A Spectral Sequence for the K-theory of Tiling Spaces

Let $\text{Tt}$ be an aperiodic and repetitive tiling of ${\text{RM}}^d$ with finite local complexity. We present a spectral sequence that converges to the $K$-theory of $\text{Tt}$ with $E_2$-page 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 $\text{Tt}$ 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 \v{C}ech cohomology of the hull of $\text{Tt}$ (a compactification of the family of its translates).