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Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which "forces its border." One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson-Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.

Abstract. We study the cohomology and hence K-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in d dimensional Euclidean space which arise as sections of higher dimensional, periodic structures. They form one of the key families of patterns used in quasicrystal physics, where their topological invariants carry quantum mechanical information. Our work develops both a theoretical framework and a practical toolkit for the discussion and calculation of their integral cohomology, and extends previous work that only successfully addressed rational cohomological invariants. Our framework unifies the several previous methods used to study the cohomology of these patterns. We discuss explicit calculations for the main examples of icosahedral patterns in R 3 -the Danzer tiling, the Ammann-Kramer tiling and the Canonical and Dual Canonical D6 tilings, including complete computations for the first of these, as well as results for many of the better known 2 dimensional examples.

Given a $\mathbb{Z}^d$ homeomorphic action, $\alpha$, on the Cantor set, $X$, we consider the higher order continuous integer valued dynamical cohomology, $H^*(X,\alpha)$. We also consider the dynamical $K$-theory of the action, the $K$-theory of the crossed product $C^*$-algebra $C(X)\times_{\alpha}\mathbb{Z}^d$. We show that these two invariants are essentially equivalent. We also show that they only take torsion free values. Our work links the two invariants via a third invariant which is based on topological complex $K$-theory evaluated on an associated mapping torus.

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