A spectral sequence for theK-theory of tiling spaces

Abstract

Let T be an aperiodic and repetitive tiling of R d with finite local complexity. We present a spectral sequence that converges to the K-theory of T with page-2 given by a new cohomology that will be called PV in reference to the Pimsner-Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of T generalizes the cohomology of the base space of a fibration with local coefficients in the K-theory of its fiber. We prove that it is isomorphic to thě Cech cohomology of the hull of T (a compactification of the family of its translates). Main ResultsLet T be an aperiodic and repetitive tiling of R d with finite local complexity (definition 3). The hull Ω is a compactification, with respect to an appropriate topology, of the family of translates of T by vectors of R d (definition 4). The tiles of T are given compatible ∆-complex decompositions (section 4.2), with each simplex punctured, and the ∆-transversal Ξ ∆ is the subset of Ω corresponding to translates of T having the puncture of one of those simplices at the origin 0 R d . The prototile space B 0 (definition 9) is built out of the prototiles of T (translational equivalence classes of tiles) by gluing them together according to the local configurations of their representatives in the tiling. The hull is given a dynamical system structure via the natural action of the group R d on itself by translation [13]. The C * -algebra of the hull is isomorphic to the crossed-productThere is a map p o from the hull onto the prototile space (proposition 1)which, thanks to a lamination structure on Ω (remark 2), resembles (although is not) a fibration with base space B 0 and fiber Ξ ∆ (remark 4). The Pimsner-Voiculescu (PV) *

On the K-theory of the stable C⁎-algebras from substitution tilings

“…Actually, in the first part of [10], we introduced a new C * -algebra, which we denote by C * r (G 0 ), associated to any tiling with finite local complexity, not necessarily a substitution one, and in this paper we will show how to compute the K-theory of these C * -algebras. Of course there are others C * -algebras associated to non-substitution tilings, and mathematicians and physicists like Bellissard, Forrest, Hunton, Kallendonk, among others, have worked on their K-theory (see, for example, [2,3,6,7,14,15,22]). In the case of an aperiodic substitution tiling, in [10], we have used the algebras C * r (G 0 ) as building blocks to another C * -algebra, which we denote by C * r (G), that is strong Morita equivalent to G s , and hence has the same K-theory as G s .…”

Tiling Groupoids and Bratteli Diagrams

“…This Theorem was first proved in [7] and it was also proved that the R daction could also be recovered from this construction (see also [38,Theorem 5]). The reader can also easily adapt the proof of Theorem 3.6 given in Sect.…”

“…The next definition is a reformulation of the AndersonPutnam space [2] that some of the authors gave in [38].…”

Homology and topological full groups of étale groupoids on totally disconnected spaces

“…Nevertheless, for certain classes of Z N -actions, it is known that the isomorphisms ( * ) hold. We refer the reader to [2], [9] and [28] for detailed information. We also remark that the isomorphisms ( * ) hold for AF groupoids andétale groupoids arising from subshifts of finite type (see Theorem 4.10, 4.11, 4.14).…”