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 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 of that point that is 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 smoothness, are imposed.
This definition of atlas is exactly analogous to the non-mathematical meaning of 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. 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 Rn, topological manifolds are obtained. If one also requires the coordinate changes to be diffeomorphisms, differentiable manifolds 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 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.
