  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).
