Topology of Tiling Spaces

Topological spaces of tilings (in the same Euclidean space) are studied that are closed under translations and complete in the tiling metric (the metric describes a closeness of two tilings using translations inside a ball). This book starts with an explanation of various constructions of tilings (substitution, cut-and-project, local match and inverse limits). The next step is to study Čech cohomology of the space of tilings (by means of cohomology of inverse limits). A new metric topology is defined on certain sets of tilings, where rotations are also considered, and its cohomology is studied. The so-called pattern-equivariant cohomology is described and its equivalence with Čech cohomology for tilings is shown (the Kellendonk-Putnam theorem). The PE cohomology assigns to every cohomology group a differential form representing the tiling. One chapter is devoted to procedures making work with tilings easier, like proper substitutions, partial collaring and Barge-Diamond collaring. In the last chapter, small motions of tiles are allowed so that tiles can meet in proper parts of edges. The third metric topology adapted to this situation is defined and briefly studied. Exercises appear through the whole text and solutions of selected ones are shown at the end of the book. No index is added.

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