Cohomology of substitution tiling spaces

Abstract

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.

Cohomology of rotational tiling spaces

“…A common topological invariant employed to study tiling spaces is Čech cohomology. For a two‐dimensional substitution tiling, Barge, Diamond, Hunton and Sadun gave a spectral sequence converging to the Čech cohomology of the Euclidean hull . The entries of the second page of this spectral sequence are determined by the number of tilings in the hull with non‐trivial rotational symmetry (assumed all to be of the same order) and the Čech cohomology of the quotient space , given by identifying tilings of the Euclidean hull which agree up to a rotation at the origin.…”

On the Pisot Substitution Conjecture

“…If Φ is unimodular, then Ξ(Φ) is a finitely generated free Z-module: let D = D(Φ) := rank( Ξ(Φ) ). The following is proved in [27] using the global shadowing technique in hyperbolic dynamics pioneered by Franks [54]. (Ω φ ) induced on first cohomology is injective and there is r ∈ N so that G is a.e.…”

“…This limits the range of the above conjectures. They can be strengthened, as can Conjectures 4.4 and 4.5, by replacing the cohomology of Ω Φ by its essential cohomology, which does not see, for example, the contributions of asymptotic cycles (see [27]). The resulting conjectures imply the non-homological conjectures in the unimodular case.…”

K-theory of the chair tiling via AF-algebras

“…In the commutative case, Barge-Diamond-Hunton-Sadun [13] have produced a machinery to compute the cohomology of a tiling space. They compute the cohomology of Ω with exact sequences arising from a filtration based on the geometry of the tiles (vertices, edges, and faces), and use relative cohomology theory.…”