doi:10.1017/S0143385708000539 Printed in the United Kingdom A spectral sequence for the K-theory of tiling spaces

Abstract. Let T be an aperiodic and repetitive tiling of Rd 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 the Čech cohomology of the hull of T (a compactification of the family of its translates). 1. Main results Let T be an aperiodic and repetitive tiling ofRd with finite local complexity (Definition 3). The hull is a compactification, with respect to an appropriate topology, of the family of translates of T by vectors of Rd (Definition 4). The tiles of T are given compatible 1-complex decompositions (§4.2), with each simplex punctured, and the 1-transversal 41 is the subset of 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 via the natural action of the group Rd on itself by translation [13]. The C∗-algebra of the hull is isomorphic to the crossed product C∗-algebra C()oRd. There is a map po from the hull onto the prototile space (Proposition 1) 41 ↪ → ↓ p