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A smooth 4-manifold is a 4-manifold with a smooth structure. 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 ...
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.
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.
An exotic R 4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R 4. The first examples were found in the early 1980s by Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds. [4]
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).
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:
The signature of the intersection form is an important invariant. A 4-manifold bounds a 5-manifold if and only if it has zero signature. Van der Blij's lemma implies that a spin 4-manifold has signature a multiple of eight. In fact, Rokhlin's theorem implies that a smooth compact spin 4-manifold has signature a multiple of 16.
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