1438 words - 6 pages

Coarse geometry

A metric space is a space equipped with a notion of distance. We use d(x, y)

to denote the distance from the point x to the point y. In topology, we study

metric spaces mainly by considering the continuous maps between them.

When deﬁning continuity of a map, we look only at very small distances.

1

We say that a map f : X → Y is uniformly continuous if for all > 0 there

exists some δ > 0 such that whenever d(x, y) < δ we have d(f (x), f (y)) < .

In contrast, coarse geometry considers the larger scale structure of a

space. A map f : X → Y is said to be coarse if

• for all R > 0 there exists some S > 0 such that whenever d(x, y) < R

we have d(f (x), f (y)) < S, ...view middle of the document...

When a manifold is equipped with such information, we can

deﬁne a sensible idea of the volume of a subspace S of that manifold. We

write this as vol(S). Note that the volume of a subspace will depend on the

metric that we have deﬁned.

When given notions of distance and angle, we can deﬁne a sensible idea of

curvature of our manifold. We will use an idea of curvature which compares

the volumes of balls in the manifold with the volume of balls of the same size

in a ’ﬂat’ space. Formally, if M is an n-dimensional Riemannian manifold

and p is some point in M , we deﬁne the ball Br (p, M ) to be the set of all

points in M which are within a distance r of the point p. We then deﬁne

the scalar curvature of M at the point p in terms of the ratio

vol(Br (p, M ))

vol(Br (0, Rn ))

which compares the volume of a ball in M with the volume of a ball of the

same radius in the Euclidean space Rn .

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If, for all possible points p, the ratio deﬁned above is greater than 1,

balls in the manifold M are of greater volume than those in Rn , and we say

that M has negative scalar curvature. If the ratio is less than 1 for all p, we

say that M has positive scalar curvature.

Note that, since the volume of balls in the manifold M depends on our

deﬁnition of distance and angle, the scalar curvature of M is also dependent

on our choice of Riemannian metric.

A fundamental question is to ﬁnd precisely when a manifold M can

be equipped with a metric of positive scalar curvature. For example, the

sphere admits a metric of positive scalar curvature (the usual notion of

distance), but it can be shown that the torus does not. There are a number

of conjectures regarding this question. Of particular interest is the GromovLawson conjecture, which states that a large class of manifolds of dimension

greater than four can not possibly admit a metric of positive scalar curvature

[4].

1.3

The Coarse Baum-Connes conjecture

A generalised homology theory associates with a topological space X a sequence Hn (X) of abelian groups in such a way that certain axioms are met.

This is useful since computations are usually easier when using these groups

than they were with the topological space. The groups retain a large amount

of the topological information about the space X.

Given a generalised homology theory H∗ , we can associate to it a coarse

coarse (see [8]) which satisﬁes coarse versions of the axioms

homology theory H∗

[13, 14].

An example of a generalised homology theory is K-homology, deﬁned by

Kasparov in [12]. A precise deﬁnition of the K-homology groups associated

with a topological space X, which we write Kn (X), can be found later in

coarse (X).

this report. By the above, we can deﬁne abelian groups Kn

Given a topological space X, one can also deﬁne the Roe C ∗ -algebra

C ∗ (X). This was introduced by Roe in [15]. Taking the analytic K-theory

groups K∗ (C ∗ (X)) gives another example of a coarse homology theory.

coarse (X) to K...

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