Monday, February 20, 2012

0505048 (Franz Gähler et al.)

Torsion in Tiling Homology and Cohomology    [PDF]

Franz Gähler, John Hunton, Johannes Kellendonk
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.
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