Coarse Geometry

In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined.

The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.

Properly, a coarse structure is not the large-scale analog of a topological structure, but of a uniform structure.

Definition

A on a set $X$ is a collection $\mathbf$ of subsets of $X \times X$ (therefore falling under the more general categorization of binary relations on $X$) called , and so that $\mathbf$ possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:

# Identity/diagonal:
#: The diagonal $\Delta = \$ is a member of $\mathbf$—the identity relation.
# Closed under taking subsets:
#: If $E \in \mathbf$ and $F \subseteq E,$ then $F \in \mathbf.$
# Closed under taking inverses:
#: If $E \in \mathbf$ then the inverse (or transpose) $E^ = \$ is a member of $\mathbf$—the inverse relation.
# Closed under taking unions:
#: If $E, F \in \mathbf$ then their union $E \cup F$ is a member of$\mathbf.$
# Closed under composition:
#: If $E, F \in \mathbf$ then their product $E \circ F = \$ is a member of $\mathbf$—the composition of relations.

A set $X$ endowed with a coarse structure $\mathbf$ is a .

For a subset $K$ of $X,$ the set E /math> is defined as $\.$ We define the of $E$ by $x$ to be the set E also denoted $E_x.$ The symbol $E^y$ denotes the set E^ These are forms of projections.

A subset $B$ of $X$ is said to be a if $B \times B$ is a controlled set.

Intuition

The controlled sets are "small" sets, or " negligible sets": a set $A$ such that $A \times A$ is controlled is negligible, while a function $f : X \to X$ such that its graph is controlled is "close" to the identity. In the bounded coarse structure, these sets are the bounded sets, and the functions are the ones that are a finite distance from the identity in the uniform metric.

Coarse maps

Given a set $S$ and a coarse structure $X,$ we say that the maps $f : S \to X$ and $g : S \to X$ are if $\$ is a controlled set.

For coarse structures $X$ and $Y,$ we say that $f : X \to Y$ is a if for each bounded set $B$ of $Y$ the set $f^(Y)$ is bounded in $X$ and for each controlled set $E$ of $X$ the set $(f \times f)(E)$ is controlled in $Y.$ $X$ and $Y$ are said to be if there exists coarse maps $f : X \to Y$ and $g : Y \to X$ such that $f \circ g$ is close to $\operatorname_Y$ and $g \circ f$ is close to $\operatorname_X.$

Examples

* The on a metric space $(X, d)$ is the collection $\mathbf$ of all subsets $E$ of $X \times X$ such that $\sup_ d(x, y)$ is finite. With this structure, the integer lattice $\Z^n$ is coarsely equivalent to $n$-dimensional Euclidean space.
* A space $X$ where $X \times X$ is controlled is called a . Such a space is coarsely equivalent to a point. A metric space with the bounded coarse structure is bounded (as a coarse space) if and only if it is bounded (as a metric space).
* The trivial coarse structure only consists of the diagonal and its subsets. In this structure, a map is a coarse equivalence if and only if it is a bijection (of sets).
* The on a metric space $(X, d)$ is the collection of all subsets $E$ of $X \times X$ such that for all $\varepsilon > 0$ there is a compact set $K$ of $E$ such that $d(x, y) < \varepsilon$ for all $(x, y) \in E \setminus K \times K.$ Alternatively, the collection of all subsets $E$ of $X \times X$ such that $\$ is compact.
* The on a set $X$ consists of the diagonal $\Delta$ together with subsets $E$ of $X \times X$ which contain only a finite number of points $(x, y)$ off the diagonal.
* If $X$ is a topological space then the on $X$ consists of all subsets of $X \times X,$ meaning all subsets $E$ such that E /math> and E^ /math> are relatively compact whenever $K$ is relatively compact.

See also

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References

* John Roe, Lectures in Coarse Geometry, University Lecture Series Vol. 31, American Mathematical Society: Providence, Rhode Island, 2003. Corrections to ''Lectures in Coarse Geometry''
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{{Topology, expanded

General topology
Metric geometry
Topology