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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.
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:
In mathematics, a piecewise linear manifold (PL manifold) is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation.
This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics . For categorical listings see Category:Manifolds and its subcategories.
For a topological manifold M, the Kirby–Siebenmann class (; /) is an element of the fourth cohomology group of M that vanishes if M admits a piecewise linear structure. It is the only such obstruction, which can be phrased as the weak equivalence T O P / P L ∼ K ( Z / 2 , 3 ) {\displaystyle TOP/PL\sim K(\mathbb {Z} /2,3)} of TOP/PL with an ...
There are three main types of structures important on manifolds. The foundational geometric structures are piecewise linear, mostly studied in geometric topology, and smooth manifold structures on a given topological manifold, which are the concern of differential topology as far as classification goes. Building on a smooth structure, there are:
This category includes maps between manifolds, smooth or otherwise, particularly in geometric topology. Pages in category "Maps of manifolds" The following 14 pages are in this category, out of 14 total.