The homology core of matchbox manifolds and invariant measures

Here we shall consider the topology and dynamics associated to a wide class of matchbox manifolds, including a large selection of tiling spaces and all minimal matchbox manifolds of dimension one. For such spaces we introduce topological invariants related to their expansions as an inverse sequence of simplicial complexes. These invariants are related to corresponding inverse sequences of groups arising from applying the top--dimension homology to these sequences. In many cases this leads to a computable invariant based on an inverse sequence of matrices. Significantly, we show that when the space is obtained by suspending a topologically transitive action of the fundamental group $\G$ of a closed orientable on a zero--dimensional compact space this invariant at the same time corresponds to the space of Borel measures on the Cantor set which are invariant under the action of $\G$. This leads to connections between the rank of homology groups we consider and the number of invariant, ergodic Borel probability measures for such actions. We illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures.