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Introduction to Topological Manifolds, Springer-Verlag, Graduate Texts in Mathematics 2000, 2nd edition 2011 [5] Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771.
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.
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.
The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the manifold concept precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract ...
Especially manifolds are of interest. Topological manifolds of dimension are always triangulable [10] [11] [1] but there are non-triangulable manifolds for dimension , for arbitrary but greater than three. [12] [13] Further, differentiable manifolds always admit triangulations. [3]
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.. The inverse function theorem implies that a smooth map : is a local diffeomorphism if and only if the derivative: is a linear isomorphism for all points .
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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 ...