On the topological structure of configuration spaces

Abstract

The presented work investigates the topological structure of the configuration spaces of mechanisms. We will demonstrate the practical importance of the considered questions by giving some motivations, especially from the viewpoint of qualitative reasoning.

Although topological invariants such as the fundamental group are computable, they cannot provide us with a useful tool for classifying arbitrary mechanisms. We will show that for any finitely represented group one can construct a simple mechanism in the plane such that the fundamental group of its configuration space is isomorphic to the given group. The construction shows the undecidability of some interesting problems, e.g. the problems of homeomorpy and homotopy equivalence of configuration spaces.