A note on homology for Smale spaces

TL;DR

This paper clarifies the homology theory for Smale spaces by demonstrating the equivalence of different complexes, introducing a simplicial framework, and discussing projective resolutions within dynamical systems.

Contribution

It provides a convergence theorem for spectral sequences, a simplicial interpretation of homology complexes, and insights into projective resolutions for Smale spaces.

Findings

All complexes yield the same homology due to spectral sequence convergence.

A simplicial framework relates complexes to symmetric Moore complexes.

Projective covers of Smale spaces are realized by shift space systems.

Abstract

We collect three observations on the homology for Smale spaces defined by Putnam. The definition of such homology groups involves four complexes. It is shown here that a simple convergence theorem for spectral sequences can be used to prove that all complexes yield the same homology. Furthermore, we introduce a simplicial framework by which the various complexes can be understood as suitable "symmetric" Moore complexes associated to the simplicial structure. The last section discusses projective resolutions in the context of dynamical systems. It is shown that the projective cover of a Smale space is realized by the system of shift spaces and factor maps onto it.