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Cohomology in one-dimensional substitution tiling spaces

Marcy Barge·Beverly Diamond
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Abstract

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which ``forces its border.'' One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. For one-dimensional substitution tiling spaces, we describe a modification of the Anderson-Putnam complex on collared tiles that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology.

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Marcy Barge, Beverly Diamond. 2007-02-22. Cohomology in one-dimensional substitution tiling spaces. https://arxiv.org/abs/math/0702669

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