Atlas (topology)
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: ''For other uses of "atlas", see [[Atlas (disambiguation)]].''
In [[topology]], a branch of [[mathematics]], an '''atlas''' describes how a complicated [[topological space|space]] called a [[manifold]] is glued together from simpler pieces. Each piece is given by a '''chart''' (also known as '''coordinate chart''' or '''local coordinate system''').
More precisely, an '''atlas''' for a complicated space is constructed out of the following pieces of information:
* A list of spaces that are considered simple.
* For each point in the complicated space, a [[neighborhood (topology)|neighborhood]] of that point that is [[homeomorphism|homeomorphic]] to a simple space. The homeomorphism is called a '''chart'''.
* Different charts being '''compatible''' is required. At the minimum, it is required that the composite of one chart with the inverse of another be a homeomorphism (known as a '''change of coordinates''' or a [[transition function]]), yet usually stronger requirements, such as [[smooth function|smoothness]], are imposed.
This definition of atlas is exactly analogous to the non-mathematical meaning of [[atlas (cartography)|atlas]]. Each individual map in an atlas of the world gives a neighborhood of each point on the globe that is homeomorphic to the [[plane (mathematics)|plane]]. While each individual map does not exactly line up with other maps that it overlaps with (because of the Earth's curvature), the overlap of two maps can still be compared (by using latitude and longitude lines, for example).
Different choices for simple spaces and compatibility conditions give different objects. For example, if one chooses for simple spaces '''R<sup>n</sup>''', [[topological manifold]]s are obtained. If one also requires the coordinate changes to be [[diffeomorphism]]s, [[differentiable manifold]]s are obtained.
Two atlases (over the same underlying topological space) are '''compatible''' if the charts in the two atlases are all compatible (or equivalently if the union of the two atlases is an atlas). Formally, (as long as the concept of compatibility for charts satisfies certain simple properties), compatibility defines an [[equivalence relation]] on the set of all atlases. Usually, we consider compatible atlases as giving rise to the same manifold (we don't care how the manifold was "glued together", only what is left after "taking away the glue"), and so each of the equivalence classes corresponds to one manifold. In fact, the union of all atlases compatible with a given atlas is itself an atlas, called a '''complete (or maximal) atlas'''. Thus every atlas is contained in a unique complete atlas ([[Zorn's lemma]] is not needed as is sometimes assumed).
By definition, a smooth [[differential structure|differentiable structure]] (or [[differential structure]]) on a manifold ''M'' is such a maximal atlas of charts, all related by smooth coordinate changes on the overlaps.
[[Category:Differential topology]]
==External links==
*[http://mathworld.wolfram.com/Atlas.html Atlas] by Rowland, Todd
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