On Topological Complexity of Billiard Configuration Spaces

Abstract

The Billiard Configuration Space was first introduced by Farber and Tabachnikov to explore the number of periodic billiard trajectories within a smooth, strictly convex domain in Euclidean space. Estimating the Lusternik–Schnirelmann category of the Billiard Configuration Space for a sphere is crucial for establishing bounds on the number of such periodic trajectories. A related concept—topological complexity—was introduced by Farber to address problems in motion planning. Farber, in collaboration with Yuzvinsky and later with Grant, computed the topological complexity of the (standard) Configuration Space in a series of papers. This thesis aims to extend these ideas by providing bounds on the topological complexity and calculating the Lusternik–Schnirelmann category of the Billiard Configuration Space using obstruction theory. Furthermore, we present two Salvetti-type cellular models: the first is applied to compute the fundamental group of the Billiard Configuration Space, while the second allows for the application of obstruction theory.