Torsion in Tiling Homology and Cohomology

Details Torsion in Tiling Homology and Cohomology Torsion in Tiling Homology and Cohomology

2005-05-19

arxiv.org/abs/math-ph/0505048v2

Physics

Mathematical Physics

This preprint contains some errors in the algebraic analysis of the calculations and the work has been corrected and substanti

Scientific paper

The first author's recent unexpected discovery of torsion in the integral cohomology of the T\"ubingen Triangle Tiling has led to a re-evaluation 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 K-theoretic 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 K-theoretic invariants, we show that cohomology and K-theory agree for all canonical projection tilings in (physical) dimension at most 3, thus proving the existence of torsion in, for example, the K-theory of the T\"ubingen Triangle Tiling. The question of the equivalence of cohomology and K-theory for tilings of higher dimensional euclidean space remains open.

Affiliated with

Also associated with

No associations

Rating

Torsion in Tiling Homology and Cohomology does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Torsion in Tiling Homology and Cohomology, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Torsion in Tiling Homology and Cohomology will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-582178