Torsion in Tiling Homology and Cohomology
Abstract
The first author's recent unexpected discovery of torsion in the integral cohomology of the Tübingen Triangle Tiling has led to a reevaluation of current descriptions of and calculational methods for the topological invariants associated with aperiodic tilings. The existence of torsion calls into question the previously assumed equivalence of cohomological and Ktheoretic invariants as well as the supposed lack of torsion in the latter. In this paper we examine in detail the topological invariants of canonical projection tilings; we extend results of Forrest, Hunton and Kellendonk to give a full treatment of the torsion in the cohomology of such tilings in codimension at most 3, and present the additions and amendments needed to previous results and calculations in the literature. It is straightforward to give a complete treatment of the torsion components for tilings of codimension 1 and 2, but the case of codimension 3 is a good deal more complicated, and we illustrate our methods with the calculations of all four icosahedral tilings previously considered. Turning to the Ktheoretic invariants, we show that cohomology and Ktheory agree for all canonical projection tilings in (physical) dimension at most 3, thus proving the existence of torsion in, for example, the Ktheory of the Tübingen Triangle Tiling. The question of the equivalence of cohomology and Ktheory for tilings of higher dimensional euclidean space remains open.
 Publication:

arXiv eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.mathph/0505048
 arXiv:
 arXiv:mathph/0505048
 Bibcode:
 2005math.ph...5048G
 Keywords:

 Mathematical Physics;
 52C23;
 37B50
 EPrint:
 This preprint contains some errors in the algebraic analysis of the calculations and the work has been corrected and substantially extended in math.kt/1202.2240 "Integral cohomology of rational projection method pattern " (same authors) which the reader should consult instead