas by Goldmann, algebraic ideas due to Gerstenhaber and an analogy to quantum field theory from physics, they constructed a framework for studying the topology of the free loop space on a manifold. They produce a number of algebraic structures on the homology of the free loop space and the homology of the its homotopy orbits under the action of the circle.
The constructions have later been greatly clarified by homotopy theoretical methods. We will focus on this point of view, and also on the related operad theoretical aspects of the situation. This will also give connections to topological field theories.
We will touch the relationship between string topology, quantum field theory and Floer homology. These relations are very exciting, but unfortunately mostly conjectural.
As another generalization of Goldman's brackets, Andersen and Reshetikhin introduced in 1995 the Poisson algebra of chord diagrams on an oriented 2-dimensional surface. This Poisson bracket and it quantization using links in a cylinder over the surface will be discussed. An understanding of the relation between these constructions and the Sullivan-Chas program would be very interesting.
Revised 19.12.2006
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