Welcome to MAT 667

Course description: In the eighties, Bill Goldman discovered a Lie algebra structure on the free abelian group with basis the free homotopy classes of closed oriented curves on an oriented surface S. In the nineties, we generalized this Lie algebra structure to families of loops (defining the equivariant homology of the free loop space of a manifold). This Lie algebra, together with other operations in spaces of loops is known now as String Topology.

Recently, the definition of the Goldman bracket was extended to two dimensional orbifolds which can be used in the classification of three dimensional manifolds.

The course will start with the discussion of the Goldman Lie bracket in surfaces, and how it "captures" the geometric intersection number between curves. Then it will continue with the study of the generalization of the Goldman Lie bracket to orbifolds. In order to do so, we will discuss intersection numbers in group theory and orbifolds. We will continue with the study of the String Topology operations on the the based and unbased loop spaces of a manifold. We will review the classification of three manifolds in order to describe several applications of String Topology to three manifolds (These are different works of Abbaspour, Basu, Gadgil, Sullivan-Sullivan and myself).

For details information, see List of Topics and References.

Announcements

Announcements are listed in reverse choronological order: most recent announcement at the top.

20/01/12

Classes will meet for the first time on Monday Jan 23, 9:10am..

17/01/12

Break Monday, April 2 - Sunday, April 8.

13/01/12

Web page created.