TOPOLOGY OF (SOME) TILING SPACES WITHOUT FINITE LOCAL COMPLEXITY

A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have"fault lines", along which tiles can slide past one another. We investigate the topology of a certain class of tiling spaces of this type. We show that they can be written as inverse limits of CW complexes, and their Cech cohomology is related to properties of the fault lines.


1.
Introduction. In discussing tilings, a standard assumption is that tiles can meet in only a finite number of ways, up to rigid motion. Equivalently, for any radius R, there are only a finite number of patches of radius R in the tiling, up to rigid motion. This condition is called finite local complexity, or FLC. 1 In this paper we consider tilings that do not meet the FLC condition. We show that spaces of such tilings can be given a natural topology in which they are compact. Many of the techniques used for FLC tilings, such as inverse limit constructions and cohomology calculations, can be modified to handle non-FLC tilings. In particular, we work out a number of simple examples, and prove theorems about a broader class of examples.
1.1. Past work. The first substantial work on non-FLC tilings was done by Kenyon [9], who considered a substitution that was combinatorially equivalent to a → a a a a a a a a a , but in which one of the columns was shifted by an irrational distance relative to the other two. (The tile itself was not a polygon. Rather, it had straight edges on the left and right, while the top and bottom had fractal pieces that looked like the devil's staircase). Upon further substitution, the shift between the columns would grow to an arbitrarily large multiple of the original irrational distance. In  the limit, an infinite line of tile edges would appear, along which tiles could face one another in an infinite variety of ways. Such a line is an example of a "fault line", which we now define.
Given a finite set of prototiles (usually assumed to be polygonal, or perhaps closed topological disks), a tiling is a covering of R d by rigid motions of copies of these prototiles that are only allowed to intersect on their boundaries. A tiling space X is any translation-invariant set of such tilings that is closed under the third topology of section 1.2. Let T ∈ X be a tiling containing an infinite line (or ray) of tile edges, and let v be a unit vector parallel to . We say that is a fault line if for every > 0 there is a tiling T ∈ X such that: 1. On one half-(or quarter-) plane with boundary , T = T , and 2. there is a t with 0 < |t| ≤ such that on the other half-(or quarter-) plane with boundary we have T = T + t v. If for every sufficiently small > 0 it is possible to choose t = , we call a regular fault line.
Fault lines play a central role in any discussion of non-FLC tilings. This is because Kenyon also proves [10] that if a tiling that is made from a finite prototile set has an infinite number of inequivalent two-tile patches, then those patches occur along straight edges, or they occur along an entire circle of tile boundaries. The former case leads to fault lines, while the latter can only occur for special prototile sets (and never, for instance, in primitive substitution tilings [5]).
In 1998, Sadun [13] proposed some generalizations of the pinwheel tiling. One such example (Til(1/2)) uses polygonal tiles (two similar right triangles) that appear in infinitely many orientations, and has a fault line. Interestingly, this example meets the conditions of Goodman-Strauss' matching rules theorem [7]. To wit: there exists a finite collection of tiles and a finite set of local matching rules such that these tiles tile the plane, but only in a manner that is locally equivalent to the generalized pinwheel. That is, a finite set of local rules forces a global hierarchical structure that in turn forces infinite local complexity.
Danzer [4] extended the theory of FLC tilings and provided additional examples. Finally, Frank and Robinson [6,5] have considered a large family of "direct product variation" (DPV) tilings. These are obtained from products of 1-dimensional substitutions by rearranging the positions of the tiles within an order-1 supertile. The examples of this paper are all DPV tilings in which the rearrangements are purely horizontal, thereby preserving the decomposition of the tiling into rows.
In all cases, the essential feature that prevents FLC is the presence of a fault line. Along the fault line, the evolution is described by two 1-dimensional substitutions. One describes what happens along one side of the fault line, while the other describes what happens along the other side. The two substitutions cannot be Pisot with the same stretching factor, because in that case the FLC property is preserved [5] and there is no possibility of a fault line. However, if the substitutions are not Pisot, then differences in distributions of lengths on opposite sides of the fault line will generally grow with successive substitution, and the FLC property can be lost. This will be discussed further in section 2.
1.2. Three Topologies. There are three metrics, and hence three topologies, that are frequently applied to tilings and tiling spaces.
In the first metric, two tilings are considered -close if they agree, up to a translation of size or less, on a ball or radius 1/ around the origin. In this topology, the closure of the (translational) orbit of a tiling is compact if and only if two conditions are met: (a) there are only a finite number of tile types, up to translation, and (b) tiles can only meet in a finite number of ways, up to translation. This is the most frequently used topology for tilings that meet these conditions. Obviously, tiling spaces without FLC are not compact in this topology, but neither are pinwheel-type spaces, which may have FLC even though the tiles may appear in infinitely many orientations.
In the second metric, one applies a metric to the group of rigid motions of the plane (e.g., defining an motion to be a translation of size ≤ followed by a rotation by ≤ about the origin). One then considers two tilings to be -close if they agree, up to an motion, on a ball of size 1/ about the origin. For tilings in which tiles appear in only finitely many orientations (e.g., the Penrose tiling), this is equivalent to the first topology. However, it also handles pinwheel-like spaces appropriately. In this topology, the orbit closure of a tiling is compact if and only if the tiling has FLC.
Finally, one can use a metric in which two tilings are -close if they contain the same tiles out to distance 1/ , and if each tile in the first tiling is within an -motion of the corresponding tile in the second tiling. Note that in this topology, an shear along a fault line yields a tiling that is -close to the original, while in the first two topologies it does not.
For tilings with FLC, the third topology is the same as the second, insofar as it is impossible to apply a small rigid motion to one tile without applying the same motion to all of its neighbors. However, in the third topology the orbit closure of any tiling with finitely many tile types is always compact. (To see sequential compactness, start with an arbitrary sequence of tilings, pick a subsequence in which the type, location and orientation of a tile near the origin converges. Then pick a subsequence in which the type, location and orientation of a second tile converges. Keep working outwards from the origin, and then apply a Cantor diagonalization argument to find a subsequence that converges everywhere.) In the third topology, FLC is not a topologically invariant property. Radin and Sadun [12] constructed a pair of spaces, one FLC and one not, that are topologically conjugate.
1.3. Outline of paper. In section 2, we study the evolution of fault lines in 2dimensional substitution tilings. This is essentially 1-dimensional dynamics, and we relate properties of the fault line to the form of the induced 1-dimensional substitutions.
In section 3, we consider a 2-dimensional substitution tiling with horizontal fault lines. The rows of this tiling are (almost) all the same, up to a horizontal shift, which is controlled by a vertical 1-dimensional substitution. We show that the resulting tiling space can be constructed as the inverse limit of compact CW complexes. We explicitly compute the cohomology of this tiling space, and discuss the meaning of each term.
In section 4, we consider a more complicated substitution, as a step towards the direct product variations considered in section 5. These direct product variations look like the product of a vertical and a horizontal 1-d substitution tiling, except that the rows are sheared by an amount governed by the vertical substitution, and exhibit horizontal fault lines. The cohomology of the resulting tiling space is computable in terms of the cohomologies of the vertical and horizontal 1-d spaces, and the combinatorics of the vertical substitution. Specifically, let µ be H 1 of the horizontal 1-d substitution tiling space, and let M be the n × n substitution matrix of the vertical substitution, as applied to collared tiles. Then • H k is trivial for k > 3.
Since H 3 is nontrivial, the 2-d tiling space is not homeomorphic to any 2-d tiling space with FLC.
Finally, in section 6 we consider open problems in the theory of tilings without FLC, and discuss our partial understanding of these problems. (1 + √ 13)/2 ≈ 2.3028, the larger root of the equation λ 2 − λ − 3 = 0. For a selfsimilar tiling, the a tile can be given length λ while the b tile has length 3. Note that for any word W , σ 2 (W ) is a cyclic permutation of σ 1 (W ), obtained by removing an a from the end and sticking it on the beginning. If W is a bi-infinite word, then σ 1 (W ) and σ 2 (W ) are the same, up to translation by the length of a. This implies that the tiling spaces defined by σ 1 and σ 2 are exactly the same.
Suppose we have a tiling with rectangular tiles a and b of widths λ and 3, respectively, and suppose that σ 1 acts as a substitution on lower edges and σ 2 acts as a substitution on upper edges. Let us construct a horizontal fault line, where the evolution above the line is governed by σ 1 and the evolution below the line is governed by σ 2 . If at some stage there is a pair of exactly aligned a tiles, one above the line and one below, then on successive substitutions we will see ba ab , aaaba abaaa , bababaaaaba abaaaababab , aaabaaaabaaaababababaaaaba abaaaababababaaaabaaaabaaa Note that in the first and third substitution, the a tiles are found more on the right of the top row and on the left of the bottom row, while in the second and fourth substitutions they are found more on the left of the top row and the right of the bottom row. The reason is that the difference between the number of a tiles up to a certain point grows like the second eigenvalue of the substitution matrix, namely 1 − λ ≈ −1.3. As we continue to iterate, this discrepancy grows without bound.
(Strictly speaking, the discrepancy gets multiplied by 1 − λ each time and then adjusted by O(1) edge effects. Once the discrepancy grows beyond a certain point, the edge effects are dominated by the multiplicative factor of 1 − λ and we have exponential growth in the discrepancy as a function of the number of substitutions.) Pick a point along the fault line. If there are m more a tiles in the top row than the bottom up to that point, then the left edges of the tiles on the top row will be offset by λm (mod 3) relative to the left edges of the tiles on the bottom row. By continuity, the discrepancy takes on all integer values between 0 and m as we move from the left edge of the pattern to the point in question. Since m is unbounded and λ is irrational, this means that the possible offsets of tiles in the top and bottom rows takes on a dense set of values in the limit of infinite substitution. In fact the left endpoints of upper a tiles are dense in the lower b tiles, because the only way for the discrepancy to grow from m to m + 1 is for an additional a tile to appear along the top with a b tile below it. Thus every possible adjacency between an upper a an a lower b can occur in the orbit closure; by primitivity of the substitutions this implies that any adjacency is possible between any upper and lower tiles. this, then Thus we have not just a fault line, but a regular fault line.
Note how the form of the fault line depends on the second eigenvalue of the substitution matrix. If the substitution were Pisot, then the discrepancy in the number of any species of tile would be bounded, and the offsets between tiles would take on only a finite number of values. As a result, the FLC condition would be preserved. For a detailed proof that Pisot substitutions do not lead to fault lines, see [5].
Finally, note that there are only two possibilities involving substitutions on two letters. If the discrepancy in the number of a tiles grows without bound, then we have a regular fault line. If the discrepancy is bounded, then we preserve FLC. It is not known whether irregular fault lines, in which the set of possible offsets is infinite but not dense, are possible in substitution tilings. In any case, they would require more than two letters.

3.
A simple 2-dimensional example. Consider a 2-dimensional tiling with two rectangular tiles. Both the A and B tiles have height 1, but the A tile has width λ = (1 + √ 13)/2 and the B tile has width 3. We consider the self-affine substitution . For any n ∈ N, an n-supertile is a collection of tiles of the form Σ n (A) or Σ n (B). A tiling is allowed by the substitution if each of its finite patches of tiles can be found in some n-supertile. The smallest closed set (under the third topology) containing all allowed tilings is called the substitution tiling space X Σ . We will see that measure-theoretically, almost all of the tilings in X Σ are allowed by the substitution, but that the ones that are not are the ones that make the topology different than in the FLC case. Note that whenever two supertiles meet along a horizontal boundary, applying Σ changes the bottom of the top supertile by σ 1 and the top of the bottom supertile by σ 2 . By the results of section 2, the substitution tiling space X Σ defined by Σ exhibits horizontal regular fault lines.
Not every row is subject to arbitrary shears. The rows themselves are labeled by points in the dyadic integers, describing their hierarchy in the vertical substitution. The label in the nth spot is 0 if the row is in the lower (n − 1)-supertile of its n-supertile and 1 if it is in the upper (n − 1)-supertile. If the labels of two adjacent rows differ only in the first digit, then the dyadic label of the upper row begins with Figure 3. The data encoded in the approximant L. The variables x, y and z are points in K that describe the rows nearest the origin, while t is the height of the origin in its row. a 0 and the label of the lower row begins with a 1. One can see that the sequence of tiles in the two rows are identical, with the upper row offset horizontally by λ. If they differ only in the first two digits, then the dyadic label of the upper row begins with 01 and the label of the lower begins with 10, and the upper row is the same as the lower row, but offset by λ 2 − λ. If they differ only in the first three digits, the upper and lower labels begin with 001 and 110 resp., and then the rows are offset by λ 3 − λ 2 − λ. If they differ in the first n digits (and agree thereafter), then the upper and lower labels begin with 0 n−1 1 and 1 n−1 0 resp., and they are offset by λ n − λ n−1 − · · · − λ. (In general, one sees the new offset as λ times the previous offset, minus λ.) However, in some tilings there exists a row with dyadic label 1111 . . . and an adjacent row above it with label 0000 . . .. These rows do not have to have the same sequence of tiles, and their offset is arbitrary.
Put another way, all tilings in the tiling space contain horizontal lines separating identical rows of tiles, offset by arbitrarily large amounts. However, in a small set of tilings (corresponding to a single orbit in the dyadic solenoid that describes the vertical hierarchical structure) there exists a fault line in which the tiling above the fault is unrelated to the tiling below the fault. These special tilings have measure zero with respect to all translation-invariant measures, and hence have no effect on measure-theoretic properties of the tiling space.
However, they have a tremendous effect on the topology of the tiling space. Indeed, this part of the tiling space is 3-dimensional! To describe such a tiling we must specify the height of the infinite fault line (a point in R 1 ), the row immediately above the fault line (a point in a 1-dimensional tiling space) and the row immediately below the fault line (a point in a 1-dimensional tiling space).
3.1. The approximant L. We will show how X Σ is the inverse limit of an approximant L under a bonding map induced by Σ (which we will again call Σ). The CW complex L is actually 4-dimensional , but Σ : L → L is not onto, and Σ(L) is only a 3-dimensional subset of L. We will see that the inverse limit of L under Σ exhibits the right combination of 2-and 3-dimensional elements.
Let σ = σ 1 , and let K be an Anderson-Putnam complex of the 1-dimensional substitution σ, obtained by using collared tiles with a sufficiently large radius D.
In order to ensure that σ 2 is a shift of σ 1 , we pick D > (λ + 3)/(λ − 1) so that λD > D + λ + 3. The 1-dimensional tiling space X σ is the inverse limit of K under a bonding map induced by σ. We then let This is understood as follows. Of the coordinates (x, y, z, t) of a point in L, t describes the height of the origin in the row containing the origin, and runs from 0 to 1. The variable y ∈ K describes a horizontal neighborhood of size D around the origin. In other words, it describes the row containing the origin. The variables x and z similarly describe the rows immediately below and above, respectively. If the origin sits exactly on the boundary of two rows, we may describe the situation either as (x, y, z, 0), with the origin sitting on the bottom of the "y" row, or as (w, x, y, 1), with the origin sitting on the top of the "x" row. Under the identification ∼, information about w or z is lost, and we know only about the two rows touching the origin ("x" and "y").
The bonding map induced from the substitution Σ is However, σ 2 (y) is just a translate of σ 1 (y) = σ(y), and likewise for σ 2 (z), so we can rewrite this as Note that, depending on the value of t, the information from either x or z is lost, so that the image of Σ is 3-dimensional. Every translate is homotopic to the identity map, so the map Σ is homotopic to
Note that we never have to explicitly construct K or computeμ. 3. Finally, take the direct limit under Σ * 2 .
Step 1. We take V to be a neighborhood of t = 0 (say, the set t < 0.2 ∪ t > 0.8) and U to be the region where t is not close to zero (say, 0.1 < t < 0.9). U retracts to K × K × K × {0.5}. Letμ x be the pullback ofμ from the x factor, and likewise forμ y andμ z . We then have Likewise, V retracts to K × K × {0} ∼ K × K × {1}. Letμ xy andμ yz denote the pullback ofμ from the first and second factors, respectively. That is,μ xy can be viewed either asμ x from t = 0 orμ y from t = 1. We then have

NATALIE PRIEBE FRANK AND LORENZO SADUN
The intersection U ∩ V retracts to two copies of K × K × K, say one at t = 0.15 and at t = 0.85, and we have The Mayer-Vietoris sequence is where ρ is restriction and ν is signed restriction. Using a basis forμ to make bases for H k (U ), H k (V ) and H k (U ∩ V ), and writing the lower copy of the basis of H k (U ∩ V ) first, the matrices of ν are where the 1's are actually identity matrices that depend on the dimension ofμ.
Note that these are all injective, except in dimension 0. As a result, the maps ρ must all be zero (except in dimension 0), and for k > 0 we have that H k (L) is the cokernel of the previous ν. To summarize, By computing ∂ * of the generators of H k (U ∩ V ), we see that the generators of H * (L) are 1 in dimension 0, dt in dimension 1, dx ∪ dt = dy ∪ dt = dz ∪ dt in dimension 2, dx ∪ dy ∪ dt = dy ∪ dz ∪ dt and dx ∪ dz ∪ dt in dimension 3 and dx ∪ dy ∪ dz ∪ dt in dimension 4. Here we have used dx, dy, dz and dt as shorthand for 1-dimensional cohomology generators in theμ x ,μ y ,μ z and circle directions.
Step 2. Taking the direct limit under Σ * 1 merely convertsμ to µ. Using 1dimensional methods [3] µ is easily shown to be the direct limit of Z 2 under the matrix 1 1 3 0 , and is isomorphic to Z[1/λ].
Step 3. From equation (6) we compute the effect of Σ * 2 on our cohomology generators.
Note that the identifications at the fault line prevent there being any contribution of µ to H 1 (X Σ ). We do get contributions from µ to H 2 and H 3 . In particular, the H 3 term is easy to understand. One factor of µ comes from the tiling above the fault line, one factor of µ comes from the tiling below the fault line, and the dt term comes from the location of the fault line. In this example, H 2 (X Σ ) equals the tensor product of µ with H 1 of the dyadic solenoid; however, this is not a general pattern. In the next section we shall see an example in which H 2 of the 2-dimensional tiling space is not the tensor product of µ with H 1 of the vertical substitution space, and in section 5 we will compute a general formula for H 2 (X Σ ).
Finally, note that the form of the answer had nothing to do with the details of the substitution σ, except that its expansion constant is not a Pisot or Salem number. Let w 1 and w 2 be any two words in the letters a and b, and consider the substitutions θ 1 (a) = w 1 a, θ 2 (b) = w 2 a, θ 2 (a) = aw 1 , θ 2 (b) = aw 2 . The substitutions θ 1 and θ 2 generate the same 1-dimensional tiling space. If the pair (θ 1 , θ 2 ) generates a fault line, as in section 2, then consider the 2-dimensional substitution Θ(A) = where W 1 and W 2 are the same as w 1 and w 2 , only written in capital letters. Θ gives rise to a 2-dimensional tiling space with horizontal fault lines, and the calculation of this section can be repeated, line by line, to show that the cohomology of X Θ is identical to that of X Σ , only with µ replaced by H 1 (X θ ).

4.
A more involved example. Our next example is a direct product variation, in the sense of Frank [6]. The vertical factor is the period-doubling substitution 0 → 01, 1 → 00, while the horizontal factor is our usual 1-dimensional substitution σ. We let A and B denote a ⊗ 1 and b ⊗ 1, and abuse notation by referring to a ⊗ 0 and b ⊗ 0 as a and b, respectively. The result is a tiling with four rectangular tiles. Note that if we ignore the difference between capital and lower case letters, we revert to the example of section 3. The period-doubling substitution space is an almost 1-1 extension of the dyadic solenoid, and the example of this section is an almost 1-1 extension of the example of section 3. Before proceding with an analysis of X Σ , we review some facts about the perioddoubling substitution. This substitution forces the border [8] on the right, since every substituted letter begins with a 0. To construct the Anderson-Putnam complex [1], we need only collar on the left. The complex (call it P , for period-doubling) is shown in figure 4. There are three collared tiles, which we call α, β and γ. The tile α is 1, preceded by a 0, the tile β is 0, preceded by a 0, and the tile γ is 0, preceded by a 1. Viewed as a map on collared tiles, the substitution sends α → γβ, β → γα, γ → βα, and interchanges the two vertices of the complex.
The 5-tuple (α, x, y, z, t) (resp. (β, x, y, z, t) or (γ, x, y, z, t)) means that the origin is sitting at height t in a row that corresponds to α (resp. β or γ) in the perioddoubling substitution. Within the row containing the origin, the horizontal position of the origin corresponds to y ∈ K. In the rows below and above that, the horizontal positions of the points one unit below and above the origin correspond to x ∈ K and z ∈ K.
We adopt the same strategy as in section 3. First we compute the cohomology of L using Mayer-Vietoris, then we take the direct limit under Σ * 1 , and then we take the direct limit under Σ * 2 . To compute H * (L), let V consist of neighborhoods of the two vertices in P , and let U be a slightly thickened complement to V . Now U retracts to three copies of K × K × K, one for each edge of P , and V retracts to two copies of K × K, one for each vertex of P . There are 6 copies of K × K × K in the retraction of U ∩ V , one at the beginning of each edge and one at the end of each edge. We then have: where the two copies in H k (V ) correspond to the two vertices in P . Since there is a copy of H k (U ) at the beginning and at the end of each edge of Writing down the generators of H k (U ∩V ) in the same order as the corresponding generators of H k (U ) and H k (V ) and keeping the copies of H k (U ) corresponding to the outgoing edges separate from those corresponding to the incoming ones, we can again compute matrices for ν in each dimension. The matrices will have two vertically aligned identity matrices in the U columns and then linearly independent vectors with two −1s in them in the V columns, so the signed restriction maps with the following generators, subject to the following constraints.
Of course H 0 (L) is generated by 1. The generators of H 1 (L) are dt α , dt β and dt γ , with dt α = dt γ . (In other words, H 1 (L) = H 1 (P ).) The products of generators ofμ x ,μ y andμ z with dt α , dt β and dt γ generate H 2 (L). Using the convention that generators ranging throughμ are referred to as simplyμ, we see the generators of H 2 (L) are subject to the four constraints The products of (μ x ∪μ y ), (μ x ∪μ z ) and (μ y ∪μ z ) with dt α , dt β and dt γ generate H 3 (L), subject to the two constraints Finally, H 4 (L) is generated byμ x ∪μ y ∪μ z ∪ (dt α , dt β , and dt γ ), with no constraints.
Note the form of the constraints on H 2 and H 3 . For H 2 , we have two constraints for each vertex of P . The sum of theμ y ∪ dt terms from the edges flowing into the vertex equals the sum of theμ x ∪ dt terms from the edges flowing out of the vertex, and the sum of theμ z ∪ dt terms from the edges flowing into the vertex equals the sum of theμ y ∪ dt terms from the edges flowing out of the vertex. These may be treated as constraints among theμ x ∪ dt andμ z ∪ dt generators, while theμ y ∪ dt generators are unconstrained. For H 3 we have one constraint per vertex, namely that the sum of theμ y ∪μ z ∪ dt terms from the edges flowing into the vertex equals the sum of theμ x ∪μ y ∪ dt terms from the edges flowing out. As in section 3, the direct limit of H * (L) under Σ * 1 takes the same form as H * (L), only withμ replaced by µ. What remains is to take the direct limit under Σ * 2 , which we do one dimension at a time.
The computation in dimension 0 is trivial, and we of course have H 0 (X Σ ) = Z. The computation in dimension 1 does not involve µ at all, and is identical to the computation of H 1 of the period-doubling substitution space. The answer is that For dimension 2, we first look at the µ y ∪ dt terms. Since Σ * 2 maps these terms to themselves, these terms give the direct limit of Z 3 under the matrix . This matrix has eigenvalues 2, -1 and -1, and the direct limit is Z[1/2] ⊕ Z 2 .
Next we look at the µ x ∪ dt terms. We compute Since the µ x ∪ dt terms map to the µ y ∪ dt terms, they do not contribute anything additional to the direct limit. Likewise, the µ z ∪ dt terms map to the µ y ∪ dt terms, and do not give any additional contributions. In dimension 3, the image of Σ * 2 consists entirely of µ x ∪ µ y ∪ dt and µ y ∪ µ z ∪ dt terms, since the image of Σ 2 never involves both x and z. Moreover, the µ x ∪ µ y ∪ dt terms map to themselves, and the µ y ∪µ z ∪dt terms map to themselves. Specifically, we have Before taking the constraints into account, the action of Σ * 2 has eigenvalues 1, 1, −1, −1, 0 and 0, making the direct limit of this action Z 4 . Because we know thatμ x ∪μ y ∪dt γ = µ y ∪μ z ∪dt α andμ x ∪μ y ∪(dt α +dt β ) =μ y ∪μ z ∪(dt β +dt γ ) in H 3 (L), and because Σ * 2 annihilates the µ x ∪µ y ∪dt α and µ y ∪µ z ∪dt γ terms, there are only two independent generators left in the direct limit for H 3 (X Σ ). These are µ x ∪ µ y ∪ dt γ = µ y ∪ µ z ∪ t α and µ x ∪ µ y ∪ dt β = µ y ∪ µ z ∪ dt β . The cohomology has registered the two possible fault lines, one between the α and γ rows, and one between two β rows.
To summarize, the cohomology of X Σ is Note that H 2 (X Σ ) is not the tensor product of µ with the first cohomology of the period-doubling substitution. Rather, it is the tensor product of µ with the direct limit of the transpose of the period-doubling substitution matrix (as applied to collared tiles), which has an additional factor of Z. The two copies of µ ⊗ µ in H 3 refer to the two types of fault lines that can occur in X Σ . One has a β row both above and below the fault line. The other has an α row below the fault line and a γ row above. Σ maps each of these situations to the other.

5.
General direct product variations with regular fault lines. The results of the previous section are suggestive of how the fault lines affect the cohomology of a tiling space. In this section we prove two theorems on the cohomology of tiling spaces with horizontal fault lines that arise as direct product variations.
Suppose we have a collection of primitive 1-dimensional substitutions σ 1 , σ 2 , ..., σ N defined on the same alphabet A. Moreover, assume that for each a ∈ A, the length of σ k (a) is the same for all k, that all of the substitutions have the same stretching factor, and that they all yield the same tiling space. (E.g., all of the σ k s might be cyclic permutations of one another). Now suppose we have another primitive 1-dimensional substitution ρ. We could then consider the direct product of the two substitutions, with σ 1 acting horizontally and ρ acting vertically. We then replace σ 1 with σ 2 , etc in various rows, so as to introduce fault lines. Furthermore, we assume that regular fault lines occur at every boundary of infinite-order ρ-supertiles.
We call the resulting 2-dimensional substitution Σ, and compute the cohomology of X Σ . Let µ = H 1 (X σ ) and let ν = H 1 (X ρ ). Let n be the number of configurations in the X ρ tiling space in which two infinite-order ρ-supertiles meet at the origin. By assumption, this is the same as the number of ways that a regular fault line can occur in an X Σ tiling. Theorem 1. Under these circumstances, the cohomology of X Σ is as follows: H k (X Σ ) = 0 for k > 3.
Proof: As usual, let P be the AP complex of X ρ , and let K be the AP complex of X σ . We take L to be one copy of K × K × K × [0, 1] for each edge in P , with the following identification. If the end of edge α meets the beginning of edge β at a vertex in P , then (α, w, x, y, 1) ∼ (β, x, y, z, 0). We compute H * (L) by Mayer-Vietoris. Let V be a union of neighborhoods of the vertices of P , and let U contain the edges. U retracts to a number of copies of K × K × K (one per edge), while V retracts to a number of copies of K × K (one per vertex). U ∩ V retracts to a number of copies of K × K × K (two per edge, one at the beginning and one at the end).
Using the same bookkeeping as in section 4, one can see that the signed restriction maps H k (U ) ⊗ H k (V ) → H k (U ∩ V ) are all injective, except in dimension 0. The cokernel is understood as follows. The image of the restriction of H k (U ) to H k (U ∩ V ) merely identifies (up to sign) the two K × K × K terms in U ∩ V that correspond to each edge. This gives obvious generators for H k (L): In dimension 1 we have dt ζ , where ζ is an edge in P , in dimension 2 we haveμ x ∪ dt ζ ,μ y ∪ dt ζ , andμ z ∪ dt ζ , in dimension 3 we haveμ x ∪μ y ∪ dt ζ ,μ x ∪μ z ∪ dt ζ , andμ y ∪μ z ∪ dt ζ , and in dimension 4 we haveμ x ∪μ y ∪μ z ∪ dt ζ , all subject to the identifications imposed by the image of H k (V ) in H k (U ∩ V ). In H 1 this is that the sum of the dt ζ s entering a vertex equals the sum of the dt ζ s coming out. In H 2 it is that the sum of theμ y ∪ dt terms from the edges flowing into the vertex equals the sum of theμ x ∪ dt terms from the edges flowing out of the vertex, and the sum of theμ z ∪ dt terms from the edges flowing into the vertex equals the sum of theμ y ∪ dt terms from the edges flowing out of the vertex. For H 3 , the the sum of theμ y ∪μ z ∪ dt terms from the edges flowing into a vertex equals the sum of theμ x ∪μ y ∪ dt terms from the edges flowing out. For H 4 , there are no constraints. This completes the computation of H k (L). Now we note that Σ is homotopic to the product of σ in the horizontal direction and ρ in the vertical direction. Taking the direct limit under σ * merely convertsμ to µ. What is left is taking the direct limit under ρ * .
In dimension 2, we note that the constraints express certain combinations of the µ x ∪ dt ζ s or the µ z ∪ dt ζ s in terms of the µ y ∪ dt ζ s. They do not constrain the µ y ∪ dt ζ s terms, which are mapped to themselves by ρ * . Furthermore, among these terms the action of ρ * is just the transpose of the substitution matrix itself (as applied to edges of P , i.e. to collared tiles). The direct limit of H 2 (L) under ρ * therefore contains the direct limit of this matrix.
In principle, the direct limit of H 2 (L) should also contain contributions from the µ x ∪ dt ζ and µ z ∪ dt ζ terms. However, we claim that these contributions are zero as a consequence of our using a complex that forces the border, as explained below.
Note that the pullback of dx ∪ dt ζ term is a dx ∪ dt ξ term for each tile ξ for which ρ(ξ) is a word beginning with ζ, plus a dy ∪ dt ξ term for each tile ξ for which ρ(ξ) contains ζ in the middle or end of the word. If the substitution forces the border in m steps, then ρ m of each edge emerging from a vertex in P is a word beginning with the same letter (call it ω). (ρ m ) * (µ x ∪ dt ω ) then equals the sum of the µ x ∪ dt terms from all the edges emerging from this vertex (plus additional µ y ∪ dt terms). However, the sum of the µ x ∪ dt terms from the edges emerging from the vertex equals the sum of the µ y ∪ dt terms from the edges entering the vertex. Likewise, if η = ω, then (ρ m ) * (µ x ∪ dt η ) contains none of the µ x ∪ dt terms from edges emerging from the vertex. Either way, the pullback by ρ m of a µ x ∪ dt term can be expressed as a sum of µ y ∪ dt terms. The same argument (applied to ends of words) shows that the pullback by ρ m of a µ z ∪ dt term can be expressed as a sum of µ y ∪ dt terms. In particular, the µ x ∪ dt and µ z ∪ dt terms that are linearly independent of the µ y ∪ dt terms do not appear in the eventual range of ρ * , and hence do not contribute to the direct limit of H 2 (L).
In dimension 3, ρ * sends µ x ∪ µ z ∪ dt terms to µ x ∪ µ y ∪ dt and µ y ∪ µ z ∪ dt terms. Therefore, we need only consider the direct limit of µ x ∪ µ y ∪ dt ζ and µ y ∪ µ z ∪ dt ζ terms. We associate µ x ∪ µ y ∪ dt ζ with the vertex in P that ζ leads into, and Figure 5. The Anderson-Putnam complex for the vertical tiling space associate µ y ∪ µ z ∪ dt ζ with the vertex it leads out of. As in dimension 2, forcing the border implies that (ρ m ) * takes each µ x ∪ µ y ∪ dt term to either all or none of the µ x ∪ µ y ∪ dt terms associated with a vertex. When dealing with µ x ∪ µ y ∪ dt, we may therefore restrict our attention to sums of all the edges emerging from a vertex, and when dealing with µ y ∪ µ z ∪ dt we may restrict our attention to sums of all the edges leading into a vertex. However, the sum of all the µ y ∪ µ z ∪ dts from the edges leading into a vertex equals the sum of all the µ x ∪ µ y ∪ dts leading out of the vertex, so we have exactly one independent term per vertex.
The substitution ρ maps vertices of P to vertices, and has a natural pullback action on the 3-forms associated to vertices. Eventually, ρ merely permutes the vertices that describe boundaries of two infinite-order supertiles. The 3-forms associated with these vertices give a basis for H 3 (X Σ ).
In dimension 4, the pullback map is zero, so H 4 (X Σ ) = 0. Finally, we consider the coefficient of µ in H 2 (X Σ ) and the coefficient of µ ⊗ µ in H 3 (X Σ ). The first is the direct limit of the transpose of the substitution matrix as applied to edges of P , and the second is the direct limit of the transpose of the substitution matrix as applied to vertices of P . In other words, they are the direct limit of 1-cochains and 0-cochains on P under substitution. These are closely related to direct limits of the cohomology of P (i.e., to the cohomology of X ρ ). Since the direct limit of the 0-cochains (namely Z n ) is Z n−1 more than H 0 (X ρ ), the direct limit of the 1-cochains must be Z n−1 more than H 1 (X ρ ).
In stating Theorem 1, we assumed that every boundary between infinite-order supertiles in X ρ led to a fault line. As the following example shows, this is not always the case.
Let A 1 , A 2 and A 3 be tiles of width λ and height 1/3, and let B 1 , B 2 , and B 3 be tiles of width 3 and height 1/3. Our substitution is Each row in a tiling contains either A 1 s and B 1 s (call this an α row), A 2 s and B 2 s (β) or A 3 s and B 3 s. The complex P consists of three edges (α, β, γ) running in a circle, with α followed by β followed by γ followed by α. The vertical substition ρ is ρ(α) = αβ, ρ(β) = γα, ρ(γ) = βγ.
Fault lines do not develop at the boundary of α and β supertiles, or at the boundary of β and γ supertiles, just at the boundary of γ and α. This example may look mysterious, but it is just a rewriting of the basic example of section 3. A 1 , A 2 , and A 3 are just the bottom, middle and top thirds of the A tile, while B 1 , B 2 and B 3 are the bottom, middle and top thirds of the B tile. The cohomology is correctly described by Theorem 1, only with n being the number of possible fault lines (namely 1), not the number of ways that two infinite-order supertiles can meet (namely 2). This observation generalizes to