Quotient cohomology of certain 1- and 2-dimensional substitution tiling spaces

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by  Enrico Paolo Bugarin

Quotient cohomology of certain 1- and 2-dimensional
substitution tiling spaces
Enrico Paolo Bugarin and Franz Ga¨hler
Fakult¨at fu¨r Mathematik, Universit¨at Bielefeld
Universit¨atsstraße 25, D-33615 Bielefeld, Germany
Thequotientcohomologyoftilingspacesisatopologicalinvariantthatrelatesatilingspacetooneofitsfactors,viewed
4
as topological dynamical systems. In particular, it is a relative version of the tiling cohomology that distinguishes
1
0 factorsoftilingspaces. Inthiswork,thequotientcohomologieswithincertainfamiliesofsubstitutiontilingspacesin1
2 and2dimensionsaredetermined. Specifically,thequotientcohomologiesforthefamilyofthegeneralisedThue-Morse
sequencesand generalised chair tilings are presented.
n
a PACS:02.40.Re, 45.30.+s
J

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1 Introduction A tiling T of Rd arising from a substitution rule ω de-

] fines a substitution tiling space as its hull. The hull of T,
T
Characterising tiling spaces through topological invari- denoted by Ω , is the closure of its translation orbit un-
T
A
ancecanprovideasystematicwayofdistinguishingtiling dera metric, wheretwo tilings are“ε-close”if they agree
h. spaces. In particular, tiling spaces with non-isomorphic ona ball of radius ε−1 aroundthe origin,after a transla-
t (Cˇech) cohomology groups are necessarily inequivalent. tion of at most ε in any direction. Two tilings T and T′
a
m The converse though is not true in general, as can be arisingfromthe samesubstitutionruleω definethesame
seen in the case of the classical Thue-Morse and period- hull, and so the tiling space is instead associated with
[
doubling sequences, which form two inequivalent tiling a substitution rule rather than a particular substitution

1 spaceswithisomorphiccohomologygroups. However,for tiling, i.e., Ωω :=ΩT =ΩT′.
v tilingspacesrelatedbyafactormap(regardingthespaces
Asubstitutiontilingspacecanberepresentedasanin-
3
as topological dynamical systems), a relative version of
verselimitofsimplerspacescalledapproximants,relative
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thetilingcohomologycanbeusedtotellthespacesapart.
5 toacontinuousbondingmapinducedbythesubstitution
Barge and Sadun [1] introduced the concept of quotient
1 rule ω, see [2, 3] for more details. When the prototiles in
1. cohomology, which is a topological invariant that distin- Ωω are homeomorphic to a disk in Rd, then the approx-
guishes factors of tiling spaces. In this paper, we present
0 imants are d-dimensional CW complexes, which are also
4 the quotient cohomologies for the families of generalised known as the Anderson-Putnam complexes. The coho-
1 Thue-Morse sequences and generalised chair tilings. In mology of Ω is then computed as the direct limit of the
ω
: recomputingthequotientcohomologiesofthegeneralised
v cohomologies of the approximants under the homomor-
i chair tilings, we find some discrepancies with [1], which phism induced by ω.
X
we address below.
Substitution tiling spaces are minimal dynamical sys-
r
a tems, and factor maps between these spaces are surjec-
1.1 Preliminaries tive and generally not injective. (And so the relative co-
homology is not immediately available.) A factor map
A primitive (tiling) substitution rule is a recipe of con-
f : Ω → Ω induces a quotient map (denoted by the
structingtilings ofRd usingonly afinite setoftile types, X Y
same symbol) on the level of approximants that is also
called prototiles. The rule prescribes on how each pro-
surjectiveandgenerallynotinjective. Thismotivatesthe
totile is scaled linearly (by a fixed inflation factor) and
following definition of the quotient cohomology [1].
then is subdivided into a collection of smaller tiles called
a supertile (of order 1), all of which are translate copies
Definition 1. Let f : X → Y be a quotient map such
of some prototiles. Applying the substitution rule to any
that f∗ is injective on cochains. Also, let Ck(X,Y) :=
(proto)tile k times produces a supertile of order k, and Q
the limit of the process produces a tiling of Rd, called a Ck(X)/f∗(Ck(Y)) be the quotient cochain groups and
take δ : Ck(X,Y) → Ck+1(X,Y) to be the usual
substitution tiling. k Q Q
coboundary operator. The (kth) quotient cohomology is
∗email: {pbugarin, gaehler}@math.uni-bielefeld.de defined as HQk(X,Y):=kerδk/imδk−1.
1
By the snakelemma, the shortexactsequence ofcochain also consider the 1-dimensional solenoid Sk+ℓ, which can
complexes beviewedasthe inverselimitof1-dimensionaltoriunder
thebondingmapsthatuniformlywrapatorusk+ℓtimes
0−→Ck(Y)−−f→∗ Ck(X)−→CQk(X,Y)−→0 arounditspredecessor. ThesolenoidSk+ℓmayberealised
astheinverselimitofthesubstitutions7−→sk+ℓ,though
induces a long exact sequence strictlyspeaking,thesolenoidisnotatilingspacebecause
tilingsgeneratedbythissubstitutionareallperiodic. The
···−→Hk−1(X,Y)−→Hk(Y)−−f→k∗ solenoidhasasadditionalinformationthe partitioningof
Q (1)
Hk(X)−→Hk(X,Y)−→··· thesetilingsintosupertilesofallorders. Forconvenience,
Q
we may nevertheless use the term ‘tiling space’ even for
that relates the cohomologies of X and Y to H∗(X,Y). solenoids in the following.
Q
The threespacesarerelatedviathe factormapsφ and
Lemma 2. Let f : X → Y be a quotient map, whose ψ, namely
pullback f∗ is injective on the cochains. If Hn+1(Y) = YTM −−φ→Ypd −−ψ→S ,
0, then Hn(X,Y) = Hn(X)/f∗(Hn(Y)). For X and Y k,ℓ k,ℓ k+ℓ
Q n
being approximant spaces for substitution tiling spaces, where φ is a sliding block map that identifies {1¯1,¯11}
H0(X,Y) = 0 if and only if f∗ : H1(Y) → H1(X) is with a, and {11,¯1¯1} with b; while ψ simply identifies a
Q 1
injective. and b with s. Note that φ is uniformly 2-to-1,whereas ψ
is a surjection that is 1-to-1 almost everywhere. Each of
Proof. If X and Y are approximant spaces for substi-
these factor maps induces a pullback on their respective
tution tiling spaces, then H0(X) = Z = h x′i =
i cohomologies given by
yhPj′’sfa∗r(eyj′t)hie=duafl0∗s(Hto0t(hYe))0-=cellfs0∗i(nZ)X, wanhderYe trheespPxe′ic’tsivaenlyd. H∗(Sk+ℓ)−−ψ→∗ H∗(Ypkd,ℓ)−−φ→∗ H∗(YTk,Mℓ ).
Thusf∗ issurjective,andsoisanisomorphism. Further,
0 As computed in [4], the cohomologies of the three tiling
the map from H0(X) to H0(X,Y) in (1) must be a zero
Q spaces are: H0 =Z and
map and so the map from H0(X,Y) to H1(Y) must be
injective. If f1∗ is injective, thQen the mapfromHQ0(X,Y) H1(Sk+ℓ)=Z[k+1ℓ], H1(Ypkd,ℓ)=Z[k+1ℓ]⊕Z,
to H1(Y) must be a zero map as well, which is already H1(YTM)=Z[ 1 ]⊕Z⊕Z[ 1 ],
showntobeinjective,forcingH0(X,Y)=0. Conversely, k,ℓ k+ℓ |k−ℓ|
if HQ0(X,Y)= 0, then f1∗ mustQbe injective since the se- where Z[01]:=0 by an abuse of notation.
quence in (1) is exact. Meanwhile, Hn+1(Y)=0 implies Theorem 3. For any k,ℓ∈N, H0 =0. Further,
Q
thatthe mapHn(X)toHn(X,Y)issurjective,andsoit
follows that Hn(X,Y)=HQn(X)/f∗(Hn(Y)). HQ1(Ypkd,ℓ,Sk+ℓ)=Z,
Q n
H1(YTM,S )=Z[ 1 ]⊕Z,
All substitution tiling spaces considered in this work Q k,ℓ k+ℓ |k−ℓ|
yield HQ0 =0. HQ1(YTk,Mℓ ,Ypkd,ℓ)=Z2⊕Z[|k−1ℓ|],
where Z[1]:=0 by an abuse of notation.
2 Generalised Thue-Morse sequences 0
Proof. The map ψ∗ embeds H1(Sk+ℓ) = Z[k+1ℓ] isomor-
For any k,ℓ∈N, the substitution rules phically onto the same summand in H1(Ypd), which
k,ℓ
the map φ∗ also embeds isomorphically onto the same
̺TM := 1 7−→ 1k¯1ℓ summand in H1(YTM). Thus, ϕ∗ := φ∗ ◦ψ∗ also em-
̺kpkd,,ℓℓ :=(cid:26)(cid:26) ¯1ab 7−7−7−→→→ ¯1bbkkk−−111ℓaabbℓℓ−−11ab (2) HHbe11d((sYYHpTkd,Mℓ1)()So.kn+tFoℓu)2rZ=thieZnkrm,[ℓHko+11rℓ(e]Y,iTsφMo∗m)mo(rsapephesi[c4tah]l)le.yTsouhnmutosm,Zathn[ekd+1mℓZ]apiinns
k,ℓ k,ℓ
define the hulls YTM (generalised Thue-Morse) and Ypd ψ1∗, φ∗1 and ϕ∗1 are all injective maps, and so HQ0 = 0
k,ℓ k,ℓ for all spaces using Lemma 2. By the same lemma, we
(generalised period doubling) respectively. The case k =
ℓ = 1 yields the classic Thue-Morse and period doubling get HQ1(YTk,Mℓ ,Ypkd,ℓ) = H1(YTk,Mℓ )/φ∗1(H1(Ypkd,ℓ)) = Z2 ⊕
sequences. For a detailed exposition on the spectral and Z[ 1 ]. The rest of the results follow similarly.
|k−ℓ|
topological properties of the generalised Thue-Morse se-
Thethreenon-trivialquotientcohomologiesarerelated
quences, we refer the readers to [4].
via the short exact sequence
Each letter in (2) becomes a tile in R by assigning the
sameconstantlengthto anyone ofthem, so thata letter 0−→H1(Ypd,S )−−×→2 H1(YTM,S )−→
becomes a closed interval in R. In turn, every bi-infinite Q k,ℓ k+ℓ Q k,ℓ k+ℓ
sequence arising from (2) tiles R in an obvious way. We HQ1(YTk,Mℓ ,Ypkd,ℓ)−→0.
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3 Generalised chair tilings
A A
ΩX,+ −−→ Ω/,+ −−→ Ω0,+
The hull of the classic chair tiling, c.f. [5], (also called
B B B
triomino tiling [2]) defined by the substitution rule
  
ΩXy,− −−A→ Ωy/,− −−A→ Ωy0,− (4)
A A C
  
 A  C 
ΩyX,0 −−→ Ωy/,0 −−→ Ωy0,0
Barge and Sadun beautifully computed the quotient co-
homologybetweenadjacenttilingspacesappearingin(4),
using a framework discussed in [1]. The quotient coho-
belongstoafamilyofsubstitutiontilingspacescalledthe
mologies are given by:
generalised chair tilings, which Barge and Sadun intro-
duced and analysed in [1]. Using decorated square tiles A: H0 =0, H1 =Z, H2 =Z[1],
Q Q Q 2
asprototiles,the mostintricateofthemdefines the tiling
B : H0 =0, H1 =Z, H2 =Z[1]⊕Z, (5)
space ΩX,+ through the substitution rule Q Q Q 2
C : H0 =0, H1 =0, H2 =Z[1]⊕Z.
Q Q Q 2
Tracing a path in (4) pertains to a factor map from
y y y y
wտ1 0ւx wց0 1րx one tiling space (starting point) to another tiling space
y 1 0 y 0 1
wտx wրx (ending point). As such, paths having identical starting
0 1 1 0
z z and ending points pertain to equivalent factor maps. In
wր0 1տx wր1 0տx
z z z z thissense,wesaythatthediagramcommutes. Wegener-
(3) alisethe resultsin(5)by givingthe quotientcohomology
betweentilingspacesin(4),dependingonthefactormap
1 0 0 1
wւ1 0տx wր0 1ցx between them as obtained by tracing an arbitrary path.
y y
wւx z z wցx z z We formalise this as the following theorem, whose calcu-
y y y y
z wց0 1ւx z wց1 0ւx lation is straightforward.
0 1 1 0
Theorem 4. The quotient cohomologies between adja-
cent tiling spaces in (4) are given in (5); for the remain-
where w,x,y,z ∈{0,1}andwith the two labels adjacent
ing pairs of tiling spaces, we have:
to the head of an arrow being the same.
Factors of ΩX,+ can be defined by removing and/or AA: HQ0 =0, HQ1 =Z2, HQ2 =Z[12]2,
identifying certain decorations on the square tiles to AB : H0 =0, H1 =Z2, H2 =Z[1]2⊕Z,
which the general substitution rule (3) applies. Tiling Q Q Q 2
spaces Ωa,b, with a ∈ {X,/,0} and b ∈ {+,−,0}, are AAB : HQ0 =0, HQ1 =Z3, HQ2 =Z[21]3⊕Z,
defined through the following: BC : H0 =0, H1 =Z, H2 =Z[1]2⊕Z2,
Q Q Q 2
AC : HQ0 =0, HQ1 =0, HQ2 =Z3⊕Z[12]2,
Index Description
a X Thefourarrowsonthesquaretilesremain. AAC : HQ0 =0, HQ1 =Z, HQ2 =Z3⊕Z[21]3,
/ Only the arrows pointing northeast or BAC : HQ0 =0, HQ1 =Z, HQ2 =Z3⊕Z[21]3⊕Z,
southwest remain, i.e., arrowheads point- ABAC : HQ0 =0, HQ1 =Z2, HQ2 =Z3⊕Z[21]4⊕Z.
ing to other directions are identified.
0 All arrows are identified/removed.
b + All four labels remain.
Note that the quotient cohomology depends only on the
− Only the labels to the left or to the right
type of path, and not necessarily on particular tiling
remain, i.e., the top and bottom labels are
spaces. Also, the quotient cohomology groups sum up
identified.
wheneverfactor maps are composed. The only exception
0 All labels are identified/removed.
is when composing A and C which produces the torsion
component Z3. For the rest of the compositions, the op-
Inparticular,ΩX,0is(equivalentto)thechairtilingspace eration is associative and commutative.
andΩ0,0isthe2-dimensionaldyadicsolenoidS2×S2. The Thefollowingpropositionsalreadyappearin[1]asThe-
scheme aboveyields nine tiling spacesthat are relatedas orems 6 and 7, although with some errors. The absolute
follows: andquotientcohomologieshavebeenrecalculatedandthe
3
corrected results appear in the following. The particular 4 Discussion and conclusion
corrections are boxed for easier identification. Proposi-
tion 6 may also be read off of Theorem 4. Determining the quotient cohomologies between 1-
dimensional substitution tiling spaces is rather straight-
Proposition 5 (cf. [1, Theorem 6]). The absolute co-
forward because of Lemma 2. In particular, it suffices to
homologies of the nine tiling spaces in (4) are given as
know f∗ to be able to compute both H0 and H1.
follows. All spaces have H0 =Z. The first cohomology is 1 Q Q
Inhigherdimensions,afirstchallengeisintheenumer-
given by
ationofinequivalentfactorsandthefactormapsbetween
Z[1]2⊕Z2 ←A−∗− Z[1]2⊕Z ←A−∗− Z[1]2⊕Z them before one can study their quotient cohomologies.
2 2 2
In the case of the generalised chair tilings, there are two
B∗ B∗ B∗ more substitution tiling spaces between ΩX,− and Ω0,0,
x x x which are inequivalent to any of those already enumer-
Z[12]2⊕Z ←A−∗− Z[12]2 ←A−∗− Z[21]2 ated,butareimpossible to obtainthroughthe identifica-
tion rules considered earlier.
A∗ A∗ C∗ Asimilaranalysisasabovehasalsobeencarriedoutfor
x x x
Z[1]2 ←A−∗− Z[1]2 ←C−∗− Z[1]2 the Squiral [6] and Chacon [7] substitution tiling spaces,
2 2 2 which are both 2-dimensional.
The second cohomology is given by
Acknowledgement
13Z[41]⊕Z[12]4 ←A−∗− 31Z[14]⊕Z[12]3 ←A−∗− Z[14]⊕Z[12]2
⊕Z ⊕Z ⊕Z2 ThisworkissupportedbytheGermanResearchFounda-
tion (DFG) via the Collaborative Research Centre (SFB
B∗ B∗ B∗ 701) at Bielefeld University.
x x x
1Z[1]⊕Z[1]3 ←A−∗− 1Z[1]⊕Z[1]2 ←A−∗− Z[41]⊕Z[12] References
3 4 2 3 4 2 ⊕Z
A∗ A∗ C∗ [1] M. Barge, L. Sadun, New
x x x York J. Math. 17, 579 (2011).
1Z[1]⊕Z[1]2 ←A−∗− Z[1]⊕Z[1]⊕Z ←C−∗− Z[1] http://nyjm.albany.edu/j/2011/17-25.html
3 4 2 4 2 4
[2] J.Anderson,I.Putnam,Ergod.Theor. Dyn.Syst.18,
Proposition 6 (cf. [1, Theorem 7]). The quotient co- 509 (1998). DOI:10.1017/S0143385798100457
homologies of the nine tiling spaces in (4), relative to
[3] L. Sadun, Topology of Tiling Spaces,
the solenoid Ω0,0, are given as follows. For all spaces,
H0 =0. The first quotient cohomology is given by http://www.ams.org/publications/authors/books/postpu
Q
American Mathematical Society, Providence, RI
Z2 ←A−∗− Z ←A−∗− Z 2008.
B∗ B∗ B∗ [4] M. Baake, F. G¨ahler, U. Grimm, J. Math. Phys. 53,
x x x 032701 (2012). DOI:10.1063/1.3688337
Z ←A−∗− 0 ←A−∗− 0
[5] E.A. Robinson Jr., Indag. Mathem. 10, 581 (1999).
A∗ A∗ C∗ DOI:10.1016/S0019-3577(00)87911-2
x x x
0 ←A−∗− 0 ←C−∗− 0 [6] M. Baake, F. G¨ahler, U. Grimm
The second quotient cohomology is given by J. I. S. 16, Article 13.2.14 (2013).
https://cs.uwaterloo.ca/journals/JIS/VOL16/Baake/baa
Z3⊕Z[12]4⊕Z ←A−∗− Z3⊕Z[12]3⊕Z ←A−∗− Z[12]2⊕Z2 [7] N.P. Frank, Exposition. Math. 26, 295 (2008).
DOI:10.1016/j.exmath.2008.02.001
B∗ B∗ B∗
x x x
Z3⊕Z[12]3 ←A−∗− Z3⊕Z[21]2 ←A−∗− Z[12]⊕Z
A∗ A∗ C∗
x x x
Z3⊕Z[12]2 ←A−∗− Z[12]⊕Z ←C−∗− 0
4