Search results
Results from the WOW.Com Content Network
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. An n-manifold will mean a topological manifold such that every point has a neighborhood homeomorphic to R n.
By definition, all manifolds are topological manifolds, so the phrase "topological manifold" is usually used to emphasize that a manifold lacks additional structure, or that only its topological properties are being considered. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space.
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.
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:
Given a topological space, it is meaningful to ask whether or not it is a topological manifold. By contrast, it is not meaningful to ask whether or not a given topological space is (for instance) a smooth manifold, since the notion of a smooth manifold requires the specification of a smooth atlas, which is an additional structure.
In mathematics, differential topology is the field dealing with the topological properties and smooth properties [a] of smooth manifolds.In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape.
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 .
In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).