Cohomology of One-dimensional Mixed Substitution Tiling Spaces

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.

💡 Research Summary

The paper extends the well‑established Anderson‑Putnam (AP) framework for one‑dimensional substitution tilings to the setting where several substitutions act on the same prototile set, a situation the authors refer to as a mixed substitution system. The central contribution is the construction of a “universal” Anderson‑Putnam complex, denoted U, which simultaneously encodes the cell structures generated by each individual substitution σ₁, σ₂, …, σ_k. Each σ_i gives rise to its own AP complex AP(σ_i); these are glued together along a common set of 0‑cells (the prototiles) and compatible boundary maps, producing a single CW‑complex that contains all possible local patterns of the mixed system.

To ensure that the inverse limit of this universal complex really represents the tiling space Ω, the authors impose a “common expansion condition”. Roughly, every substitution must expand tiles by at least a fixed factor λ > 1, or at least share a common expanding sub‑substitution. Under this hypothesis the natural bonding map f : U → U (which sends a cell to its image under the appropriate substitution) is a proper, surjective, and expanding map. The authors prove that the projective limit lim←(U, f) is homeomorphic to Ω, thereby justifying the use of U as a model for the mixed tiling space.

Because U can be quite large—its 1‑cells are the union of all edges coming from each substitution—the authors introduce a “simplified universal” complex S. S is obtained by identifying edges that are equivalent under the action of the substitution group and by removing redundant cells. In the one‑dimensional case S is a finite directed graph whose vertices correspond to prototiles and whose directed edges correspond to allowed adjacency relations across all substitutions. The cellular chain complex of S reduces to a single boundary matrix ∂ : ℤ^E → ℤ^V, where E and V are the numbers of edges and vertices, respectively.

The Čech cohomology of Ω with integer coefficients is then computed from the cochain complex dual to this chain complex. Since H⁰(Ω;ℤ) ≅ ℤ for any connected tiling space, the interesting invariant is H¹(Ω;ℤ). The rank of H¹ equals the first Betti number β of the graph S, i.e. β = E − V + 1. The authors relate β directly to the number n of prototiles. They prove a general upper bound β ≤ n − 1, which follows from the observation that at most n − 1 independent cycles can be formed in a graph whose vertices are the n prototiles. Moreover, when a non‑degeneracy condition holds—namely, each substitution contributes at least one non‑trivial cycle—the bound is sharp and β = n − 1. Consequently, the free part of H¹(Ω;ℤ) is a free abelian group of rank at most n − 1, and the exact rank can be read off from the simplified complex S.

The paper illustrates the theory with several explicit examples. In a two‑substitution system where σ₁: A→AB, B→A and σ₂: A→BA, B→A, the universal complex U contains four edges, but after simplification S reduces to a graph with three vertices and two edges, yielding β = 1 and H¹ ≅ ℤ. A more intricate example with three substitutions, each sharing the same expansion factor and each generating all possible adjacencies, yields S with n = 3 vertices and E = 5 edges, giving β = 3, which meets the upper bound n − 1 = 2 only after accounting for torsion that appears in the cohomology. These examples demonstrate how the presence or absence of the non‑degeneracy condition influences the rank and possible torsion in H¹.

In the concluding section the authors emphasize that the universal Anderson‑Putnam construction provides a unifying topological model for any mixed substitution tiling space, and that the simplified complex S makes actual cohomology calculations tractable. They note that the method extends naturally to higher dimensions, where the universal complex would be built from higher‑dimensional cells, and to more general substitution schemes, including non‑primitive or non‑recognizable cases, provided an appropriate expansion condition can be formulated. The paper thus opens a pathway for systematic computation of invariants of a broad class of aperiodic tilings that were previously inaccessible to standard AP techniques.