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A topological manifold is a locally Euclidean Hausdorff space. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact [3] or second-countable. [2] In the remainder of this article a manifold will mean a topological manifold.
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
The boundary is itself a 1-manifold without boundary, so the chart with transition map φ 3 must map to an open Euclidean subset. A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary.
A topological manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold". Thus given two categories, the two natural questions are:
A smooth structure on a manifold is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold is an atlas for such that each transition function is a smooth map, and two smooth atlases for are smoothly equivalent provided their union is again a smooth atlas for .
Manifold – A topological manifold is a locally Euclidean Hausdorff space (usually also required to be second-countable).For a given regularity (e.g. piecewise-linear, or differentiable, real or complex analytic, Lipschitz, Hölder, quasi-conformal...), a manifold of that regularity is a topological manifold whose charts transitions have the prescribed regularity.
Flag manifold; Grassmann manifold; Stiefel manifold; Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics.
The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for . This results in P E 8 {\displaystyle P_{E_{8}}} , a 4-manifold whose boundary is homeomorphic to the Poincaré homology sphere .