Orbifold String Topology I

In these two lectures I will survey my recent work with Lupercio and Xicotencatl on the properties of the homology of the free loop space of an orbifold. In the first lecture I will review the work of Chas and Sullivan on string topology and I will describe how this theory could be generalized to the space of free loops on an orbifold. I will also review the Hochschild version of string topology due to Cohen and Jones and how this construction could also be generalized to the orbifold case. At the end of the lecture I will show some simple examples to get a flavor of the theory. In the second lecture I will show how the orbifold string topology on the symmetric product gives rise to a ring structure on the cohomology of the inertia orbifold; this we have coined the virtual cohomology. Then I will explain the relation between the virtual cohomology and the Chen-Ruan cohomology. To finish I will show how to calculate in general the virtual cohomology, and I will give some examples.