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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.
After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the unit circle, x 2 + y 2 = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1).
Brieskorn, Egbert V. (1966), "Examples of singular normal complex spaces which are topological manifolds", Proceedings of the National Academy of Sciences of the United States of America, 55 (6): 1395–1397, doi: 10.1073/pnas.55.6.1395, MR 0198497, PMC 224331, PMID 16578636
For more examples see 3-manifold. 4-manifolds ... Spin(7) manifold; Categories of manifolds ... Topological manifold;
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:
Example 1. If closed 2-manifolds M and N are homotopically equivalent then they are homeomorphic. Moreover, any homotopy equivalence of closed surfaces deforms to a homeomorphism. Example 2. If a closed manifold M n (n ≠ 3) is homotopy-equivalent to S n then M n is homeomorphic to S n.
The geometry and topology of three-manifolds is a set of widely circulated notes for a graduate course taught at Princeton University by William Thurston from 1978 to 1980 describing his work on 3-manifolds. They were written by Thurston, assisted by students William Floyd and Steven Kerchoff. [1]
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 ...