Classical Invariants for Global Actions and Groupoid Atlases

A global action is the algebraic analogue of a topological manifold. This construction was introduced in first place by A. Bak as a combinatorial approach to K-Theory and the concept was later generalized by Bak, Brown, Minian and Porter to the notion of groupoid atlas. In this paper we define and investigate homotopy invariants of global actions and groupoid atlases, such as the strong fundamental groupoid, the weak and strong nerves, classifying spaces and homology groups. We relate all these new invariants to classical constructions in topological spaces, simplicial complexes and simplicial sets. This way we obtain new combinatorial formulations of classical and non classical results in terms of groupoid atlases.