A Spectral Sequence for the K–theory of Tiling Spaces

Let T be an aperiodic and repetitive tiling of R with finite local complexity. We present a spectral sequence that converges to the K-theory of T with page-2 isomorphic to the Pimsner cohomology of T . It is a generalization of Serre spectral sequence to a class of spaces which are not fibered. The Pimsner 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 the Čech cohomology of the hull of T (the closure for an appropriate topology of the family of its translates). 1 Main Results Let T be an aperiodic and repetitive tiling of R with finite local complexity (definition 3). The hull Ω is a compactification, for an adequate topology, of the family of translates of T by vectors of R (definition 4). The tiles of T are given a simplicial decomposition, with each simplex punctured, and the simplicial transversal Ξs is the subset of Ω corresponding to translates of T having the puncture of one of those simplices at the origin 0Rd. The prototile space B0 (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 with the homeomorphic action of the group R by translation [13]. The C∗-algebra of the hull is the crossed-product C∗-algebra C(Ω)⋊R. There is a projection (proposition 1) ∗Work supported by the NSF grants no. DMS-0300398 and no. DMS-0600956.