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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).