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, and

• whenever a subset B of Y is bounded, its preimage f −1 (B) is bounded

in X.

In topology, we often consider spaces up to homeomorphism. Two spaces

X and Y are homeomorphic if there is a continuous map f : X → Y which

has a continuous inverse f −1 : Y → X. In coarse geometry, we consider

spaces up to coarse equivalence. The spaces X and Y are considered coarsely

equivalent if there are coarse maps f : X → Y and g : Y → X which are

inverses up to a uniformly bounded error. For example, the real line R

and the space of integers Z are coarsely equivalent. Note that R and Z

are very diﬀerent when considering their small scale structure, but could be

considered the same at a large scale.

1.2

Scalar curvature

An n-dimensional manifold is a space which locally resembles the Euclidean

space Rn . For example, a sphere in three-dimensional space locally resembles

the plane R2 . To illustrate this example, consider the surface of the Earth.

The Earth is (homeomorphic to) a sphere but, when looked at locally, seems

to be ﬂat, like the plane R2 .

A Riemannian metric on a manifold is a notion of both distance and

angle. A manifold equipped with a Riemannian metric is called a Riemannian manifold. When a manifold is equipped with such information, we can

deﬁne a...

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, and

• whenever a subset B of Y is bounded, its preimage f −1 (B) is bounded

in X.

In topology, we often consider spaces up to homeomorphism. Two spaces

X and Y are homeomorphic if there is a continuous map f : X → Y which

has a continuous inverse f −1 : Y → X. In coarse geometry, we consider

spaces up to coarse equivalence. The spaces X and Y are considered coarsely

equivalent if there are coarse maps f : X → Y and g : Y → X which are

inverses up to a uniformly bounded error. For example, the real line R

and the space of integers Z are coarsely equivalent. Note that R and Z

are very diﬀerent when considering their small scale structure, but could be

considered the same at a large scale.

1.2

Scalar curvature

An n-dimensional manifold is a space which locally resembles the Euclidean

space Rn . For example, a sphere in three-dimensional space locally resembles

the plane R2 . To illustrate this example, consider the surface of the Earth.

The Earth is (homeomorphic to) a sphere but, when looked at locally, seems

to be ﬂat, like the plane R2 .

A Riemannian metric on a manifold is a notion of both distance and

angle. A manifold equipped with a Riemannian metric is called a Riemannian manifold. When a manifold is equipped with such information, we can

deﬁne a...