Tiling Spaces Are Cantor Set Fiber Bundles

We prove that fairly general spaces of tilings of R are fiber bundles over the torus T , with totally disconnected fiber. This was conjectured (in a weaker form) in [W3], and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a Z subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that 1) the tiles are assumed to be polygons (polyhedra if d > 2) that meet full-edge to full-edge (or full-face to full-face), 2) only a finite number of tile types are allowed, and 3) each tile type appears in only a finite number of orientations. The proof is constructive, and we illustrate it by constructing a “square” version of the Penrose tiling system.