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Cohomology of substitution tiling spaces

Published online by Cambridge University Press:  04 November 2009

MARCY BARGE ,
BEVERLY DIAMOND ,
JOHN HUNTON  and
LORENZO SADUN
MARCY BARGE
Affiliation:
Department of Mathematics, Montana State University, Bozeman, MT 59717, USA (email: [email protected])
BEVERLY DIAMOND
Affiliation:
Department of Mathematics, College of Charleston, Charleston, SC 29424, USA (email: [email protected])
JOHN HUNTON
Affiliation:
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK (email: [email protected])
LORENZO SADUN
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA (email: [email protected])
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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. In earlier work, Barge and Diamond described a modification of the Anderson–Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 6 , December 2010 , pp. 1607 - 1627
Copyright
Copyright © Cambridge University Press 2009

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