"Classifying homeomorphism of tiling spaces"Sadun, LorenzoA basic goal of algebraic topology is to classify maps between spaces up to homotopy, isotopy, or other equivalence. We show that homeomorphisms $h: \Omega \to \Omega'$ of spaces of tilings with "finite local complexity" are classified, up to homotopy and local relabeling, by the first Cech cohomology $\check H^1(\Omega, \mathbb{R}^d)$ of $\Omega$ with values in $\mathbb{R}^d$. This implies that all homeomorphisms of FLC tiling spaces can be decomposed as the composition of three pieces: $\bullet$ A map from the source space $\Omega$ to itself that is homotopic to the identity map. $\bullet$ A shape change, in which the shapes and sizes of the tiles are altered but the combinatorics of the tilings is preserved. $\bullet$ A local relabeling (e.g. breaking each "A'' tile into three smaller pieces). This is joint work with Antoine Julien. |
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