Atlas (topology)

Formal definition of atlas

An atlas for a topological space ${\displaystyle M}$ is an indexed family ${\displaystyle \{(U_{\alpha },{\phi }_{\alpha }):\alpha \in I\}}$ of charts on ${\displaystyle M}$ which covers ${\displaystyle M}$ (that is, $\bigcup _{\alpha \in I}{U}_{\alpha }=M$). If for some fixed n, the image of each chart is an open subset of n-dimensional Euclidean space, then ${\displaystyle M}$ is said to be an n-dimensional manifold.

The plural of atlas is atlases, although some authors use atlantes.

An atlas ${\displaystyle {\left(U_i,{\phi }_i\right)}_{i\in I}}$ on an ${\displaystyle n}$-dimensional manifold ${\displaystyle M}$ is called an adequate atlas if the following conditions hold:

• The image of each chart is either ${\displaystyle {\mathbb{R}}^n}$ or ${\displaystyle {\mathbb{R}}_{+}^n}$, where ${\displaystyle {\mathbb{R}}_{+}^n}$ is the closed half-space,
• ${\displaystyle {\left(U_i\right)}_{i\in I}}$ is a locally finite open cover of ${\displaystyle M}$, and
• $M=\bigcup _{i\in I}{\phi }_i^{-1}\left(B_1\right)$, where ${\displaystyle B_1}$ is the open ball of radius 1 centered at the origin.

Every second-countable manifold admits an adequate atlas. Moreover, if ${\displaystyle \mathcal{V}={\left(V_j\right)}_{j\in J}}$ is an open covering of the second-countable manifold ${\displaystyle M}$, then there is an adequate atlas ${\displaystyle {\left(U_i,{\phi }_i\right)}_{i\in I}}$ on ${\displaystyle M}$, such that ${\displaystyle {\left(U_i\right)}_{i\in I}}$ is a refinement of ${\displaystyle \mathcal{V}}$.

Transition maps

A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)

To be more precise, suppose that ${\displaystyle (U_{\alpha },{\phi }_{\alpha })}$ and ${\displaystyle (U_{\beta },{\phi }_{\beta })}$ are two charts for a manifold M such that ${\displaystyle U_{\alpha }\cap U_{\beta }}$ is non-empty. The transition map ${\displaystyle {\tau }_{\alpha ,\beta }:{\phi }_{\alpha }(U_{\alpha }\cap U_{\beta })\to {\phi }_{\beta }(U_{\alpha }\cap U_{\beta })}$ is the map defined by $${\displaystyle {\tau }_{\alpha ,\beta }={\phi }_{\beta }\circ {\phi }_{\alpha }^{-1}.}$$

Note that since ${\displaystyle {\phi }_{\alpha }}$ and ${\displaystyle {\phi }_{\beta }}$ are both homeomorphisms, the transition map ${\displaystyle {\tau }_{\alpha ,\beta }}$ is also a homeomorphism.

More structure

One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable. Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

If each transition function is a smooth map, then the atlas is called a smooth atlas, and the manifold itself is called smooth. Alternatively, one could require that the transition maps have only k continuous derivatives in which case the atlas is said to be ${\displaystyle C^k}$.

Very generally, if each transition function belongs to a pseudogroup ${\displaystyle \mathcal{G}}$ of homeomorphisms of Euclidean space, then the atlas is called a ${\displaystyle \mathcal{G}}$-atlas. If the transition maps between charts of an atlas preserve a local trivialization, then the atlas defines the structure of a fibre bundle.