Atlas (topology)

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For other uses of "atlas", see Atlas (disambiguation).

In topology, an atlas describes how a complicated space 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.
  • We require the different charts to be compatible. At the minimum, we require that the composite of one chart with the inverse of another be a homeomorphism (known as a change of coordinates or a transition function), but we usually impose stronger requirements, such as smoothness.

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 we choose for our simple spaces Rn, we get topological manifolds. If we also require the coordinate changes to be diffeomorphisms, we get differentiable manifolds.

We call two atlases compatible if the charts in the two atlases are all compatible (or equivalently if the union of the two atlases is an atlas). Usually, we want to consider two compatible atlases as giving rise to the same space. Formally, (as long as our concept of compatibility for charts has certain simple properties), we can define an equivalence relation on the set of all atlases, calling two the same if they are compatible. 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 (N.B. we don't need Zorn's lemma 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.

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