String Topology and Cacti

Specialeforsvar: Erik Søndergaard Gimsing

Titel: String Topology and Cacti

Abstract: Let M be a finite dimensional Riemannian manifold. We describe the circle action, the Chas-Sullivan product and the Goresky-Hingston coproduct geometrically and make a computation via geometric intersection of chains. Pursuing a proof of the BV-algebra structure given by the circle action and the Chas-Sullivan product we describe the basics of operads. We then rigorously construct the topological quasi-operad Cacti1 in terms of graph theory and prove results related to its cellular structure. We proceed to utilize this cell structure to sketch a possible way to construct an action of H∗(Cacti1) on H∗(LM) reproducing the circle action and the Chas-Sullivan product appropriately thereby showing the existence of the BV-algebra structure. Finally we briefly discuss how the sketch might be completed to a full proof.

Vejleder: Nathalie Wahl
Censor:    Iver Mølgaard Ottosen, DTU