Automorphisms of topological Markov shifts and Wagoner’s complexes

A topological Markov shift is the set of two sided inifinite paths in a finite directed graph endowed with the product topology and with the left shift acting on this space. The automorphisms of the space are the shift commuting self-homeomorphisms. Wagoner realized the automorphism group of a topological Markov shift as the fundamental group of a certain CW complex. This construction has been crucial in many results regarding automorphisms and isomorphism in symbolic dynamics. We give a simplified construction of this complex, which also works in more general contexts, and sketch some applications.