Atlas (topology)

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, particularly topology

Topology

Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...

, an atlas describes how a manifold

Manifold

In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....

 is equipped with a differential structure

Differential structure

In mathematics, an n-dimensional differential structure on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold...

. Each piece is given by a chart (also known as coordinate chart or local coordinate system).

Before giving the formal definition of an atlas, we recall that a chart on a manifold M is defined to be a homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between two topological spaces that has a continuous inverse function...

  from an open subset

Open set

In mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...

 U of M to an open subset V of . If and are two charts on M such that is non-empty, then define the transition map
Note that since and are both homeomorphisms, the transition maps are also homeomorphisms. So, the transition maps are already endowed with a kind of compatibility in the sense that changing from the coordinate system on one chart to the coordinate system on another chart is continuous.

Then an atlas on a manifold M is a collection of charts on M whose domains cover M.

Now, we say that two overlapping charts and are smoothly compatible if the transition map between them is infinitely differentiable as a map from Euclidean space to itself.

Having defined these notions, a smooth atlas on M is an atlas where we make the additional requirement that, for any two overlapping charts on M, the transition maps between them are smoothly compatible.

Two atlases and on M are smoothly compatible if all charts in which overlap charts in are smoothly compatible.

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, particularly topology

Topology

Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...

, an atlas describes how a manifold

Manifold

In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....

 is equipped with a differential structure

Differential structure

In mathematics, an n-dimensional differential structure on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold...

. Each piece is given by a chart (also known as coordinate chart or local coordinate system).

Before giving the formal definition of an atlas, we recall that a chart on a manifold M is defined to be a homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between two topological spaces that has a continuous inverse function...

  from an open subset

Open set

In mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...

 U of M to an open subset V of . If and are two charts on M such that is non-empty, then define the transition map
Note that since and are both homeomorphisms, the transition maps are also homeomorphisms. So, the transition maps are already endowed with a kind of compatibility in the sense that changing from the coordinate system on one chart to the coordinate system on another chart is continuous.

Then an atlas on a manifold M is a collection of charts on M whose domains cover M.

Now, we say that two overlapping charts and are smoothly compatible if the transition map between them is infinitely differentiable as a map from Euclidean space to itself.

Having defined these notions, a smooth atlas on M is an atlas where we make the additional requirement that, for any two overlapping charts on M, the transition maps between them are smoothly compatible.

Two atlases and on M are smoothly compatible if all charts in which overlap charts in are smoothly compatible. If this is the case then is also a smooth atlas on M. This gives a natural equivalence relation

Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets...

, from which we can consider an equivalence class

Equivalence class

In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...

 of smoothly compatible atlases, which we call the maximal atlas. A manifold M together with a maximal atlas is said to have a smooth structure. There are, in higher dimensions, examples of topological manifolds with multiple different smooth structures. One of the first examples was John Milnor's discovery of an exotic sphere

Exotic sphere

In mathematics, an exotic sphere is a differentiable manifold that is homeomorphic to the standard Euclidean n-sphere, but not diffeomorphic. That means that such a manifold M is a sphere from a topological point of view, but not from the point of view of its differential structure...

, a 7-manifold which is homeomorphic to the 7-sphere but not diffeomorphic.

In general, doing computations with the maximal atlas of a manifold is unwieldy and we need only choose one particular smooth atlas to work with. Maximal atlases are needed for the unambiguous definition of smooth maps from one manifold to another.

The differentiability requirements on the transition functions can be weakened, so that we only require the transition maps to be k-times continuously differentiable; or strengthened, so that we require the transition maps to real-analytic. Accordingly, this gives a or analytic structure on the manifold rather than a smooth one. Similarly, we can define a complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cⁿ, such that the transition maps are holomorphic....

by requiring the transition maps to be holomorphic.