Tiling deformations, cohomology, and orbit equivalence of tiling spaces

We study homeomorphisms of tiling spaces with finite local complexity (FLC), of which suspensions of $d$-dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the FLC case, we construct a cohomological invariant of homeomorphisms, and show that all homeomorphisms are a combination of tiling deformations, translations, and local equivalences (MLD). In the ILC case, we construct a cohomological invariant in the so-called weak cohomology, and show that all orbit equivalences are combinations of tiling deformations, translations, and topological conjugacies. These generalize results of Parry and Sullivan to higher dimensions. When the tiling spaces are uniquely ergodic, we show that homeomorphisms (FLC) or orbit equivalences (ILC) are completely parametrized by the appropriate cohomological invariants. We also show that, under suitable cohomological conditions, continuous maps between tiling spaces are homotopic to compositions of tiling deformations and either local derivations (FLC) or factor maps (ILC).