INTRODUCTION

3

the non-commutative topology of the C*-algebra of observables and

the topological invariants of T are the invariants of AT- In particu-

lar, we shall be interested in the K-theory of AT- Its i^o-group has

direct relevance to physics through the gap-labelling. In fact, Jean

Bellissard's K-theoretic formulation of the gap-labelling stands at the

beginning of this approach [Bl].

The second way of looking at the topological invariants of a dis-

crete point set T begins with the construction of the continuous hull

of T. There are various ways of defining a metric on a set of patterns

through comparison of their local configurations. Broadly speaking,

two patterns are deemed close if they coincide on a large window

around the origin 0 E R

d

up to a small discrepancy. It is the way

the allowed discrepancy is quantified which leads to different metric

topologies and we choose here one which has the strongest compact-

ness properties, though we have no intrinsic motivation for this. The

continuous hull of T is the closure, MT', of the set of translates of

T with respect to this metric. We use the notation MT because it

is essentially a mapping torus construction for a generalized discrete

dynamical system:

Rrf

acts on MT by translation and the set fir of

all elements of MT which (as point sets) contain 0 forms an abstract

transversal called the discrete hull. If d were 1 then fir would give

rise to a Poincare section, the intersection points of the flow line of

the action of R with fir defining an orbit of a Z action in fir, and

A4T would be the mapping torus of that discrete dynamical system

(fir,Z). For larger d one cannot expect to get a

Zd

action on fir

in a similar way but finds instead a generalized discrete dynamical

system which can be summarized in a groupoid QT whose unit space

is the discrete hull fir- Topological invariants for T are therefore the

topological invariants of MT and of QT and we shall be interested in

particular in their cohomologies. We define the cohomology of T to be

that of QT. Under a finite type condition, namely that for any given

r there are only a finite number of translational congruence classes

of subsets which fit inside a window of diameter r, the algebra AT

sketched above is isomorphic to the groupoid C*-algebra of QT. This

links the two approaches.

Having outlined the general philosophy we hasten to remark that

we will not explain all its aspects in the main text. In particular, we

have nothing to say there about the physical aspects of the theory and

the description of the algebra of observables, referring here the reader

to [BZH] [KePu], or to the more original literature [Bl] [B2] [Kl].

Instead, our aim in this memoir is to discuss and compare the different