Cohomology of One-dimensional Mixed Substitution Tiling Spaces

📝 Abstract

We compute the Cech cohomology with integer coefficients of one-dimensional tiling spaces arising from not just one, but several different substitutions, all acting on the same set of tiles. These calculations involve the introduction of a universal version of the Anderson-Putnam complex. We show that, under a certain condition on the substitutions, the projective limit of this universal Anderson-Putnam complex is isomorphic to the tiling space, and we introduce a simplified universal Anderson-Putnam complex that can be used to compute Cech cohomology. We then use this simplified complex to place bounds on the rank of the first cohomology group of a one-dimensional substitution tiling space in terms of the number of tiles.

💡 Analysis

We compute the Cech cohomology with integer coefficients of one-dimensional tiling spaces arising from not just one, but several different substitutions, all acting on the same set of tiles. These calculations involve the introduction of a universal version of the Anderson-Putnam complex. We show that, under a certain condition on the substitutions, the projective limit of this universal Anderson-Putnam complex is isomorphic to the tiling space, and we introduce a simplified universal Anderson-Putnam complex that can be used to compute Cech cohomology. We then use this simplified complex to place bounds on the rank of the first cohomology group of a one-dimensional substitution tiling space in terms of the number of tiles.

📄 Content

COHOMOLOGY OF ONE-DIMENSIONAL MIXED SUBSTITUTION TILING SPACES FRANZ G¨AHLER AND GREGORY R. MALONEY Abstract. We compute the Cech cohomology with integer coef- ficients of one-dimensional tiling spaces arising from not just one, but several different substitutions, all acting on the same set of tiles. These calculations involve the introduction of a universal version of the Anderson-Putnam complex. We show that, under a certain condition on the substitutions, the projective limit of this universal Anderson-Putnam complex is isomorphic to the tiling space, and we introduce a simplified universal Anderson-Putnam complex that can be used to compute Cech cohomology. We then use this simplified complex to place bounds on the rank of the first cohomology group of a one-dimensional substitution tiling space in terms of the number of tiles.

1. Introduction and Definitions The purpose of this work is to investigate the cohomology groups of tilings obtained by mixing several different substitutions. Many of the results can be proven for tilings in arbitrary dimension, so the notation and definitions for general tiling spaces will be introduced in Sections 1.1, 1.2, and 1.3. Nevertheless, special attention will be given to the class of one-dimensional tilings, and in this context it is often easier to work with symbolic substitutions and subshifts; accordingly, the relevant notions for symbolic substitutions will be introduced in Section 1.4. This work follows the paper [1] very closely, and many of the defini- tions and notations relating to tilings are taken from that source. An excellent introduction to the theory of topology of tiling spaces can be found in [7]. The definitions and notations relating to symbolic shift spaces are standard; see [6] for an introduction. 1.1. Tilings. A tile is a subset of Rd that is homeomorphic to the closed unit ball. A partial tiling T is a set of tiles, any two of which intersect only on their boundaries (let us denote the boundary of a set Date: March 5, 2012. 2010 Mathematics Subject Classification. Primary: 37B10, 55N05 Secondary: 54H20, 37B50, 52C23. Key words and phrases. Cohomology, tiling spaces, substitution. Both authors were partly supported by the German Research Council (DFG), via CRC 701. 1 arXiv:1112.1475v2 [math.DS] 6 Mar 2012 2 FRANZ G¨AHLER AND GREGORY R. MALONEY S by ∂(S)). The support of T, denoted Supp(T), is the union of its tiles. A tiling is a partial tiling, the support of which is Rd. When we need different tiles that look alike, let us associate a label with each tile; in such cases, let us consider a tile to be an ordered pair consisting of the set and the label. Given a partial tiling T and a vector u ∈Rd, define the translation of T by u to be T + u = {t + u : t ∈T}, where, for a tile t, t + u = {x + u : x ∈t}. Any set of tilings of Rd can be equipped with a metric, in which two tilings are close if, up to a small translation, they agree on a large ball around the origin. There are several ways to define a metric in this way, all of which give rise to the same topology. Let us use the metric defined in [1]: for any two tilings T, T ′ of Rd, D(T, T ′) := inf  {1/ √ 2} ∪{ϵ : T + u and T ′ + v agree on B1/ϵ(0) for some ∥u∥, ∥v∥< ϵ}) . With respect to the topology arising from this metric, the action of Rd by translation is continuous. 1.2. Substitutions. Let {p1, . . . , pk} be a finite set of tiles, called pro- totiles. Let ˜Ωdenote the set of all partial tilings that contain only trans- lates of these prototiles. A substitution φ is a map from {p1, . . . , pk} to ˜Ωfor which there exists an inflation constant λ > 1 such that, for all i ≤k, the support of φ(pi) is λpi. Then φ can be extended to a map φ : ˜Ω→˜Ωby φ(T) = [ pi+u∈T (φ(pi) + λu). Then the tiling space or hull Ωφ is the set of all tilings T ∈˜Ωsuch that, for any partial tiling P ⊆T with bounded support, we have P ⊆φn(pi + u) for some prototile pi and some vector u. Note that φ(Ωφ) ⊆Ωφ. Remark 1.1. The substitution tiling spaces considered in [1] all satisfy the following three conditions. (1) φ is one-to-one on Ωφ. This is required in order for φ|Ωφ to have an inverse. By [8], φ is one-to-one on Ωφ if and only if Ωφ consists only of non-periodic tilings. (2) φ is primitive. This means that there exists some n ≥1 such that, for any two prototiles pi, pj, some translate of pi appears in φn(pj). (3) Ωφ has finite local complexity (FLC). This means that, for each positive real number R, there are, up to translation, only finitely many partial tilings that are subsets of tilings in Ωφ and that have supports with diameter less than R. COHOMOLOGY OF MIXED SUBSTITUTION TILING SPACES 3 Let us consider only substitutions that satisfy these three conditions, in addition to the following extra condition, which is a hypothesis of some of the theorems in [1]. (4) The prototiles of φ have a CW-structure with respect to which the tilings in Ωφ are edge-to-edge, which means that, given any two subcells of tiles in the same

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