On the topological structure of configuration spaces

We investigate the topological structure of configuration spaces of kinematic scenes, i.e., mechanisms consisting of rigid, independently movable objects in a 2- or 3-dimensional environment. We demonstrate the practical importance of the considered questions by giving a motivation from the viewpoint of qualitative reasoning. Especially, we investigate topological invariants of the configuration space as a means for characterizing and classifying mechanisms. This paper focuses on the topological invariants homeomorphy, homotopy equivalence and fundamental group. We describe a procedure for computing a finite representation of the fundamental group of a given kinematic scene, and investigate its possible structure and complexity for simple classes of scenes. We show that for any finitely represented group one can construct a kinematic scene in the plane such that the fundamental group of its configuration space is isomorphic to the given group. This construction shows the undecidability of a variety of problems concerning the topological structure of configuration spaces, and reveals that the considered invariants in their general form do not provide an algorithmic method for characterizing or classifying arbitrary mechanisms.