Torsion in Tiling Homology and Cohomology
Franz G\"ahler
The first author's recent unexpected discovery of torsion in the integral cohomology of the T\"ubingen Triangle Tiling has led to a re-evaluation of current descriptions of and calculational methods for the topological invariants associated with aperiodic tilings. The existence of torsion calls into question the previously assumed equivalence of cohomological and $K$-theoretic invariants as well as the supposed lack of torsion in the latter. In this paper we examine in detail the topological invariants of canonical projection tilings; we extend results of Forrest, Hunton and Kellendonk to give a full treatment of the torsion in the cohomology of such tilings in codimension at most 3, and present the additions and amendments needed to previous results and calculations in the literature. It is straightforward to give a complete treatment of the torsion components for tilings of codimension 1 and 2, but the case of codimension 3 is a good deal more complicated, and we illustrate our methods with the calculations of all four icosahedral tilings previously considered. Turning to the $K$-theoretic invariants, we show that cohomology and $K$-theory agree for all canonical projection tilings in (physical) dimension at most 3, thus proving the existence of torsion in, for example, the $K$-theory of the T\"ubingen Triangle Tiling. The question of the equivalence of cohomology and $K$-theory for tilings of higher dimensional euclidean space remains open.
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\begin{document}
%\thispagestyle{empty}
\title{Torsion in Tiling Homology and Cohomology}

\date{version 19 May 2005}

\author{Franz G\"ahler}
\address{Institut f\"ur Theoretische und Angewandte Physik, 
Universit\"at Stuttgart, D-70550 Stuttgart, Germany}

\email{gaehler@itap.physik.uni-stuttgart.de}

\thanks{The second author was partially supported by EPSRC grant GR/S08046/01}

\author{John Hunton}
\address{The Department of Mathematics, University of
Leicester, Leicester, LE1 7RH, England}

\email{j.hunton@mcs.le.ac.uk}

\author{Johannes Kellendonk}
\address{Institut Camille Jordan, Universit\'e Claude Bernard Lyon 1, 
F-69622 Villebanne cedex, France}

\email{kellendonk@igd.univ-lyon1.fr}

%\date{\today}
\keywords{cohomology groups, $K$-groups, tilings, aperiodic order}

\begin{abstract}
The first author's recent unexpected discovery of torsion in the integral
cohomology of the T\"ubingen Triangle Tiling has led to a re-evaluation
of current descriptions of and calculational methods for the topological
invariants associated with aperiodic tilings. The existence of torsion
calls into question the previously assumed equivalence of cohomological
and $K$-theoretic invariants as well as the supposed lack of torsion in
the latter. In this paper we examine in detail the topological invariants
of canonical projection tilings; we extend results of Forrest, Hunton and
Kellendonk to give a full treatment of the torsion in the cohomology of
such tilings in codimension at most 3, and present the additions
and amendments needed to previous results and calculations in the
literature. It is straightforward to give a complete treatment of the
torsion components for tilings of codimension 1 and 2, but the case of
codimension 3 is a good deal more complicated, and we illustrate our
methods with the calculations of all four icosahedral tilings previously
considered. Turning to the $K$-theoretic invariants, we show that
cohomology and $K$-theory agree for all canonical projection tilings in
(physical) dimension at most 3, thus proving the existence of torsion in,
for example, the $K$-theory of the T\"ubingen Triangle Tiling. The
question of the equivalence of cohomology and $K$-theory for tilings of
higher dimensional euclidean space remains open.
\end{abstract}

\maketitle

\bibliographystyle{amsalpha}

\section{Introduction}\label{Intro}

For large classes of aperiodic tilings various methods have been
devised to compute topological invariants such as the Cech cohomology or
topological $K$-theory of the continuous hull, the cohomology of the
discrete tiling groupoid or the $K$-theory of various associated
$C^*$-algebras. All these invariants are closely related, many essentially
equivalent, although, as a consequence of the work below, the equivalence
of the cohomological and $K$-theoretic invariants is now only {\em a
priori\/} true for tilings in dimension at most 3.

For tilings generated by a  primitive substitution, Kellendonk
\cite{KeCMP} showed how to compute the top dimension cohomology via the
groupoid $C^*$-algebra, while Anderson and Putnam
(AP) \cite{AP} constructed  the hull as an inverse limit of a sequence of
finite CW-complexes, allowing the determination of the cohomology and
$K$-theory as direct limits. These direct limits can be computed explicitly,
although in more than one dimension the procedure quickly becomes
tedious. For some of the simpler two-dimensional examples it can still be
carried out  with the aid of a computer \cite{FGBanff}. 

For canonical projection tilings, Forrest, Hunton and Kellendonk
(FHK) \cite{FHKmem} related the Cech cohomology and $K$-theory to
certain group or dynamical homology theories.  For codimension up to
three, they provided explicit formul\ae\ 
\cite{FHKcmp,FHKmem}, expressing the (rational) ranks of 
the homology groups in terms of the data defining the tiling.
These formul\ae\ are straightforward to evaluate \cite{GK,FHKicmp}, 
and provide an efficient means to compute the cohomology and other 
related topological invariants. Moreover, from an earlier result 
\cite{FoHu} it was concluded that the integral cohomology  groups
for canonical projection tilings would always be free, and so they 
would be completely determined by their ranks, hence equivalently
by their rational ranks. However, this premise was not correct in
the generality needed, and thus there is more to the determination of the
cohomology of a canonical projection tiling than its rational rank.
The main purpose of this article is to provide a proper treatment of the 
integral cohomology; equivalently, we provide a discussion of
the torsion part of the integral tiling homology. 

A third approach, appearing in a recent paper of Kalugin
\cite{Kalugin}, gives yet another method to determine the (rational)
Cech cohomology of canonical  projection tilings. Although not stated in
this generality, his method  can be applied in essentially the same
situations as the FHK method
\cite{FHKcmp,FHKmem}. 


There are important differences, however, between these three
approaches.  Canonical projection tilings are obtained by projection
from a periodic  structure in $(n+d)$-dimensional space to
$d$-dimensional physical  space. The complementary $n$-dimensional
space is the so-called internal  or perpendicular space. Whereas FHK
work entirely with internal space,  Kalugin's method is set up in
the $(n+d)$-dimensional space (wrapped  onto a torus), and the AP
method works exclusively in physical space (and knows nothing about
internal space).

The classes of primitive substitution tilings on the one hand, and 
of canonical projection tilings on the other hand, have a non-trivial 
overlap, and for tilings in the overlap all three methods can be 
used for the computation of topological invariants. 
Exploiting this as a consistency check, the AP method had been 
implemented and applied to a number of popular two-dimensional 
tilings \cite{FGBanff}, whose cohomology had been computed previously 
by the FHK method \cite{GK}. 
Whereas in all cases considered, the free part of the cohomology
agreed with the previous results, for the T\"ubingen Triangle Tiling 
(TTT) an additional 
torsion part $\Z_5^2$ was found with the AP method in the 
top-dimensional cohomology group, which is in contradiction with
the assumed freeness of cohomology in \cite{FHKcmp,FHKmem}. The question
of torsion therefore had to be reconsidered. 
For the codimension two case, to 
which the TTT belongs, it is indeed easy to reconcile the FHK theory 
with the new result. The correct torsion is in fact present in 
the formul\ae\ of \cite{FHKcmp}, but instead of reading it off, 
its authors neglected it, assuming it must vanish by appealing to
the result of \cite{FoHu}, which apparently is not applicable in the
required generality. So, if properly interpreted, the results
obtained with both methods agree in all known cases, and the results
of \cite{FHKcmp} only have to be complemented with an explicit
expression for the torsion part of the cohomology. 

For codimension three the situation is somewhat more involved. 
Though it is still possible to use essentially the methods of 
\cite{FHKmem}, extra care is needed in order to keep track 
of torsion. The essence of the problem is that in the top
dimension torsion can arise in more than one way, and there are
potential extension problems to consider; these problems disappear
for lower dimensional cohomology groups. Under some extra conditions,
however, the  torsion can still be determined formulaically, and even
if these conditions are not satisfied, it is still possible, in
favourable cases, to solve the extension problems and so determine the
torsion completely, but only by appealing further to the geometry of the
explicit example under consideration. We shall see an example of
this in Section~\ref{sec:ex3d}, where we compute for the dual canonical 
$D_6$ tiling. 

An alternative perspective that is of value for codimension three schemes
(and in principle is equally applicable for higher codimensions) is to
work with homology or cohomology over a field of characteristic $p$. Here
the formalism and methods of \cite{FHKcmp,FHKmem}, which essentially
worked with homology over a field of characteristic zero, carries over
directly. By these methods we can obtain precise expressions for the
{\em rank,\/} though not the {\em order,\/} of the $p$-torsion in integral
cohomology, without having to deal with any of the extension problems.

Although in the codimension three case the characterisation of
torsion is not as straightforward as one might have hoped, this is
nevertheless the first time that a detailed and transparent
characterisation of torsion has been obtained for a rather large class of
tilings. In previous work, torsion was either believed to be  absent
\cite{FHKcmp,FHKmem}, or was present only implicitly,  and had to be
computed tediously on a case by case basis \cite{AP}. Nevertheless,
we have as yet little to say about the geometrical interpretation of 
torsion for these tilings. 

A further point worth emphasizing is that, for canonical projection
tilings in dimension $d\leqslant3$ with finitely generated
cohomology, the $K$-theory is still isomorphic to the cohomology, whether
or not there is torsion in the cohomology. Previous arguments had
relied on there being no torsion in the cohomology. As a result we have
specific examples in which the $K$-theory of the tiling (including the
$K$-theory of the associated non-commutative $C^*$-algberas) has torsion.
We note that the question of whether there are tilings in higher
dimensions
$d\geqslant4$ for which the cohomology and $K$-theory differ remains open.

With this note we also take the opportunity to clarify two further
discrepancies between Kalugin's results \cite{Kalugin} and our 
previous work \cite{GK,FHKmem}. Kalugin finds that all generalized 
Penrose tilings have isomorphic cohomology. We entirely agree with 
this; the deviating results in \cite{GK} are due to an incorrect 
parametrisation of the generalized Penrose tilings. The discrepancy 
in the cohomology of the icosahedral Ammann-Kramer tiling is due to 
an incorrect ``simplification'' in \cite{FHKmem}, which is not valid 
in general. If this simplification is not made, both results 
agree. We present corrected formul\ae\ in Section~\ref{sect-correction}.

The arrangement of this article is as follows. The core of our
approach to the cohomology and $K$-theory of canonical projection
tilings is the evaluation of certain group homologies
$H_*(\Gamma;C^n)$, and the technique as developed in
\cite{FHKcmp,FHKmem} is to compute these via subsidiary homologies
$H_*(\Gamma;C^r)$ for $r<n$. We briefly review this material, introducing
our main notation and explaining the role of these various homology
groups in Section~\ref{Prelim}. This section is a  summary of
the relevant ideas of \cite{FHKcmp,FHKmem}, and the reader may refer
to those articles for a more complete background. In
Section~\ref{RightWrong} we present an overview of which parts of
previous articles still stand, irrespective of potential torsion,
and which need alteration in the light of torsion. In this section
we also observe that many of the previous formul\ae, which were
based on calculations over the rationals, remain valid when worked
(with care) over a field of positive characteristic, such as the field of
$p$ elements $\F_p$, and in so doing give a powerful tool for the
computation of the rank of the 
$p$-torsion when this occurs; we see an example of such a method used
in Section~\ref{sec-examples}, where some explicit examples are
computed. Section~\ref{Low} provides the body of work, giving
formul\ae\ for the homology of a canonical projection tiling in the
cases of codimension 1, 2 and 3, including formul\ae\ for any
torsion in these homology groups. In Section~\ref{sect-correction}
we give the correction to the codimension 3 formula of \cite{FHKcmp}
mentioned above. In Section~\ref{sec-examples} we work some examples and
provide a complete table of all previously published cohomology
calculations, now amended so as to include their torsion components,
where they exist. Finally, in Section~\ref{Gen} we finish with some general
results on where torsion can (or cannot) occur for an arbitrary
canonical projection tiling, and prove our theorem relating the
$K$-theory to the cohomology.




\section{Preliminaries}\label{Prelim}
We start by briefly summarising our notation and the main objects of
computation for our discussion of torsion in the cohomology and
$K$-theory of canonical projection tilings. We use, generally, the
notation and ideas of \cite{FHKcmp,FHKmem}, and we refer the reader
to those articles for motivation and a discussion of the background
material which gives a rationale for the following assertions. 

Our set-up 
is as follows. We are aiming to compute the topological
invariants for a canonical projection tiling or pattern in $\R^d$
with an effective codimension of $n$. Our most basic object to
compute is a group homology $H_*(\Gamma;C^n)$, where $\Gamma$ is a
free abelian group of rank $d+n$ which can be identified with a dense
lattice of the internal space $\R^n$. The coefficients 
$C^n$ can be seen as the $\Z\Gamma$ module
generated by indicator functions on compact polytopes, 
so-called $\mathcal{C}$-topes
\cite{FHKmem}, which are intersections of $\Gamma$-translates of
the acceptance domain and can be most conveniently described with the
help of singular spaces (see next subsection).
We have the
related coefficient groups $C^r$ for $0\leqslant r\leqslant n$ which
fit into an exact sequence 
\begin{equation}\label{coeffs}
0\to C^n\to C^{n-1}\to \cdots\to C^0\to\Z\to0\,.
\end{equation}
This sequence allows direct calculations for $H_*(\Gamma;C^n)$ at
least for small $n$, as in \cite{FHKcmp}, which we investigate in
greater detail below for
$n=2,3$, while for general $n$ it gives rise to the spectral
sequence of \cite{FHKmem}
\begin{equation}\label{specseq}
E^1_{r,s}=H_s(\Gamma;C^r)\Longrightarrow
H_{r+s}(\Gamma;\Z)=\Z^{{n+d}\choose{r+s}}\,.
\end{equation}

Whether the analysis of $H_*(\Gamma;C^n)$ via the sequence
(\ref{coeffs}) is done by hand or via this spectral sequence, the
principle in both cases is to get at these homology groups by
computing the lower terms $H_*(\Gamma;C^r)$ for $r<n$. Just as the
$C^n$ have a geometric origin, so do the $C^r$: these are modules of
functions on $r$-dimensional subsets (singular $r$-planes) of
$\R^n$, essentially the
$r$-dimensional facets of the $\mathcal{C}$-topes. A key
observation,  which makes this method a practical one, is that the
homology groups $H_*(\Gamma;C^r)$ split as
\begin{equation}
H_*(\Gamma;C^r)=\bigoplus_\Theta H_*(\Gamma^\Theta;C_\Theta^r)
\end{equation}
where $C_\Theta^r$ is an analogous module of functions on a
representative of a $\Gamma$-orbit of singular $r$-planes, 
and $\Gamma^\Theta$ is the stabiliser subgroup of this object.

We shall concentrate entirely on the case when these homologies are
finitely generated as groups. Necessary and sufficient conditions
for this are given in \cite{FHKmem}, and in particular it is
necessary that the number $\nu={{(n+d)}\over n}$ is an integer (and
hence $n$ must divide $d$). In this case, the
ranks of the stabiliser groups of the singular
$r$-planes are then forced to be $r\nu$, and an important consequence
of this for us here is the following bound on the non-zero ranks of
the homology groups: 
\begin{equation}\label{heights}
H_k(\Gamma;C^r)=0\quad \mbox{\rm unless}\quad 0\leqslant
k\leqslant r(\nu-1).
\end{equation}
This puts, for example, significant constraints on the non-zero terms
in the spectral sequence (\ref{specseq}).

Given a computation of the homology $H_*(\Gamma;C^n)$, there is a
duality which tells us how to compute the Cech cohomology of the
hull $MP$ of the pattern by the relation
\begin{equation}
H^i(MP;\Z)=H_{d-i}(\Gamma;C^n)\,;
\end{equation}
in particular, note that this cohomology is potentially non-zero
only for $0\leqslant i\leqslant d$, and moreover $H^0(MP;\Z)=\Z$. 

The main aim of Section~\ref{Low} is to give an analysis of the torsion in
the homology $H_*(\Gamma;C^n)$ for low values of $n$, basically
$n\leqslant3$. This is done most easily by navigating the coefficient
sequence (\ref{coeffs}) by hand, much as it was done in the computations
carried out in \cite{FHKcmp}. However, for our general results in
Section~\ref{Gen} it is best to appeal to the spectral sequence
approach (\ref{specseq}); this should be thought of as morally the same
approach, but using a more sophisticated method of book-keeping to keep
track of $n$ long exact sequences and their interactions all at
once. We record a couple of observations that will be useful later.
A good general reference to the ideas of spectral sequences and
their application can be found in \cite{Mc}.

Firstly, we note that the differentials act $d^j\colon E^j_{r,s}\to 
E^j_{r-j,s+j-1}$. The constraints on where the the homology groups
$H_*(\Gamma;C^r)$ are non-zero noted above will often tell us that
these differentials are trivial.

Secondly, the convergence of this spectral sequence means that the
limit groups $E^\infty_{r,s}$ give the composition quotients of a
filtration of $H_{r+s}(\Gamma;\Z)$. Of particular interest to us
below will be the cases where there are only one or two non-zero
terms $E^\infty_{r,s}$ for given value of $r+s$. In particular, for
given $t$, if there is only one non-zero $E^\infty_{r,s}$ with
$r+s=t$ then $E^\infty_{r,s}=H_t(\Gamma;\Z)$, and if there are two
such non-zero groups, say $E^\infty_{r_1,s_1}$ and
$E^\infty_{r_2,s_2}$ with $r_1<r_2$, then there is a short exact
sequence
\begin{equation}\label{compser}
0\to E^\infty_{r_1,s_2}\to H_t(\Gamma;\Z)\to 
E^\infty_{r_2,s_2}\to 0\,.
\end{equation}

We finish our preliminary observations with a note about the
deduction of the $K$-theory from the homology and cohomology
calculations. This is via a second spectral sequence, a
version we shall call $A^{**}_*$ say, of the Atiyah-Hirzebruch
spectral sequence (AHSS) computing the topological
$K$-theory of $MP$ (and thence for various non-commutative theories
too)
\begin{equation}
A^{r,s}_2=H^r(MP;K^s)\Longrightarrow K^{r+s}(MP)
\end{equation}
where we recall that $K^s=\Z$ if $s$ is even, and is zero if $s$ is
odd. This sequence is further described in \cite{FoHu} and can also
be considered as a simple case of the Kasparov spectral sequence.



\subsection{Notation for geometric data}

We record the detailed notation used below and in
\cite{FHKcmp,FHKmem} to describe the $\Z\Gamma$ modules $C^r$ and
various associated objects used in our computations. 

Given a canonical projection scheme, the basic data consists of a
triple $(V,\Gamma,\W)$. Here $V$ is a euclidean space of dimension
$n$ (the internal space), $\Gamma$ a free abelian group of
rank $n+d$ (the super-lattice), 
which we identify
with a specific dense lattice in $V$, and a finite family
$\W=\{W_i\}_{i}$ of affine hyperplanes whose normals span
$V$ (the spans of the linear components of the boundary of 
the acceptance domain; the
latter is assumed to be polytopal). 
Furthermore, $\W$ cannot be written as a
union $\W=\W_1\cup \W_2$ such that the normals of $\W_i$ span
complementary spaces $V_i$. 

The homology groups $H_*(\Gamma,C^r)$ are defined as homologies
of dynamical systems derived from the data $(V,\Gamma,\W)$.
The coefficient groups $C^r$ are the $\Z\Gamma$-modules generated by
indicator functions on $r$-dimensional facets of $\mathcal C$-topes,
which now can be described as
compact polytopes in $V$ whose boundary belongs to some union
$\bigcup_{(W,x)\in A}(W+x)$, where $A$ 
is a finite subset of $\W\times\Gamma$.
The homology groups depend therefore on the geometry
and combinatorics of the intersections 
$ \bigcap_{(W,x)\in A}(W+x)$, 
where $A$ is some finite subset of $\W\times\Gamma$. We
call such an affine subspace a singular $r$-space if its dimension is $r$.
Let $\JJ_r$ be the set of singular $r$-spaces and denote the orbit
space under the action by translation $I_r:=\JJ_r/\Gamma$. The
stabilizer $\{x\in\Gamma|\hat\Theta+x=\hat\Theta\}$ of a singular
$r$-space $\hat\Theta$ depends only on the orbit class $\Theta\in
I_r$ of $\hat\Theta$ and we denote it $\Gamma^\Theta$. Fix
$\hat\Theta\in \JJ_{k}$ for $r<k<\dpe$ and let
$\JJ_r^{\hat\Theta}:=\{\hat\Psi\in\JJ_r|{\hat\Psi}\subset
{\hat\Theta}\}$. Then $\Gamma^\Theta$ ($\Theta$ the orbit class of
$\hat\Theta$) acts on $\JJ_r^{\hat\Theta}$ and we let
$I_r^{\hat\Theta}=\JJ_r^{\hat\Theta}/\Gamma^\Theta$. We can
naturally identify $I_r^{\hat\Theta}$ with $I_r^{\hat\Theta'}$ if
$\hat\Theta$ and $\hat\Theta'$ belong to the same $\Gamma$-orbit
and so we define $I_r^{\Theta}$, for the class $\Theta\in I_{k}$. Then
$I_r^{\Theta}\subset I_r$ consists of those orbits of singular
$r$-spaces which have a representative that lies in a singular
space of class $\Theta$. Finally we use the notation $ L_r =
|I_r|,\quad L^\Theta_r = |I^\Theta_r|$,
and $C^r_\Theta$ for the submodule of $C^r$ generated by indicator
functions on $r$-dimensional facets lying in a singular $r$-space
representing $\Theta$. 
Below, and as in \cite{FHKcmp,FHKmem}, 
we shall use $\alpha$ to index codimension $1$ singular spaces
and $\theta$ to index codimension $2$ singular spaces.

\begin{theorem} [\cite{FHKmem}] A necessary and sufficient condition
for the homology groups $H_*(\Gamma;C^n)$ of a canonical projection
tiling to be finitely generated is that the number $L_0$ is finite.
\end{theorem}




\section{What's right and what's wrong from before}\label{RightWrong}

The origins of this article lie with the discovery \cite{FGBanff}
of 5-torsion in the homology of the T\"ubingen Triangle Tiling
(TTT), a property that had been thought to be not possible in
earlier work, including \cite{FHKcmp, FHKmem}, and hence had 
erroneously been overlooked in previous work. Torsion phenomena of various
orders have now been discovered in other examples, and we give more
details in Section~\ref{sec-examples}.

The oversight of torsion in \cite{FHKcmp, FHKmem} is not due to 
errors {\em per se\/} in those articles, but rather to their use of
the claim in \cite{FoHu} Proposition~3.2 that for $(X,\Z^d)$ a
minimal Cantor dynamical system, there was no torsion in the
homology groups $H_*(\Z^d;C(X,\Z))$; the TTT example shows that this
result cannot hold in the generality needed for projection tiling
systems. The lack of torsion was also used in previous work to
deduce the $K$-theory of the hull; the presence of torsion means
that the computation of the $K$-theory from the tiling homology groups
is a lot more subtle than it would have been without torsion.

The existence of torsion thus requires discussion on two fronts: the
elaboration of the work of \cite{FHKcmp, FHKmem} so as to include
the overlooked torsion, and a more detailed examination of
the resulting  $K$-theory of the hull. In this section we outline which
aspects of the calculations in print stand as they are, and which need
modification in the light of potential torsion.



\subsection{Torsion-free part of tiling homology}
The main theoretical work on the homology associated to
canonical projection patterns is to be found in \cite{FHKcmp,
FHKmem}, and explicit applications to compute particular examples
have appeared in \cite{FHKicmp,  GK}. The corrections needed to
\cite{FHKcmp, FHKmem} concern  only the explicit formul\ae\ for
$H_*(\Gamma;C^n)$ of
\cite{FHKcmp} Theorem 64 and \cite{FHKmem} Theorems 2.5, 
2.6 and 2.7. We note that the calculations of any codimension 1
system remain correct, as there can be no torsion  for such systems;
we give details in Section \ref{Low}. This remark includes all the
work of \cite{FHKmem}  Chapter III. The statements about the Euler
characteristic in
\cite{FHKmem} Chapter V (in particular Theorem 
2.8) also remains correct: the Euler characteristic is essentially a
rational invariant and so does not see torsion.

The strategy of deriving the formul\ae\ in \cite{FHKcmp, FHKmem} for
higher codimension systems basically yields the torsion free part of the
homology, as it essentially works by deriving the rationalised homology
groups $H_*(\Gamma;C^n)\otimes\Q=H_*(\Gamma;C^n\otimes\Q)$. Indeed in
\cite{FHKmem} page 94 we define the numbers $D_k$ 
as the ranks of these rational vector spaces, and the computions are
correct as they stand with this 
interpretation of
$D_k$: the problem is with deducing that once one has computed the
rank of the free part, then one has the whole 
homology group computed. In the paper \cite{FHKcmp} the proof of
Theorem 64 is correct up to the final line where the claim that the
homology groups are torsion free enters for the first time. In
particular, the formula
$$H_k(\Gamma;C^2)\cong
H_{k+1}(\Gamma;C^0_0)/\mbox{im}\beta_{k+1}\oplus
H_k(\Gamma;C^1)\cap\mbox{ker}\beta_k$$ 
is correct as it stands and we shall generalise this in Section
\ref{Low}. It is not correct, however, to 
assume that the quotient in this expression,
$H_{k+1}(\Gamma;C^0_0)/\mbox{im}\beta_{k+1}$, is torsion free, and
in  fact the TTT explicitly shows that there can be torsion in this term.

The consequence of this for the published calculations such as
\cite{FHKicmp,GK} is that, apart from a computational error which we
correct in Section~\ref{sect-correction}, the ranks stated there are
correct as regards the torsion free part of the homology, but there
may be torsion terms that need to be added to give a complete
computation for the integral theory. We note also that as all these
calculations are for patterns with finitely generated homology, any
potential torsion will necessarily be a direct summand. Explicitly,
with the above interpretation of $D_k$ we have
\begin{equation}\label{defTk}
H_k(\Gamma;C^n)=\Z^{D_k}\oplus \tau_k
\end{equation}
where each $\tau_k$ is a finite abelian group. One of the objectives
of Section~\ref{Low} may be seen as giving explicit descriptions of
the groups $\tau_k$.


\subsection{Homology over finite fields}
In \cite{FHKcmp, FHKmem}, the assumption of torsion-freedom allowed
one to effectively work with coefficients in
a field, and use linear algebra; in the absence of torsion, it
suffices to compute ranks of free abelian groups, which are
equivalent to the dimensions of the corresponding vector spaces
$H_*(\Gamma;C^n\otimes\F)$, where $\F$ is any field (in the
definition of $D_k$ this was taken to be $\Q$, but if there is no
torsion it could just as well be
$\F_p$, the field with $p$ elements for some prime $p$).

In the presence of torsion, while the linear algebra fails to
compute the full integral homology, it is no longer the case that
the choice of field $\F$ is irrelevant, and this insight allows us 
to use the formul\ae\ of \cite{FHKmem} to get a hold on the torsion. 
We first introduce some notation. For $A$ a finitely generated 
abelian group and $p$ any prime, the $p$-torsion of $A$ is the 
subgroup of those elements of $A$ which are annihilated by 
multiplication by some power of $p$. We continue to write $D_k$ 
for the $\Q$ rank of $H_*(\Gamma;C^n\otimes\Q)$. Let us also 
write $D_k^p$ for the $\F_p$-rank of $H_*(\Gamma;C^n\otimes\F_p)$,
and $T^p_k$ for the rank of the $p$-torsion in $H_*(\Gamma;C^n)$;
with the notation of equation (\ref{defTk}) this means 
$T^p_k=\mbox{dim}_{\F_p}(\tau_k\otimes\F_p)$. Interpret
$T^p_{-1}$ as zero. We make two observations.

First of all, the numbers $D^p_k$ for low codimensions are
computable from the formul\ae\ in \cite{FHKmem}. In fact, the
formul\ae\ given in the Theorems 2.6 and 2.7 hold exactly for
computations of $D_k^p$, provided the ranks $r_k$ and $R_k$ are
interpreted correctly: these numbers are defined in terms of the
ranks of certain sublattices of exterior powers $\Lambda_r\Gamma$
and should be interpreted as the ranks of the same objects after
tensoring with the field $\F_p$. In the presence of torsion these ranks
depend on the value of the prime $p$ considered; for example, the rank of
the submodule $5\Z$ in $\Z$ will be 1 after tensoring with $\F_p$ or
$\Q$, unless $p=5$, when the rank will be $0$.

Secondly, the standard machinary of homological algebra gives us
explicit relations (\lq universal coefficient formul\ae\rq\ \cite{Bro})
between the homology groups $H_*(\Gamma;C^n)$ and groups
$H_*(\Gamma;C^n\otimes R)$ for any commutative ring $R$. In our
notation these yield the following result.

\begin{prop}\label{torrank}{\em 
Assume that the homology groups $H_*(\Gamma;C^n)$ are finitely
generated. Then 
$$D_k^p=D_k+T_k^p+T_{k-1}^p\,.$$
In particular, $H_*(\Gamma;C^n)$ is free of $p$-torsion if and only
if $D_k=D_k^p$ for all $0\leqslant k\leqslant d$.}
\end{prop}

\noindent Note that the Euler characteristic $\sum_k (-1)^kD_k=\sum_k (-1)^kD_k^p$ is independent of the field
$\F$ chosen.

\smallskip
\begin{rem}
While this proposition gives a strong and computable hold on the
torsion in $H_*(\Gamma;C^n)$, it will not tell the whole story
in general. Basically, this result tells us only of the $p$-rank, and
not of the order of $p$-torsion; for example, if it tells us that
$T_k^p=1$ we cannot deduce whether the $p$-part of $\tau_k$ is
$\Z_{p}$ or $\Z_{p^s}$ for some other positive integer $s$. To handle
this we develop explicit formul\ae\ for the groups $\tau_k$ in the
case of low codimension patterns in Section \ref{Low}. Nevertheless, the
same idea can be generalised further by considering homology over a more
general ring $R$ and it turns out that this also gives us a useful
computational tool. For example, although working over
$\F_2$ will not distinguish between $\Z_2$ and
$\Z_4$ in the homology of the tiling, working over $R=\Z_4$ will. The
disadvantage of this is that for rings $R$ which are not fields, there is
no simple analogue of the formul\ae\ of \cite{FHKmem}, and computations
must be hand-crafted to the specific case under consideration. We shall
see an explicit example of this method in Section~\ref{sec-examples} when
we compute the homology of the dual canonical $D_6$ tiling.
\end{rem}


\subsection{Consequences of torsion for $K$-theory}
As noted before, one role of the supposed lack of torsion in tiling
homology was its use in the deduction of the behavour of the AHSS
computing
$K^*(MP)$. With torsion in $H_*(\Gamma;C^n)$ there is the potential
of non-trivial differentials in $A^{**}_*$, and so a failure of the
claim that the $K$-theory of $MP$ is the direct sum of homology
groups. In fact, things are not as bad as they might be, and we shall
show in Section \ref{Gen}, that for canonical projection patterns in
$\R^d$ with $d\leqslant 3$ and finitely generated homology, there are
isomorphisms
\begin{equation}
K^0(MP)\cong \bigoplus_{r} H^{2r}(\Gamma;C^n)\qquad
K^1(MP)\cong \bigoplus_{r} H^{2r+1}(\Gamma;C^n)\,.
\end{equation}
Similar results for higher values of $d$ also hold when there is no
$p$-torsion for small primes $p$, and such results hold for all $d$
under the assumptions that the homology groups $H_*(\Gamma;C^n)$ are
finitely generated and torsion free.





\section{Low codimension formul\ae}\label{Low}
In this section we produce specific descriptions for the homology of
canonical projection tilings of codimension 1, 2 and 3, including as
far as possible explicit expressions for torsion components in those
degrees where the homology is potentially non-free. The codimension
3 case is complemented by the formul\ae\ of
Section~\ref{sect-correction}, replacing those of Theorem~2.7
of~\cite{FHKmem}. 


\subsection{No torsion for Codimension 1}
Here we prove the assertion of the title, that for a codimension 1
(that is, $n=1$) projection 
pattern, $H_k(\Gamma;C^1)$ has no torsion. In fact, this follows
also from the analysis of the hull $MP$ for an 
arbitrary codimension 1 pattern in Chapter III of \cite{FHKmem}, where
it is shown that $MP$ can be modelled by 
a punctured torus, but the argument below will demonstrate the spirit
of the method used for higher 
codimensions. We shall work in greater generality than just the
canonical patterns, and assume that there are 
potentially a number of distinct $\Gamma$-orbits of \lq cut points\rq\
in the internal space; as before, we shall also call 
this number $L_0$. See \cite{FHKmem} Chapter III for further
discussion of the projection scheme. 

\smallskip\noindent{\em Proof.}
As in Section \ref{Prelim}, we compute the groups $H_*(\Gamma;C^1)$
where there is a short exact sequence of 
$\Z\Gamma$-modules
$$0\longrightarrow C^1\longrightarrow
C^0\longrightarrow\Z\longrightarrow 0\,.$$ In this sequence, $\Z$
carries the trivial $\Gamma$ action, while the action of $\Gamma$ on
$C^0$ is free. This  sequence gives rise to a long exact sequence in
homology
$$\cdots\to H_{k+1}(\Gamma;\Z)\to H_k(\Gamma;C^1)\to
H_k(\Gamma;C^0)\to\cdots\,.$$ 
Now $H_k(\Gamma;\Z)\cong\Lambda_k\Gamma$ 
is just the homology of a $(d+1)$-torus, so
$H_k(\Gamma;\Z)$ is free abelian of rank 
${d+1}\choose{k}$. 
Meanwhile, the freeness of $\Gamma$ on $C^0$ means that the homology
groups $H_k(\Gamma;C^0)$ 
are zero for $k>0$ and $H_0(\Gamma;C^0)=\Z^{L_0}$.

Our long exact sequence now tells us in dimensions $k>0$ that
$H_k(\Gamma;C^1)\cong\Lambda_{k+1}\Gamma\cong
\Z^{{{d+1}\choose{k+1}}}$ and for  dimension 0 there is an exact
sequence
$$0\to\Lambda_1\Gamma \to H_0(\Gamma;C^1)\to
\Z^{L_0}\stackrel{\epsilon}{\to}\Z\to0\,$$
where $\epsilon$, which takes the sum over the coefficients in 
$\Z^{L_0}$, is onto.
Hence $H_0(\Gamma;C^1)\cong\Lambda_1\Gamma \oplus \ker \epsilon$,
and so in particular it is free abelian of rank
$L_0+d$. \null\hfill$\square$

\subsection{Codimension 2}\label{cd2}
The codimension 2 theory follows pretty much the same analysis,
via a pair of exact sequences, as in the
proof of Theorem 64 in \cite{FHKcmp}. The exact sequence of
coefficients
\ref{coeffs} breaks into  two short exact sequences
\begin{equation}\label{2ses}
0\to C^2\to C^1\to C^0_0\to0\qquad0\to C^0_0\to C^0\to\Z\to0\,.
\end{equation}
The right hand sequence behaves identically to the calculations above
for codimension 1 and we obtain 
$$H_k(\Gamma;C_0^0) \cong \left\{\begin{array}{ll}
\Z^{\left(d+2 \atop k+1\right)}=\Lambda_{k+1}\Gamma & \mbox{for}
\quad k>0,\\
\Z^{d+L_0+1}=\Lambda_{1}\Gamma\oplus \ker\epsilon & \mbox{for}\quad
k=0
\end{array}\right.
$$
A similar calculation based on the exact sequences
\begin{equation}\label{sesal}
0\to C^1_\alpha \to C^0_\alpha \to\Z\to 0
\end{equation}
gives
$$H_k(\Gamma;C^1) \cong \left\{\begin{array}{ll}
\bigoplus_{\alpha\in I_1}\Lambda_{k+1}\Gamma^\alpha & 
\mbox{for}\quad k>0,\\
\bigoplus_{\alpha\in I_1}(\Lambda_{1}\Gamma^\alpha\oplus 
\ker\epsilon^\alpha) 
& \mbox{for}\quad k=0
\end{array}\right.
$$
where $\Gamma^\alpha$ 
denotes the stabiliser subgroup of a cut line represented by $\alpha$
and $\epsilon^\alpha : H_0(\Gamma^\alpha;C^0_\alpha)\cong
\Z^{L_0^\alpha} \to \Z$.
We note by \cite{FHKmem} page 97 that $H_*(\Gamma;C^2)$ is 
finitely generated if and only if $L_0$ is finite, and in this case
$\nu={{d+2}\over2}$ is an integer and the rank of each
$\Gamma^\alpha$ is $\nu$. The number $L_0^\alpha$ is the number of
$\Gamma^\alpha$ orbits of cut points on a cut line represented by $\alpha$.

Denote by $\beta_k$ the homomorphism in $H_k(\Gamma;-)$ induced by
the module homomorphism $C^1\to C^0_0$. For 
dimensions $k>0$ this identifies with the  homomorphism
$$\bigoplus_{\alpha\in I_1}\Lambda_{k+1}\Gamma^\alpha\to\Lambda_{k+1}\Gamma$$
induced by the inclusions $\Gamma^\alpha\to\Gamma$. Similarly, 
$\beta_0$ identifies with the homomorphism
$$\bigoplus_{\alpha\in I_1}(\Lambda_{1}\Gamma^\alpha\oplus\ker\epsilon^\alpha)
\to\Lambda_{1}\Gamma\oplus\ker\epsilon $$
induced by the inclusions
$\Gamma^\alpha\to\Gamma$ and $C^0_\alpha\subset C^0$.
The left hand sequence in (\ref{2ses}) now gives a long exact sequence
in homology, which at $H_k(\Gamma;C^2)$ 
may be written as the short exact sequence
$$0\to\mbox{coker }\beta_{k+1}\to H_k(\Gamma;C^2)\to\mbox{ker
}\beta_k\to0\,.$$ This sequence splits as $\mbox{ker }\beta_k$ is a
subgroup of  a free abelian group. We obtain

\begin{theorem}\label{thm51}
Assuming we work with a canonical projection pattern with codimension
2 and finitely generated homology, then 
$$H_k(\Gamma;C^2)=\coker\beta_{k+1}\oplus \ker \beta_k$$
and so $\tau_k$, the torsion part of $H_k(\Gamma;C^2)$, is given by
the torsion part of the cokernel 
of
$$\beta_{k+1}\colon\bigoplus_{\alpha\in I_1}\Lambda_{k+2}
\Gamma^\alpha\to\Lambda_{k+2}\Gamma\,.$$  
In particular, $H_k(\Gamma;C^2)$ is torsion free for $k\geqslant
d/2$.
\end{theorem}

\smallskip\noindent{\em Proof.} We have justified all but the final
sentence. We have observed that the torsion 
in $H_k(\Gamma;C^2)$ is equivalent to the torsion in
coker$\,\beta_{k+1}$. Clearly, there will be no torsion in this
cokernel if each
$\Lambda_{k+2}\Gamma^\alpha$ is zero, and this will be 
the case if $k+2$ exceeds the rank of 
$\Gamma^\alpha$. Thus there is no torsion in
$H_k(\Gamma;C^2)$ if 
$k+1\geqslant{{d+2}\over2}$, from which the inequality
follows.\hfill$\square$

\medskip
This final result, putting bounds on where torsion may appear, will be
seen to be a special case of a result for 
arbitrary codimension in Section~\ref{Gen}. Examples suggest that
these bounds are best possible. For now we 
note that for canonical projection patterns with $n=d=2$ and finitely
generated homology (equivalently, $L_0$ finite), 
torsion can only appear in $H_0(\Gamma;C^2)$. 



\subsection{Codimension 3}\label{cd3}
The codimension 3 case is a good deal more complex than the
codimension 2 theory, though the principles of 
computation remain the same. We consider here only the physically
interesting case
$d=3$ for simplicity, though  the method can be repeated to cover
other codimension 3 cases. Note that the condition for finitely
generated  homology (that $\nu={{n+d}\over{n}}$ be an integer) means
that the only canonical projection patterns with $d=3$ and 
with finitely generated homology will be of codimension 1 or 3.

Computation proceeds as before. We could compute via the spectral
sequence of \cite{FHKmem}, and in a sense this 
is what we do, but the need to describe the torsion terms via explicit
homomorphisms, kernels and cokernels means 
that we must examine the details of the underlying exact couple, which
amounts to saying that we compute the 
long exact sequences associated to the tower \ref{coeffs}, which here
breaks into the three short exact sequences 
$$\begin{array}{ll}
&0\to C^0_0\to C^0\to\,\Z\,\to0\,,\\
&0\to C^1_0\to C^1\to C^0_0\to0\,,\\
&0\to C^3\to C^2\to C^1_0\to0\,.
\end{array}
$$
The dimension 3 internal space now has families of singular lines and
singular planes. As in \cite{FHKmem}, and below in 
Theorem~\ref{thm-corrections}, we shall index by 
$\theta$ the lines and by $\alpha$ the planes. The rank of the main
group $\Gamma$ is 6. Finite generation of 
homology implies that the rank of the stabiliser $\Gamma^\alpha$ of a
singular plane is 4, while the stabiliser $\Gamma^\theta$ of a singular 
line has rank 2.

The third of the short exact sequences above gives us a long exact
sequence computing $H_*(\Gamma;C^3)$ with sections
\begin{equation}
\cdots\to H_{s}(\Gamma;C^3) \to H_{s}(\Gamma;C^2)
\buildrel\phi_{s}\over\longrightarrow H_{s}(\Gamma;C^1_0)\to
H_{s-1}(\Gamma;C^3)\to\cdots
\end{equation}
telling us we have short exact sequences
\begin{equation}\label{SES}
0\to\mbox{coker }\phi_{s+1}\to H_s(\Gamma;C^3)\to\mbox{ker
}\phi_s\to0\,.
\end{equation}

In order to describe these kernel and cokernel terms, we must first
describe the homologies $H_*(\Gamma;C^1_0)$ 
and $H_*(\Gamma;C^2)$. Computations for each are essentially
identical to those for a codimension 2 system, as 
presented in the previous subsection. In particular, using
the exact sequence of coefficients
$$0\to C^1_0\to C^1\to C^0\to\,\Z\,\to0\,$$ 
we compute
\begin{equation}\label{HC10}
H_s(\Gamma;C^1_0)=\left\{\begin{array}{ll}
0& \mbox{for}\quad s\geqslant5,\\
\Lambda_{s+2}\Gamma& \mbox{for}\quad s\geqslant2,\\
\Lambda_3\Gamma\oplus\mbox{ ker }\gamma_1& \mbox{for}\quad s=1,\\
\mbox{coker }\gamma_1\oplus\mbox{ ker }\gamma_0& \mbox{for}\quad s=0,
\end{array}\right.
\end{equation}
where 
$$\gamma_s\colon\bigoplus_{\theta\in
  I_1}\Lambda_{s+1}\Gamma^\theta\to\Lambda_{s+1}\Gamma 
\quad (s>0),\quad
\gamma_0\colon\bigoplus_{\theta\in
  I_1}\big(\Lambda_1\Gamma^\theta\oplus
\ker\epsilon^\theta \big)
\to\Lambda_1\Gamma\oplus\ker\epsilon$$
are both induced by the inclusions $\Gamma^\theta\to\Gamma$ and
$C_\theta^0\subset C^0$. Here
$L_0$ denotes as usual the number of 
$\Gamma$-orbits of cut points, and $L^\theta_0$ denotes the number of
$\Gamma^\theta$-orbits of cut points on the 
line indexed by $\theta$; finite generation of homology implies all
these numbers to be finite \cite{FHKmem}. 
Note that all the terms in (\ref{HC10}) are free of torsion except
possibly the coker$\,\gamma_1$ summand.  

For each singular plane we have an analogous sequence
$$0\to C^2_\alpha\to C^1_\alpha\to C^0_\alpha\to\,\Z\,\to0$$
and the homology $H_*(\Gamma;C^2)$ splits as $\bigoplus_{\alpha\in I_2}
H_*(\Gamma^\alpha;C^2_\alpha)$. We obtain 
\begin{equation}\label{HC2}
H_s(\Gamma^\alpha;C^2_\alpha)=\left\{\begin{array}{ll}
0& \mbox{for}\quad s\geqslant3,\\
\Lambda_{s+2}\Gamma^\alpha=\Z& \mbox{for}\quad s=2,\\
\Lambda_3\Gamma^\alpha\oplus\mbox{ ker }\beta^\alpha_1& 
\mbox{for}\quad 
s=1,\\
\mbox{coker }\beta_1^\alpha\oplus\mbox{ ker }\beta_0^\alpha&
\mbox{for}\quad s=0\,. 
\end{array}\right.
\end{equation}
Here
$$\begin{array}{rl}
\beta^\alpha_s\colon&\bigoplus_{\theta\in
I_1^\alpha}\Lambda_{s+1}\Gamma^{\theta}
\to\Lambda_{s+1}\Gamma^\alpha\qquad\ \quad (s>0),\\
\beta^\alpha_0\colon&\bigoplus_{\theta\in 
I_1^\alpha}\big(\Lambda_1\Gamma^{\theta}
\oplus\ker\epsilon^\theta\big)
\to\Lambda_1\Gamma^\alpha\oplus\ker\epsilon^\alpha.\end{array}$$ 
The $\beta_s^\alpha$ are again induced by inclusion, and the
only potential torsion term in (\ref{HC2}) arises from the
cokernel$\,\beta^\alpha_1$ expression;  all other terms are free
abelian.

The expression for $\phi_s$ under the identifications
(\ref{HC10},$\,$\ref{HC2}) can be
obtained as in \cite{FHKmem}:
$\phi_s$ is a 
sum of morphisms $\phi_s^\alpha = (\tilde\delta_{s 2}^\alpha\otimes
1)_*$,  and the latter is determined by the diagram 
\begin{equation}
\begin{array}{rcccccl}
{\to} & H_s(\Gamma;C_\alpha^2\otimes\Z[\Gamma/\Gamma^\alpha]) & \to &
\bigoplus_{\theta\in I_1^\alpha} \Lambda_{s+1}\Gamma^{\theta} &
\bra{\beta^\alpha_s} & \Lambda_{s+1}\Gamma^{\alpha} & {\to}\\
&\bigg\downarrow \phi_s^\alpha & & \bigg\downarrow
j^\alpha_{s} & & \bigg\downarrow \imath^\alpha_s& \\ {\to} &
H_s(\Gamma;C^1_0) & \to & \bigoplus_{\theta\in I_1}
\Lambda_{s+1}\Gamma^{\theta} & \bra{\gamma_s} &
\Lambda_{s+1}\Gamma & {\to} 
\end{array}
\end{equation}
for $s>0$, and
\begin{equation}
\begin{array}{rcccccl}
{\to}&\!\!\!H_0(\Gamma;C_\alpha^2\otimes\Z[\Gamma/\Gamma^\alpha])\!\!\! &
\to &
\!\!\!\bigoplus_{\theta\in I_1^\alpha}(\Lambda_{1}\Gamma^{\theta}
\oplus\ker\epsilon^\theta)\!\!\!  &
\bra{\beta^\alpha_0} & \!\!\!\Lambda_{1}\Gamma^{\alpha}
\oplus\ker\epsilon^\alpha\!\!\!& 
{\to0}\\
&\bigg\downarrow \phi_s^\alpha & & \bigg\downarrow
j^\alpha_{0} & & \bigg\downarrow \imath^\alpha_0& \\ {\to} &
H_0(\Gamma;C^1_0) & \to & \!\!\!\bigoplus_{\theta\in I_1}
(\Lambda_{1} \Gamma^{\theta}\oplus\ker\epsilon^\theta)\!\!\! &
\bra{\gamma_0} &
\!\!\!\Lambda_{1}\Gamma\oplus\ker\epsilon\!\!\! & {\to0} 
\end{array}
\end{equation}
where $j_s^\alpha$ and $\imath^\alpha_s$ are the obvious inclusions.

\begin{lemma}
Consider a commutative diagram of abelian groups,
\begin{equation}
\begin{array}{rcccccl}
0 {\to} & A & \longrightarrow & B & 
\buildrel f\over\longrightarrow & C & \to 0 \\
& \downarrow \psi'  & & \downarrow \psi
& & \downarrow\psi'' & \\ 
0 {\to} & X & \longrightarrow & Y & 
\buildrel g\over\longrightarrow & Z & \to 0,
\end{array}
\end{equation}
in which the rows are short exact. If we assume that $\psi''$ is
injective and $C$ and $Z$ are free abelian then $B\cong A\oplus C$,
$Y\cong X\oplus Z$ and with respect to this decomposition 
$\psi = \psi' \oplus \psi''$.
\end{lemma}

\noindent{\em Proof.} Freeness of $C$ and $Z$ ensures
that the exact rows split, and we can consider $B$ and
$Y$ as direct sums as indicated. The issue is to show
that with respect to such a decomposition the resulting
matrix expression for $\psi$ is diagonal. Simple
diagram chasing shows that one off-diagonal term (the
map from $A$ to $Z$) will always be zero; thus $\psi$
will have an expression
$$\left(\begin{array}{cc}
\psi'&\zeta\\0&\psi''
\end{array}\right)$$
for some homomorphism $\zeta\colon C\to X\subset Y$. If $\zeta$ is not the
zero map, then there is an element $c\in C$ with $\psi(0,c)=\zeta(c)$ a
non-zero element of the image of $X$ in $Y$. Then 
$$0=g\psi(0,c)=\psi''(f(0,c))=\psi''(c)$$
contradicting $\psi''$ being injective.
 \hfill$\square$

\smallskip\noindent
We apply this lemma to $\psi=\phi_s^\alpha$ to obtain 
$$\phi_s^\alpha \cong \tilde i_{s+1}^\alpha \oplus
{j_s^\alpha}\left|_{\ker\beta_s^\alpha}\right.
$$
where 
$\tilde
i_{s+1}^\alpha:\coker\beta_{s+1}^\alpha\to\coker\gamma_{s+1}$  is
induced by $i_{s+1}^\alpha$. The conditions are satisfied as
$\ker\beta_s^\alpha$ and $\ker\gamma_s$ are free abelian and
  ${j_s^\alpha}\left|_{\ker\beta_s^\alpha}\right.$ injective.
Since $\phi_s$ is the direct sum of the $\phi_s^\alpha$, we obtain
the following description of the long exact sequence
(only $s\leqslant 2$ is relevant for our considerations)
\begin{equation}\label{dia1}
\begin{array}{rcccl}
\to H_{2}(\Gamma;C^3)\to
& H_2(\Gamma;C^2)& \stackrel{\phi_2}{\longrightarrow} &
  H_2(\Gamma;C^1_0) &\to  
\\
& \| & & \| & \\
& \bigoplus_{\alpha\in I_2} \Lambda_{4}\Gamma^\alpha &
\stackrel{\phi_2'}{\longrightarrow} & \Lambda_{4}\Gamma \\
\\ \\
\to H_{1}(\Gamma;C^3)\to
& H_1(\Gamma;C^2)& \stackrel{\phi_1}{\longrightarrow} &
  H_0(\Gamma;C^1_0) &\to  
\\ 
& \| & & \| & \\
& \!\!\!\!\!\!\bigoplus_{\alpha\in I_2} (\Lambda_{3}\Gamma^\alpha \oplus
\ker\beta_1^\alpha) &
\stackrel{\phi_1'\oplus\phi_1''}{\longrightarrow} & 
\Lambda_{3}\Gamma \oplus 
\ker\gamma_1 
\\ \\ \\ 
\to H_{0}(\Gamma;C^3)\to
& H_0(\Gamma;C^2)& \stackrel{\phi_0}{\longrightarrow} &
  H_0(\Gamma;C^1_0) &\to 0 \\
& \| & & \| & \\
& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\bigoplus_{\alpha\in I_2}
(\coker\beta_1^\alpha
\oplus
\ker\beta_0^\alpha)\!\!\! &
\stackrel{\phi_0'\oplus\phi''_0}{\longrightarrow} & \!\!\!\coker\gamma_1
\oplus
\ker\gamma_0\!\!\!\!\!\! \end{array} 
\end{equation}
where $\phi'_s$ and $\phi''_s$ represent the homomorphisms
$\tilde i_{s+1}^\alpha$ and
${j_s^\alpha}\left|_{\ker\beta_s^\alpha}\right.$, respectively.

\begin{theorem}\label{codim3}
Consider a canonical projection pattern with $n=d=3$ and assume the
homology is finitely generated 
(equivalently \cite{FHKmem} that $L_0$ is finite). Then 
$$H_s(\Gamma;C^3)=\left\{\begin{array}{ll}
0& \mbox{for}\quad s\geqslant4,\\
\Z& \mbox{for}\quad s=3,\\
\Z^6\oplus\ker\left\{\phi_2'\colon\bigoplus_{\alpha\in I_2}
(\Lambda_4\Gamma^\alpha)\to\Lambda_4\Gamma\right\} 
&\mbox{for}\quad s=2,
\end{array}\right.$$
and so there is no torsion in these degrees. There is potential
torsion in degrees 0 and 1, and we have  
$$\mbox{{\em Torsion}}\left(H_1(\Gamma;C^3)\right)=\mbox{{\em Torsion}}
(\coker\phi_2')$$
and an exact sequence (note the right hand map is not necessarily onto)
$$0\to\mbox{{\em Torsion}}\left(\coker \phi_1'\oplus \coker \phi_1''\right)
\to\mbox{{\em Torsion}}\left(H_0(\Gamma;C^3)\right)
\to\mbox{{\em Torsion}}\left(\ker(\phi_0')\right).$$
\end{theorem}

\smallskip\noindent{\em Proof.} These results can be read off from the
long exact sequence
(\ref{dia1}) together with the observation that
$\ker\phi_0''$ and $\ker\phi_1$ are torsion free. The last exact
sequence arises, as taking the torsion is a left exact
functor.\hfill$\square$
\bigskip

Note that the torsion of a codimension $3$ tiling is determined
by the above theorem without additional information only if 
$\ker \phi_0'$ is torsion free. Otherwise the exact sequence
(\ref{SES}) for $s=0$ 
$$0\to\mbox{coker }\phi_{1}\to
H_0(\Gamma;C^3)\to\mbox{ker }\phi_0\to0$$
presents an extension problem, whose solution requires further
geometric input from the projection data. We see an example of such
a problem in Section~\ref{sec-examples}, where we compute the
homology of the dual canonical $D_6$ tiling.






\section{Corrected formul\ae\ for the rational ranks of codimension 3
systems}\label{sect-correction}

We correct the formul\ae\ of Theorem~2.7 of
\cite{FHKmem}. The formul\ae\ given in \cite{FHKmem} are correct only if
the equation (\cite{FHKmem}, page 112, bottom half)
$ \erz{\im
j^\alpha_s\cap\ker \gamma_s:\alpha\in I_2} = \ker \gamma_s $
is valid. For the Ammann-Kramer tiling and the
dual canonical $D_6$ tiling the rank of the left hand
side is, however, one less than the rank of the right hand side. 

The corrected formul\ae\ for the ranks is obtained by replacing 
$$\rk\ker\gamma_s = 
L_1 -\rk\erz{\Lambda_{s+1}\Gamma^{\Theta}:\Theta\in I_1}$$
with 
$$\rk \erz{\im j^\alpha_s\cap\ker \gamma_s:\alpha\in I_2} = 
\rk\erz{(\bigoplus_{\theta\in I_1^\alpha}
\Lambda_{s+1}\Gamma^\theta)\cap
  \ker\gamma_s^\alpha : \alpha\in I_2} $$
where
$$\gamma_s^\alpha:\bigoplus_{\theta\in
  I_1^\alpha}\Lambda_{s+1}\Gamma^\theta\to
\Lambda_{s+1}\Gamma $$
is the direct sum of the inclusions, i.e.\
$\gamma_s^\alpha (x_1,\cdots,x_{L_1^\alpha})=x_1+\cdots +
x_{L_1^\alpha}$ (in the notation of \cite{FHKmem} $\gamma_s^\alpha =
\gamma_s\circ j_s^\alpha$.) This replacement is straightforward,
yielding:

\begin{theorem}[Erratum to Theorem~2.7 of
\cite{FHKmem}] \label{thm-corrections}
Given a projection method pattern with codimension $3$
and finite $L_0$ then, for $s>0$, 
\begin{eqnarray*} 
D_s &=& \left( 3\nu\atop s+3\right)
+L_2 \left( 2\nu\atop s+2\right) + \sum_{\alpha\in I_2}L_1^\alpha
\left( \nu\atop s+1\right) + L_1\left( \nu\atop s+2\right)\\
&& - R_s - R_{s+1}, \\ 
D_0 & =& \sum_{j=0}^3 (-1)^j
\left( 3\nu\atop 3-j\right)+ L_2\sum_{j=0}^2 (-1)^j \left(
2\nu\atop 2-j\right) \\
&& + \sum_{\alpha\in I_2}L_1^\alpha \sum_{j=0}^1 (-1)^j
\left(\nu\atop 1-j\right) + L_1\sum_{j=0}^2 (-1)^j
\left(\nu\atop 2-j\right) + e  - R_1 
\end{eqnarray*} where 
\begin{eqnarray*}
R_s &=& \rk\erz{\Lambda_{s+2}\Gamma^{\alpha}:\alpha\in I_2}
+\sum_{\alpha\in I_2}
\rk\erz{\Lambda_{s+1}\Gamma^{\theta}:\theta\in I_1^\alpha}\\
& &+\rk\erz{(\bigoplus_{\theta\in
I_1^\alpha}\Lambda_{s+1}\Gamma^\theta)\cap
  \ker\gamma_s^\alpha : \alpha\in I_2}
\end{eqnarray*}
and the Euler characteristic is $$ e:=\sum_{s} (-1)^s D_{s} =
L_0-\sum_{\alpha\in I_2} L^\alpha_0 +\sum_{\alpha\in
I_2}\sum_{\theta\in I_1^\alpha}L^\theta_0 -\sum_{\theta\in
I_1}L^\theta_0. $$
\end{theorem}
The above formul\ae\ applied to
the Ammann-Kramer tiling agree with the results of \cite{Kalugin}. 
The corrected rational ranks for the tilings of \cite{FHKicmp} appear
in Section~\ref{sec-examples}.





\section{Examples}
\label{sec-examples}

\subsection{Remarks on the computations}
\label{sec:excomp}

All examples discussed below have to some 
extent been calculated by computer. For this purpose, we have used
the computer algebra system GAP \cite{GAP}, the GAP package Cryst
\cite{EGN97,Cryst}, as well as further software written in the GAP 
language. It should be emphasized that these computations are not 
numerical, but use integers and rationals of arbitrary size or 
precision. Neglecting the possibility of programming errors, 
they must be regarded as exact. 

One piece of information that needs to be computed is the set of
all intersections of singular affine subspaces, along with
their incidence relations. This is done with code based on the
Wyckoff position routines from the Cryst package. The set of
singular affine subspaces is invariant under the action of a
space group. Cryst contains routines to compute intersections
of such affine subspaces and provides an action of space group
elements on affine subspaces, which allows to compute space group
orbits. These routines, or variants thereof, are used to determine 
the space group orbits of representatives of the singular affine 
subspaces, and to decompose them into translation orbits. The 
intersections of the affine subspaces from two translation orbits 
is the union of finitely many translation orbits of other affine 
subspaces. These intersections can be determined essentially by 
solving a linear system of equations modulo lattice vectors,
or modulo integers when working in a suitable basis. With these 
routines, it is possible to generate from a space group and a 
finite set $\mathcal{W}$ of representative singular affine spaces 
the set of all singular spaces, their intersections, and their 
incidences.

A further task is the computation of ranks, intersections, and 
quotients of free $\Z$-modules, and of homomorphisms between such 
modules, including their kernels and cokernels. These are standard 
algorithmic problems, which can be reduced to the computation of 
Smith and Hermite normal forms of integer matrices, including the 
necessary unimodular transformations \cite{Coh95}. GAP already 
provides such routines, which are extensively used.

\subsection{The first appearance of torsion}
\label{sec:extor}

The first hint for the possible existence of torsion in tiling
(co)homology was obtained \cite{FGBanff} by applying the AP method 
\cite{AP} to several substitutional canonical projection tilings of 
dimension two. The purpose of this endeavour was to determine the 
action of the substitution on the tiling cohomology. We only sketch 
the general procedure here. In a first step, a large piece of the 
tiling is generated by substitution, and from the local tile 
neighborhoods a finite CW space $\Omega_{1}$ is constructed, 
as detailed in \cite{AP}. The tiles are labelled according to their 
first corona, and each translation class of labelled tiles gives 
rise to a 2-cell in $\Omega_{1}$. Boundaries of such 2-cells are 
identified whenever corresponding labelled tiles share such a boundary 
somewhere in the tiling. There is a natural action of the substitution 
on the (integral) co-chain groups of $\Omega_{1}$, which induces a
corresponding action on the cohomology groups $H^{*}(\Omega_{1})$. 
The cohomology of the hull of the tiling is then obtained by taking 
the direct limit of $H^{*}(\Omega_{1})$ under the iterated substitution
action \cite{AP}.

This procedure was applied to the Penrose tiling \cite{deBruijn}, 
the (undecorated) Ammann-Beenker tiling \cite{Beenker,AGS}, and 
the T\"ubingen Triangle Tiling (TTT) \cite{BKSZ,KSB}. Especially 
for the latter, it is a computational tour de force. For the TTT, 
$\Omega_{1}$ turned out to have 860 2-cells, 1710 1-cells, and 
880 0-cells. Computing the (integral) cohomology of such a large 
cell complex is computationally quite demanding, and it seems 
hardly possible to consider more complicated cases like the 
decorated versions of the Ammann-Beenker tiling \cite{AGS,Soc89} 
and the Socolar tiling \cite{Soc89}, let alone the three-dimensional 
examples. In all cases considered, all 
rational ranks of the cohomology groups agreed with the results 
obtained earlier by different methods \cite{GK}. The only 
difference was a torsion part $\Z_{5}^{2}$ in $H^{2}$ ($\cong H_{0}$) 
of the TTT, which according to \cite{FHKcmp,FHKmem} should not be there, 
and which did not go away even after an extensive review of the 
(rather complicated) computer program. This was the starting point 
of the revised theory of tiling (co)homology presented here.

\subsection{Codimension 2 examples with dihedral symmetry}
\label{sec:ex2d}

The codimension 2 examples discussed in the following have
dihedral symmetry of order $2n$, with $n$ even. The lattice 
$\Gamma_{n}$ is given by the $\Z$-span of the vectors in the 
star $e_{i}=(\cos(\frac{2\pi i}{n}),\sin(\frac{2\pi i}{n}))$,
$i=0,\ldots,n-1$. The singular lines have special orientations 
with respect to this lattice. They are parallel to mirror lines 
of the dihedral group, which means that they are either {\em along}
the basis vectors $e_{i}$, or {\em between} two neighboring basis 
vectors, i.e., along $e_{i}+e_{i+1}$. In all examples, one line 
from each translation orbit passes through the origin. We denote 
the sets of representative singular lines by $\mathcal{W}_{n}^{a}$
and $\mathcal{W}_{n}^{b}$, for lines along and between the basis 
vectors $e_{i}$. Denoting by $V$ the internal space 
containing the singular lines, we can now describe the defining 
data of several well-known tilings, whose cohomology was already
given in \cite{GK} (without torsion). The Penrose tiling \cite{deBruijn}
is defined by the triple $(V,\Gamma_{10},\mathcal{W}_{10}^{a})$,
the T\"ubingen Triangle Tiling (TTT) \cite{BKSZ,KSB} by the triple
$(V,\Gamma_{10},\mathcal{W}_{10}^{b})$, the undecorated octagonal 
Ammann-Beenker tiling \cite{Beenker} by the triple 
$(V,\Gamma_{8},\mathcal{W}_{8}^{a})$, and the undecorated Socolar
tiling \cite{Soc89} by the triple 
$(V,\Gamma_{12},\mathcal{W}_{12}^{a})$. For the decorated versions
of the Ammann-Beenker \cite{Soc89,AGS,G93} and Socolar tilings 
\cite{Soc89}, the set of singular lines $\mathcal{W}_{n}^{a}$ 
has to be replaced by $\mathcal{W}_{n}^{a}\cup\mathcal{W}_{n}^{b}$, 
$n=8$ and 12, respectively. Finally, the heptagonal tiling in \cite{GK}
is given by the triple $(V,\Gamma_{14},\mathcal{W}_{14}^{a})$.
For all these tilings, the rational ranks given in \cite{GK}
remain valid. The results of the 2 dimensional tilings are listed 
in Table~\ref{tab2d}.

The generalized Penrose tilings \cite{PenGen} are somewhat different 
from the tilings discussed above. They are built upon the decagonal
lattice $\Gamma_{10}$, too, but have only fivefold rotational symmetry.
The singular lines do not pass through the origin in general,
and their positions depend on a continuous parameter $\gamma$.
For instance, the representatives lines of the two translation 
orbits of lines parallel to $e_{0}$ pass through the points 
$-\gamma e_{1}$ and $\gamma(e_{1}+e_{2})$. It turns out that
these shifts of line positions always lead to the same line 
intersections and incidences. Even multiple intersection points
remain stable, and are only moved around if $\gamma$ is varied.
Consequently, all generalized Penrose tilings have the same
homology, except for $\gamma\in\Z[\tau]$, which corresponds
to the real Penrose tilings \cite{deBruijn}. This had already been 
observed by Kalugin \cite{Kalugin}, and is in contradiction with 
the results given in \cite{GK}, which were obtained due to a 
wrong parametrisation of the singular line positions. Corrected 
results are given in Table~\ref{tab2d}.

\begin{table}
\caption{Homology of codimension 2 tilings with dihedral symmetry.}
\label{tab2d}
\begin{center}
\small
\newcommand{\op}{\oplus}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|l|c|c|c|}
\hline%\rule{0ex}{2.8ex}
Tiling                       & $H_0$                &$H_1$      & $H_2$  \\
\hline%\rule{0ex}{2.8ex} 
Ammann-Beenker (undecorated) & $\Z^{ 9}$            & $\Z^{ 5}$ & $\Z^1$ \\
Ammann-Beenker (decorated)   & $\Z^{23}$            & $\Z^{ 9}$ & $\Z^1$ \\
Penrose                      & $\Z^{ 8}$            & $\Z^{ 5}$ & $\Z^1$ \\
generalized Penrose          & $\Z^{34}$            & $\Z^{10}$ & $\Z^1$ \\
T\"ubingen Triangle          & $\Z^{24}\op\Z_5^{2}$ & $\Z^{ 5}$ & $\Z^1$ \\
Socolar (undecorated)        & $\Z^{28}$            & $\Z^{ 7}$ & $\Z^1$ \\
Socolar (decorated)          & $\Z^{59}$            & $\Z^{12}$ & $\Z^1$ \\
\hline
\end{tabular}
\end{center}
\end{table}

Among the tilings discussed above, only the TTT has torsion in 
its homology. The set of singular lines of the TTT is constructed 
from the lines $\mathcal{W}_{10}^{b}$. The translation stabilizers 
$\Gamma^{\alpha}$ of all these lines are contained in a common 
sublattice $\Gamma'_{10}$ generated by the star of vectors 
$e_{i}+e_{i+1}$; it has index 5 in $\Gamma_{10}$.
It is therefore not too surprising that $\coker\beta_{1}$
(Theorem~\ref{thm51}) develops a torsion component $\Z_{5}^{2}$, 
which shows up in the homology group $H_{0}$ of the TTT, in 
agreement with the results obtained by the AP method.
In much the same way, a torsion component $\Z_{2}$ in $H_{0}$
is obtained also for the octagonal tilings described by the data 
$(V,\Gamma_{8},\mathcal{W}_{8}^{b})$, but these tilings have not 
been considered before in the literature. The same applies also
to the four-dimensional codimension 2 tilings with data
$(V,\Gamma_{14},\mathcal{W}_{14}^{b})$, which have a torsion
component $\Z_{7}^{4}$ in $H_{0}$, and $\Z_{7}^{3}$ in $H_{1}$,
in agreement with the bounds given in Theorem~\ref{thm51}.

There is an interesting relation of the TTT to the Penrose
tiling. Since the lattice $\Gamma'_{10}$ is rotated by $\pi/10$ 
with respect to $\Gamma_{10}$, the TTT can also be constructed
from the triple $(V,\Gamma'_{10},\mathcal{W}_{10}^{a})$. However,
the singular set $\Gamma'_{10}+\mathcal{W}_{10}^{a}$ is even
invariant under all translations from $\Gamma_{10}$, so that it
is equal to $\Gamma_{10}+\mathcal{W}_{10}^{a}$, which defines
the Penrose tiling. In other words, the TTT and the Penrose tiling 
have the same set of singular lines, only the lattice $\Gamma$ 
acting on it is different. The TTT is obtained by breaking the 
translation symmetry of the Penrose tiling to a sublattice of 
index 5. This explains why the Penrose tiling is locally derivable 
from the TTT, but local derivability does not hold in the opposite
direction \cite{BSJ91}. A broken symmetry can be restored in a local way, 
but the full lattice symmetry cannot be broken to a sublattice 
in any local way, because there are no local means to distinguish 
the five cosets of the sublattice. Tilings whose set of singular 
lines accidentally has a larger translation symmetry are likely 
candidates for having torsion in their homology.


\subsection{Codimension 3 examples with icosahedral symmetry}
\label{sec:ex3d}

There are four icosaheral tilings whose homology has been discussed
in the literature so far: the Ammann-Kramer tiling \cite{AmmannKramer}, 
the Danzer tiling \cite{Dan89}, the canonical $D_{6}$ tiling \cite{KP95}, 
and the dual canonical $D_{6}$ tiling \cite{KP95}. We give here an 
update to those results. In particular, we add the missing 
information on torsion, and also correct the rational ranks 
as outlined in Section~\ref{sect-correction}. These examples also give a 
good overview of the different phenomena that can occur in the 
determination of torsion.

We start by describing the relevant lattices $\Gamma$ and families of
singular planes $\mathcal{W}$. In three dimensions, there are three 
inequivalent icosahedral lattices of minimal rank 6. The primitive
lattice $\Gamma_{P}$ is generated by a star of vectors pointing from
the center to the vertices of a regular icosahedron. We choose any
basis $e_{1},\ldots,e_{6}$ from this vector star. The lattice 
$\Gamma_{F}$ is then the sublattice of those integer linear combinations
of the $e_{i}$, whose coefficients add up to an even integer. 
The lattice $\Gamma_{I}$ is given by the $\Z$-span of the vectors in
$\Gamma_{P}$, and the additional vector $\frac12(e_{1}+\ldots+e_{6})$.
These lattices are analogues of the the primitive, F-centered, and 
I-centered cubic lattices.\footnote{strictly speaking, $\frac12\Gamma_{F}$
is a centering of $\Gamma_{P}$} $\Gamma_{F}$ is an index-2 sublattice
of $\Gamma_{P}$, which in turn is an index-2 sublattice of $\Gamma_{I}$.
The action of the icosahedral group $A_{5}$ on the three lattices gives
rise to three integral representations, which are inequivalent under
conjugation in $GL_{6}(\Z)$. The singular planes of all four examples 
have special orientations, being perpendicular either to a 5-fold,
a 3-fold, or a 2-fold axis of the icosahedron (the latter are also parallel
to a mirror plane). Moreover, each $\Gamma$-orbit of singular planes 
contains a representative which passes through the origin. We therefore
define the families of planes $\mathcal{W}^{n}$, $n=5,3,2$, consisting
of all planes perpendicular to an $n$-fold axis, and passing through
the origin. Denoting by $V$ the internal space 
containing the lattices and singular planes, the Ammann-Kramer tiling 
is then defined by the triple $(V,\Gamma_{P},\mathcal{W}^{2})$, 
the dual canonical $D_{6}$ tiling by the triple $(V,\Gamma_{F},
\mathcal{W}^{2})$, the Danzer tiling by the triple $(V,\Gamma_{F},
\mathcal{W}^{5})$, and the canonical $D_{6}$ tiling by the triple 
$(V,\Gamma_{F},\mathcal{W}^{5}\cup\mathcal{W}^{3})$. Interestingly,
the sets $\Gamma_{P}+\mathcal{W}^{2}$ and $\Gamma_{F}+\mathcal{W}^{2}$
are invariant even under all translations from $\Gamma_{I}$, which
means that they are both equal to $\Gamma_{I}+\mathcal{W}^{2}$. In 
other words, the sets of singular planes of the Ammann-Kramer tiling 
and the dual canonical $D_{6}$ tiling are the same, only the lattices 
acting on them by translation are different. Conversely, the sets
of singular planes of the Danzer tiling and the canonical $D_{6}$ tiling
have a lattice of translation symmetries which is equal to the lattice
$\Gamma_{F}$ they are constructed from. With these data, and the 
methods described in section \ref{sec:excomp}, is it now straightforward 
to evaluate the formul\ae\ (Theorem~\ref{thm-corrections}) 
for the ranks of the rational homology groups. 
The results are summarized in Table~\ref{tabico}. As can be seen, compared 
to previously published results the rational ranks of $H_{0}$ and $H_{1}$ of 
the Ammann-Kramer tiling and the dual canonical $D_{6}$ tiling have 
been increased by 1, in agreement with Kalugin \cite{Kalugin}, whereas 
all other rational ranks remain the same.

Next, we discuss the determination of torsion, which is possibly
non-trivial only for $H_{0}$ and $H_{1}$. The torsion in $H_{1}$
is given by the torsion of $\coker\phi'_{2}$, which is straightforward
to compute. The results are given in Table~\ref{tabico}. For
the torsion in $H_{0}$, however, we only have the exact sequence
in Theorem~\ref{codim3}. As $\phi'_{1}$ and $\phi''_{1}$ are maps between
free abelian groups, the torsions of $\coker\phi'_{1}$ and  
$\coker\phi''_{1}$ are not difficult to compute. The results are
shown in Table~\ref{tabico}. It is interesting to note that for all
four examples considered, $\coker\phi''_{1}$ is not free, which 
will result in all four tilings having torsion in $H_{0}$. This 
is so even for a tiling as simple as the Danzer tiling, which has, 
like the Penrose tiling, only a single $\Gamma$-orbit of singular 
points. In fact, we do not know of any 3D icosahedral tiling having 
vanishing torsion in $H_{0}$. To determine the torsion in $H_{0}$,
we further need the torsion of $\ker\phi'_{0}$, which is a map from 
$\oplus_{\alpha\in I_{2}}\coker\beta_{1}^{\alpha}$ to 
$\coker\gamma_{1}$, both of which are potentially non-free. 
In three of our four examples, $\coker\beta_{1}^{\alpha}$ 
is free for all $\alpha\in I_{2}$, hence also $\ker\phi'_{0}$ is free,
and the torsion of $H_{0}$ is given by the
direct sum of the torsions of $\coker\phi'_{1}$ and $\coker\phi''_{1}$.
Our fourth example, the dual canonical $D_6$ tiling, is the
only one which needs a more detailed analysis. 
In this case, $\coker\beta_{1}^{\alpha}$ is not free, but 
$\coker\gamma_{1}$ is, so that the torsion of $\ker\phi'_{0}$ can 
still be determined; 
in fact, $\tor(\ker\phi'_{0})=\Z_{2}^{15}$, because
$\tor(\coker\beta_{1}^{\alpha})=\Z_{2}$, and there are 15 planes
$\alpha$. We further have $\tor(\coker\phi'_{1})=\Z_{2}^{6}$ and
$\tor(\coker\phi''_{1})=\Z_{2}^{7}$. 
As an additional piece of information, we can compute 
the rank $T_{0}^{2}$ of the torsion subgroup of $H_{0}(\Gamma,C^3)$, 
which by Proposition~\ref{torrank} is the difference of the ranks of
$H_0(\Gamma;C^3\otimes \F_2)$ and $H_0(\Gamma;C^3\otimes\Q)$
(note that 2 is the only prime $p$, for which $T_{0}^{p}$ 
can be non-zero). We find
$$
H_0(\Gamma;C^3\otimes \Q) = \Q^{D_0},\quad 
H_0(\Gamma;C^3\otimes \F_2) = \F_2^{D_0+27},
$$
so that $T_{0}^{2}=27$ ($D_{0}=331$). 

\begin{table}
\caption{Homology of codimension 3 tilings with icosahedral symmetry,
and data required for the determination of its torsion component. 
We use the abbreviations $t'_1 = \tor(\coker \phi'_1)$, 
$t''_1 = \tor(\coker \phi''_1)$, and $t'_0 = \tor(\ker \phi'_0)$.}
\label{tabico}
\begin{center}
\small
\newcommand{\op}{\oplus}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
  \hline%\rule{0ex}{2.8ex}
  Tiling           & $t'_{1}$
                   & $t''_{1}$ 
                   & $t'_{0}$  & $H_{0}$ & $H_{1}$ & $H_{2}$ & $H_{3}$ 
                   \\
  \hline%\rule{0ex}{2.8ex} 
  Ammann-Kramer    & 0 & $\Z_{2}$ & 0 
                   & $\Z^{181}\op\Z_{2}$             
                   & $\Z^{ 72}\op\Z_{2}$              
                   & $\Z^{12}$ & $\Z^1$ \\
  dual canonical $D_6$  & $\Z_{2}^{6}$ & $\Z_{2}^{7}$ & $\Z_{2}^{15}$ 
                   & $\Z^{331}\op\Z_{2}^{26}\op\Z_4$ 
                   & $\Z^{102}\op\Z_{2}^{4}\op\Z_{4}$ 
                   & $\Z^{12}$ & $\Z^1$ \\
  Danzer            & 0 & $\Z_{2}$ & 0  
                   & $\Z^{ 20}\op\Z_{2}$             
                   & $\Z^{ 16}$                       
                   & $\Z^7$ & $\Z^1$\\
  canonical $D_6$  & 0 & $\Z_{2}$ & 0 
                   & $\Z^{205}\op\Z_{2}^{2}$         
                   & $\Z^{ 72}$                       
                   & $\Z^7$ & $\Z^1$ \\
 \hline
 \end{tabular}
%\vspace{0.3cm}
\end{center}
\end{table}

Taking everything together, this means that the exact sequence
(\ref{SES}) for $s=0$ is of the form
$$0\to \Z_2^{13}\oplus\Z^l\to T\oplus\Z^{D_0} \to
\Z_2^{15}\oplus\Z^{D_0-l}\to0$$
for some finite abelian group $T$ with $T\otimes\F_2=\Z_2^{27}$;
moreover, it is immediate that $T$ can contain only elements 
of order 2 and 4. Let us
write $T=\Z_2^a\oplus\Z_4^b$ for non-negative integers $a$ and $b$. As
$T\otimes\F_2=\Z_2^{27}$ we must have $a+b=27$, but computing the 
order of $T$ we also need $a+2b=28-z,$ where $z$ is the rank of the
cokernel of the inclusion $\Z^l\to\Z^{D_0}$. Together these say $b=1-z$,
hence $b=0$ or 1, and inspection shows the only two possible
extensions are as follows. The first is
\begin{equation}\label{ext1}
0 \to \Z^{12}_2\oplus  \Z_2 \oplus \Z^l \stackrel{\varphi}{\to}  
\Z^{26}_2\oplus  \Z_4 \oplus \Z^{D_0} \to
 \Z^{14}_2\oplus  \Z_2 \oplus \Z^{D_0-l} \to 0
\end{equation}
where $\varphi = \varphi_1\oplus \varphi_2\oplus \varphi_3$ respect
the direct sums indicated, $\varphi_1$ and $\varphi_3$ are inclusions 
as direct summands, and $\varphi_2:\Z_2\to\Z_4$ is multiplication
by $2$. The second possible extension is
\begin{equation}\label{ext2}
0 \to \Z^{13}_2\oplus  \Z \oplus \Z^{l-1} \stackrel{\varphi'}{\to}  
\Z^{27}_2\oplus  \Z \oplus \Z^{D_0-1} \to
 \Z^{14}_2\oplus  \Z_2 \oplus \Z^{D_0-l} \to 0
\end{equation}
where $\varphi' = \varphi_1'\oplus \varphi_2'\oplus \varphi_3'$ respects
the direct sums indicated, $\varphi_1'$ and $\varphi_3'$ are inclusions 
as direct summands, and $\varphi_2':\Z\to\Z$ is multiplication
by $2$. We note that the first extension corresponds to the case where 
one gets an exact sequence also of the torsion parts, that is, the
right map of the last sequence in Theorem~\ref{codim3} is onto. This
could be used to distinguish the two cases, but we reason in another
way. 

The short exact sequence 
$$0\to\ker\phi_0\to
H_0(\Gamma;C^2)\stackrel{\phi_0}{\to}H_0(\Gamma;C^1_0)\to0$$
upon tensoring with a commutative ring $R$ gives (by standard homological
algebra, for example see \cite{HS}) an exact sequence
$$\cdots\to\mbox{\rm Tor}^1(H_0(\Gamma;C^1_0),R)\to\ker\phi_0\otimes R\to
H_0(\Gamma;C^2)\otimes R\stackrel{\phi_0}{\to}H_0(\Gamma;C^1_0)\otimes
R\to0\,.$$
However, by the universal coefficent theorem \cite{Bro}, we also have 
$H_0(\Gamma;C^2\otimes R) = H_0(\Gamma;C^2)\otimes R$ and
$H_0(\Gamma;C^1_0\otimes R) = H_0(\Gamma;C^1_0)\otimes R$. 
Moreover, since $H_0(\Gamma;C^1_0)$ is torsion free, we have 
$\mbox{\rm Tor}^1(H_0(\Gamma;C^1_0),R)=0$. Thus we can make the 
identification
$$\ker_{R}\phi_0  = (\ker \phi_0)\otimes R$$
where $\ker_{R}\phi_0 : =
\ker\{\phi_0 : H_0(\Gamma;C^2\otimes R) \to H_0(\Gamma;C^1_0\otimes R)\}$.
In general, however,   
$$\coker_{R} \phi_1  \neq (\coker \phi_1)\otimes R$$
and so the above two extensions can be distinguished by comparing
$\coker_{\Z_4}\phi_1$ with  
$(\coker \phi_1)\otimes \Z_4$. In fact, we deduce from the
above that (\ref{ext1}) implies
\begin{equation}\label{ext1b}
0 \to   \coker_{\Z_4}\phi_1 \to
\Z^{26}_2\oplus \Z_4^{D_0+1} \to
 \Z^{15}_2 \oplus \Z_4^{D_0-l} \to 0
\end{equation}
and (\ref{ext2}) implies
\begin{equation}\label{ext2b}
0 \to \coker_{\Z_4}\phi_1 \to
\Z^{27}_2\oplus  \Z^{D_0}_4 \to
 \Z^{15}_2 \oplus \Z^{D_0-l} \to 0.
\end{equation}
In the first case, $ \coker_{\Z_4}\phi_1$ has $2^{2l+13}$ elements and
in the second case $ \coker_{\Z_4}\phi_1$ has $2^{2l+12}$ elements. 
The number of elements of $ \coker_{\Z_4}\phi_1''$ has been determined
by computer; we find that there are $2^{9}$ elements, and
since $ \coker_{\Z_4}\phi_1'= (\coker\phi_1')\otimes\Z_4$, as one
easily checks, the first extension (\ref{ext1}) is the right one. Hence 
$$
H_0(\Gamma;C^3)\cong \Z_2^{26}\oplus\Z_4\oplus \Z^{D_0}.
$$

Summarizing, if $\ker\phi'_{0}$ is free, the torsion in $H_{0}$ can be 
determined relatively easily. Otherwise, an extension problem has to
be solved. In the case of the dual canonical $D_{6}$ tiling, there
were only two possible extensions, which were not too difficult to
distinguish. It remains to be seen whether such an approach can
still be successful also in more complicated cases.








\section{General results}\label{Gen}
We conclude with two general results, one which limits the
homological degrees in which torsion can occur, and the second which
discusses the relation between the integral (co)homology of a canonical
projection tiling and its $K$-theory.

\subsection{Bounds on where torsion can occur} We consider a general
canonical projection pattern of dimension 
$d$, codimension $n$, and assume the homology is finitely generated. By
\cite{FHKmem} homology will be non-zero 
only in dimensions $s=0,\ldots,d$.

\begin{theorem}\label{bounds} 
For a canonical projection pattern of dimension $d$, codimension $n$, 
and with finitely generated homology, there is no torsion in
homology dimensions $s\geqslant {{(n-1)d}\over 
n}$. Moreover, in homology dimensions ${{(n-2)d}\over n}\leqslant
s<{{(n-1)d}\over n}$ the torsion is given by 
the torsion part of 
$$\coker\left\{\bigoplus_{\alpha\in
    I_{n-1}}\Lambda_{n+s}\Gamma^\alpha\to
\Lambda_{n+s}\Gamma\right\}$$  where $\alpha$ indexes the orbit
classes of the $(n-1)$-dimensional singular hyperplanes in the
internal space $\R^n$.
\end{theorem}

\smallskip\noindent{\em Proof.} This is an extension of the
results in dimensions $s\geqslant1$ in Theorem~\ref{codim3} and may
be proved in a similar fashion by induction on the codimension $n$,
examining the top dimensions of the corresponding long exact
sequences.  

However, the most direct proof utilises the spectral sequence
of \cite{FHKmem}. Recall that the $r^{\rm th}$ column, $E^1_{r,*}$
for $0\leqslant r\leqslant n$ is
$H_*(\Gamma;C^r)=\bigoplus_{\Theta\in I_r}
H_*(\Gamma^\Theta;C_\Theta^r)$ and, as noted in (\ref{heights}),
these columns are only non-zero in respective degrees $0\leqslant
t\leqslant r{d\over n}$. The differentials act $d^j\colon
E^j_{r,s}\to  E^j_{r-j,s+j-1}$. 

First observe that for $s>{{(n-1)d}\over n}$, the only
non-zero entries in $E^1_{*,s}$ lie in the column $*=n$; thus these
cannot be the source or target of any differential and as noted
in Section~\ref{Prelim} we read off 
$$H_s(\Gamma;C^n)=E^1_{n,s}=E^\infty_{n,s}=H_{n+s}(\Gamma;\Z)
=\Z^{{{n+d}\choose{n+s}}}$$
which of course is torsion free.

For the range of degrees, ${{(n-2)d}\over n}<
s\leqslant{{(n-1)d}\over n}$, there are precisely two non-zero terms
in $E^1_{*,s}$ and these are for $*=n$ and $n-1$. Here we have the
possibility of only one non-trivial differential, namely $d^1$, and
just two non-zero groups $E^\infty_{a,b}$ on the line $a+b=s$. Thus
we obtain a long exact sequence
$$\cdots\to
H_{s+1}(\Gamma;C^{n-1})\buildrel\psi_{s+1}\over\longrightarrow
H_{n+s}(\Gamma;\Z)\to
H_s(\Gamma;C^{n})\buildrel d^1\over\longrightarrow
H_s(\Gamma;C^{n-1})\buildrel\psi_{s}\over\longrightarrow
\cdots$$
and the $\psi_s$ are interpreted in terms of the maps 
$$\bigoplus_{\alpha\in I_{n-1}}\Lambda_{n+s-1}\Gamma^\alpha\to 
\Lambda_{n+s-1}\Gamma\,,$$
in direct analogy to the maps $\beta_s$ of section
\ref{cd2} and the $\phi_s$ of \ref{cd3}. The result now follows as
the only possibility for torsion in $H_s(\Gamma;C^n)$ must arise
from coker$(\psi_{s+1})$. Note that for $s=(n-1)d/n$ this cokernel is
torsion free since the source of $\psi_{(n-1){d\over n}+1}$ is zero. The
result for $s={{(n-2)d}\over n}$ is similar but more involved as there
can also be a non-trivial differential
$d^1\colon E^1_{n-1,s}\to E^1_{n-2,s}$ for this $s$.\hfill$\square$


\smallskip
\begin{cor} {\em For a system as in Theorem~\ref{bounds}, 
$$H_s(\Gamma;C^n)=\Z^{{n+d}\choose d-s}\quad \text{for} \quad 
s>{{(n-1)d}\over n}.$$
}\hfill$\square$
\end{cor}


\smallskip
\subsection{$K$-theory} Our most straightforward results deal with
the case of tilings in dimensions $d\leqslant 3$.

\begin{theorem}\label{Kgen}
For canonical projection patterns in $\R^d$ with $d\leqslant 3$ and
finitely generated homology there are 
isomorphisms
$$K^0(MP)\cong \bigoplus_{r} H^{2r}(\Gamma;C^n)\qquad
K^1(MP)\cong \bigoplus_{r} H^{2r+1}(\Gamma;C^n)\,.
$$
\end{theorem}

\smallskip\noindent{\em Proof.} We have noted before that for
finitely generated homology, $n$ must divide $d$, so for $d=3$ we
must have $n=1$ or $3$, and for $d=2$ we must have $n=1$ or $2$. If
$n=1$ we have seen the computations are essentially those for the
homology of a punctured torus and it is immediate that the
$K$-theory and the cohomology will agree.

For the larger codimension cases, we consider the Atiyah-Hirzebruch
type spectral sequence linking $H^*(\Gamma;C^n)$ with the $K$-theory
$K^*(MP)$. This has an $E_2$-term 
$$A_2^{r,s}=\left\{\begin{array}{ll}
H^r(\Gamma;C^n)=H_{d-r}(\Gamma;C^n)& \mbox{for $s$ even,}\\
0& \mbox{for $s$ odd,}
\end{array}\right.$$
and differentials $d_j\colon A_j^{r,s}\to A_j^{r+j,s-j+1}$. Of
course $A_2^{r,s}=0$ if $r>d$.

Thus, if $d=2$ there can be no non-trivial differentials as there
is no appropriate pair of non-zero entries between which a
differential can act. For $d=n=3$ there is at first sight the
possibility of a non-trivial differential
$$d_3\colon \Z=A_2^{0,2}\to A_2^{3,0}=H_0(\Gamma;C^3),$$
but in fact this cannot be non-zero: by comparison with the
Atiyah-Hirzebruch type spectral sequence for a point we see that the
column $A_*^{0,*}$ can support no non-trivial differentials. Thus in
all cases $A_2^{*,*}=A_\infty^{*,*}$.

It remains to check that there are no extension problems. In the
case $d=n=3$ we have extensions
\begin{equation}\label{exts}
\begin{array}{ccccccccc}
0&\to&H^2(\Gamma;C^3)&\to&K^0(MP)&\to&H^0(\Gamma;C^3)&\to&0\\ \\
0&\to&H^3(\Gamma;C^3)&\to&K^1(MP)&\to&H^1(\Gamma;C^3)&\to&0
\end{array}
\end{equation}
The quotient groups in each exact sequence are
$H^0(\Gamma;C^3)=H_3(\Gamma;C^3)=\Z$ and
$H^1(\Gamma;C^3)=H_2(\Gamma;C^3)$, which by Section~\ref{Low} are
torsion free and, by assumption, finitely generated. Thus they are
free abelian and so the exact sequences~(\ref{exts}) split as
claimed. The case $d=n=2$ is similar, though simpler, as the
exact sequence for $K^1(MP)$ directly gives an isomorphism
$K^1(MP)=H^1(\Gamma;C^2)$.\hfill$\square$

\smallskip
This, together with the calculations of Section~\ref{sec-examples}, yields
the following, showing that there are examples of torsion in the
$K$-theory of an aperiodic tiling. The same result holds for the
$K$-theory of the non-commutative $C^*$-algebras associated to this
tiling, as considered in
\cite{FHKcmp,FHKmem}.

\begin{cor} {\em For the T\"ubingen Triangle Tiling,
$K^0(MP)=\Z^{25}\oplus\Z^2_5$ and $K^1(MP)=\Z^5$.}\hfill$\square$
\end{cor}


\smallskip
Theorem \ref{Kgen} admits a number of possible generalisations in
particular cases. A classical result tells that the differentials in
the Atiyah-Hirzebruch spectral sequence are always torsion valued,
and moreover the first differential that can non-trivially take a
$p$-torsion value is $d_{2p-1}$. Thus, for example, if
$H_*(\Gamma;C^n)$ is torsion free, this spectral sequence will
collapse and an assumption of finitely generated homology will mean
that all extension problems are trivial, yielding the following result,
originally claimed in \cite{FHKcmp,FHKmem}. 
\begin{theorem}
For canonical projection tilings with finitely generated, torsion free
homology there is an isomorphism 
$$K^*(MP)\cong\bigoplus H^*(\Gamma;C^n)\,.$$\hfill$\square$
\end{theorem}

More subtly, if the lowest primary torsion in $H_*(\Gamma;C^n)$ is
$p$ and $d\leqslant 2p-1$, then again we can deduce that the
spectral sequence collapses. Depending on which cohomological
degrees the torsion occurs in, there may or may not be extension
problems.


\bigskip\noindent
{\bf Acknowledgement.} The authors wish to thank the Banff International 
Research Station for their 
hospitality and support while some of this research was undertaken.





\begin{thebibliography}{10}


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\bibitem[BKSZ90]{BKSZ} M.~Baake, P.~Kramer, M.~Schlottmann, and
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\bibitem[BSJ91]{BSJ91} M.~Baake, M.~Schlottmann, P.D.~Jarvis.
\newblock Quasiperiodic tilings with tenfold symmetry and equivalence
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\newblock {\em J.\ Phys.\ A: Math.\ Gen.} 24 (1991) 4637--4654.

\bibitem[Ben82]{Beenker} F.P.M.~Beenker,
\newblock Eindhoven University of Technology Report 82-WSK-04 (1982).

\bibitem[Br]{Bro} K. Brown,
\newblock Cohomology of Groups,
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\bibitem[deB81]{deBruijn} N.G.~de Bruijn,
\newblock Algebraic theory of Penrose's aperiodic tilings of the plane,
\newblock {\em Math.\ Proc.} A84 (1981) 39-66.

\bibitem[Coh95]{Coh95} H.~Cohen,
\newblock A course in computational algebraic number theory, 2nd edition,
\newblock Springer, Berlin, Heidelberg 1995.

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\newblock Three dimensional analogues of the planar Penrose tilings and
quasicrystals,
\newblock {\em Discrete Math}. 76 (1989) 1-7.

\bibitem[EGN97]{EGN97} B.~Eick, F.~G\"ahler, W.~Nickel.
\newblock Computing Maximal Subgroups and Wyckoff Positions of Space Groups
\newblock {\em Acta Cryst.} A53 (1997) 467--474.

\bibitem[Cryst]{Cryst} B.~Eick, F.~G\"ahler, W.~Nickel.
\newblock Cryst -- Computing with crystallographic groups, 
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\bibitem[FHK01]{FHKicmp} A.H. Forrest, J.R. Hunton and J.~Kellendonk.
\newblock Topological invariants for projection method patterns.
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London 2000 Proceedings, pp333-339, Int. Press, Boston, MA, 2001,  Ed. A.
Fokas, A. Grigoryan, T. Kibble and B. Zegarlinski. ISBN 1-57146-085-3.

\bibitem[FHK02a]{FHKcmp} A.H. Forrest, J.R. Hunton and J.~Kellendonk.
\newblock Cohomology of canonical projection tilings.
\newblock {\em Comm. Math. Phys.} 226 (2002) 289-322.

\bibitem[FHK02b]{FHKmem} A.H. Forrest, J.R. Hunton and J.~Kellendonk.
\newblock Topological invariants for projection method patterns.
\newblock {\em Mem. Amer. Math. Soc.} 758 (2002).

\bibitem[G04]{FGBanff} F.~G\"ahler,
\newblock Lecture given at PIMS Workshop on {\em Aperiodic Order: 
Dynamical Systems, Combinatorics, and Operators}, Banff, June 2004.

\bibitem[G93]{G93} F.~G\"ahler.
\newblock Matching Rules for Quasicrystals: 
  The Composition-Decomposition Method
\newblock {\em J.\ Non-Cryst.\ Solids} 153 \& 154 (1993) 160--164.

\bibitem[GK00]{GK}  F. G\"ahler and J.~Kellendonk.
\newblock Cohomology groups for projection tilings of codimension 2.
\newblock {\em Materials Science and Engineering.} 438-440 (2000) 294-296.

\bibitem[GAP]{GAP} The GAP~Group 
\newblock GAP -- Groups, Algorithms, and Programming, Version 4.4 (2004)
{\tt (http://www.gap-system.org)}.

\bibitem[HiSt]{HS} P. J. Hilton and U. Stammbach. 
\newblock A course in homological algebra.
\newblock Springer GTM 4 (1971).

\bibitem[Ka05]{Kalugin} P. Kalugin.
\newblock Cohomology of quasiperiodic patterns and matching rules,
\newblock J. Phys. A: Math. Gen. 38 No 14 (2005) 3115-3132.

\bibitem[K97]{KeCMP} J.~Kellendonk.
\newblock The local structure of tilings and their integer group of
  coinvariants.
\newblock {\em Commun. Math. Phys.}, 187(1):115--157, 1997.

\bibitem[KP95]{KP95} P.~Kramer and Z.~Papadopolos, \emph{Models of icosahedral
quasicrystals from 6d  lattices}, 
Proceedings of the International conference on Aperiodic Crystals,
  Aperiodic '94 (G.~Chapuis et~al., ed.), World Scientific, Singapore, 1995,
  pp.~70--76.

\bibitem[KSB93]{KSB} R.~Klitzing, M.~Schlottmann, and M.~Baake,
\newblock  Perfect matching rules for undecorated triangular tilings
  with 10-, 12-, and 8-fold symmtery,
\newblock  {\em Int.\ J.\ Mod.\ Phys.\ B} 7 (1993) 1455--1473.

\bibitem[KN84]{AmmannKramer} P.~Kramer and R.~Neri, 
\newblock On periodic and non-periodic space fillings of $\mathcal{E}^{m}$
  obtained by projection,
\newblock {\em Acta Cryst A} 40 (1984) 580--587.

\bibitem[Mc01]{Mc} J. McCleary,
\newblock A user's guide to spectral sequences,
\newblock $2^{\rm nd}$ edition, Cambridge Studies in Advanced
Mathematics 58, Cambridge University Press 2001.

\bibitem[PK87]{PenGen} A.~Pavlovitch and M.~Kleman,
\newblock Generalised 2D Penrose tilings: structural properties,
\newblock {\em J.\ Phys.\ A: Math.\ Gen.} 20 (1987) 687--702.

\bibitem[Soc89]{Soc89} J.E.S.~Socolar.
\newblock Simple octagonal and dodecagonal quasicrystals
\newblock {\em Phys.\ Rev.\ B 39} (1989) 10519--10551.

\end{thebibliography}




\end{document}


math-ph0505048.extracted_bib
\begin{thebibliography}{10}


\bibitem[AGS92]{AGS} R.~Ammann, B.~Gr\"unbaum, and G.C. Shephard,
\newblock Aperiodic Tiles,
\newblock {\em Discrete Comput.\ Geom.} 8 (1992) 1--25.

\bibitem[AP98]{AP} J.E.~Anderson and I.F.~Putnam,
\newblock Topological invariants for substitution tilings and their
  associated $C^{*}$-algebras,
\newblock {\em Ergod.\ Th.\ \& Dynam.\ Syst.} 18 (1998) 509--537.

\bibitem[BKSZ90]{BKSZ} M.~Baake, P.~Kramer, M.~Schlottmann, and
D.~Zeidler,
\newblock Planar patterns with fivefold symmetry as sections of
  periodic structure in 4-space,
\newblock {\em Int.\ J.\ Mod.\ Phys.\ B} 4 (1990) 2217--2268.

\bibitem[BSJ91]{BSJ91} M.~Baake, M.~Schlottmann, P.D.~Jarvis.
\newblock Quasiperiodic tilings with tenfold symmetry and equivalence
  with respect to local derivability
\newblock {\em J.\ Phys.\ A: Math.\ Gen.} 24 (1991) 4637--4654.

\bibitem[Ben82]{Beenker} F.P.M.~Beenker,
\newblock Eindhoven University of Technology Report 82-WSK-04 (1982).

\bibitem[Br]{Bro} K. Brown,
\newblock Cohomology of Groups,
\newblock Springer 1982.

\bibitem[deB81]{deBruijn} N.G.~de Bruijn,
\newblock Algebraic theory of Penrose's aperiodic tilings of the plane,
\newblock {\em Math.\ Proc.} A84 (1981) 39-66.

\bibitem[Coh95]{Coh95} H.~Cohen,
\newblock A course in computational algebraic number theory, 2nd edition,
\newblock Springer, Berlin, Heidelberg 1995.

\bibitem[Dan89]{Dan89} L. Danzer,
\newblock Three dimensional analogues of the planar Penrose tilings and
quasicrystals,
\newblock {\em Discrete Math}. 76 (1989) 1-7.

\bibitem[EGN97]{EGN97} B.~Eick, F.~G\"ahler, W.~Nickel.
\newblock Computing Maximal Subgroups and Wyckoff Positions of Space Groups
\newblock {\em Acta Cryst.} A53 (1997) 467--474.

\bibitem[Cryst]{Cryst} B.~Eick, F.~G\"ahler, W.~Nickel.
\newblock Cryst -- Computing with crystallographic groups, 
  Version 4.1.4 (2004))
  {\tt (http://www.gap-system.org/Packages/cryst.html)}.

\bibitem[FH99]{FoHu} A.H. Forrest and J.R.~Hunton.
\newblock The cohomology and {K}-theory of commuting homeomorphisms of the
  {C}antor set.
\newblock {\em Ergod. Th. and Dynam. Sys.}, 19 (1999) 611-625.

\bibitem[FHK01]{FHKicmp} A.H. Forrest, J.R. Hunton and J.~Kellendonk.
\newblock Topological invariants for projection method patterns.
\newblock {\em XIIIth International Congress of Mathematical Physics},
London 2000 Proceedings, pp333-339, Int. Press, Boston, MA, 2001,  Ed. A.
Fokas, A. Grigoryan, T. Kibble and B. Zegarlinski. ISBN 1-57146-085-3.

\bibitem[FHK02a]{FHKcmp} A.H. Forrest, J.R. Hunton and J.~Kellendonk.
\newblock Cohomology of canonical projection tilings.
\newblock {\em Comm. Math. Phys.} 226 (2002) 289-322.

\bibitem[FHK02b]{FHKmem} A.H. Forrest, J.R. Hunton and J.~Kellendonk.
\newblock Topological invariants for projection method patterns.
\newblock {\em Mem. Amer. Math. Soc.} 758 (2002).

\bibitem[G04]{FGBanff} F.~G\"ahler,
\newblock Lecture given at PIMS Workshop on {\em Aperiodic Order: 
Dynamical Systems, Combinatorics, and Operators}, Banff, June 2004.

\bibitem[G93]{G93} F.~G\"ahler.
\newblock Matching Rules for Quasicrystals: 
  The Composition-Decomposition Method
\newblock {\em J.\ Non-Cryst.\ Solids} 153 \& 154 (1993) 160--164.

\bibitem[GK00]{GK}  F. G\"ahler and J.~Kellendonk.
\newblock Cohomology groups for projection tilings of codimension 2.
\newblock {\em Materials Science and Engineering.} 438-440 (2000) 294-296.

\bibitem[GAP]{GAP} The GAP~Group 
\newblock GAP -- Groups, Algorithms, and Programming, Version 4.4 (2004)
{\tt (http://www.gap-system.org)}.

\bibitem[HiSt]{HS} P. J. Hilton and U. Stammbach. 
\newblock A course in homological algebra.
\newblock Springer GTM 4 (1971).

\bibitem[Ka05]{Kalugin} P. Kalugin.
\newblock Cohomology of quasiperiodic patterns and matching rules,
\newblock J. Phys. A: Math. Gen. 38 No 14 (2005) 3115-3132.

\bibitem[K97]{KeCMP} J.~Kellendonk.
\newblock The local structure of tilings and their integer group of
  coinvariants.
\newblock {\em Commun. Math. Phys.}, 187(1):115--157, 1997.

\bibitem[KP95]{KP95} P.~Kramer and Z.~Papadopolos, \emph{Models of icosahedral
quasicrystals from 6d  lattices}, 
Proceedings of the International conference on Aperiodic Crystals,
  Aperiodic '94 (G.~Chapuis et~al., ed.), World Scientific, Singapore, 1995,
  pp.~70--76.

\bibitem[KSB93]{KSB} R.~Klitzing, M.~Schlottmann, and M.~Baake,
\newblock  Perfect matching rules for undecorated triangular tilings
  with 10-, 12-, and 8-fold symmtery,
\newblock  {\em Int.\ J.\ Mod.\ Phys.\ B} 7 (1993) 1455--1473.

\bibitem[KN84]{AmmannKramer} P.~Kramer and R.~Neri, 
\newblock On periodic and non-periodic space fillings of $\mathcal{E}^{m}$
  obtained by projection,
\newblock {\em Acta Cryst A} 40 (1984) 580--587.

\bibitem[Mc01]{Mc} J. McCleary,
\newblock A user's guide to spectral sequences,
\newblock $2^{\rm nd}$ edition, Cambridge Studies in Advanced
Mathematics 58, Cambridge University Press 2001.

\bibitem[PK87]{PenGen} A.~Pavlovitch and M.~Kleman,
\newblock Generalised 2D Penrose tilings: structural properties,
\newblock {\em J.\ Phys.\ A: Math.\ Gen.} 20 (1987) 687--702.

\bibitem[Soc89]{Soc89} J.E.S.~Socolar.
\newblock Simple octagonal and dodecagonal quasicrystals
\newblock {\em Phys.\ Rev.\ B 39} (1989) 10519--10551.

\end{thebibliography}