An algebra structure on the homology of free loop spaces

Abstract

String topology studies the spaces of paths and loops in a manifold. An interesting topic in this theory is the existence of an algebra structure on the homology of free loop spaces. This algebra structure comes from the Chas-Sullivan product. On based loop spaces there is a simply way to construct a product on its homology, by concatenation of loops, however concatenation of loops is not defined in the free loop space. In this thesis we define the Chas-Sullivan product on the homology of free loop spaces and explain how this induces an algebra structure. Secondly, we will compute this algebra structure for spheres and projective spaces.