Course Geometry Essay

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.


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 .

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


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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