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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).
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.
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:
3-manifolds are an interesting special case of low-dimensional topology because their topological invariants give a lot of information about their structure in general. If we let M {\displaystyle M} be a 3-manifold and π = π 1 ( M ) {\displaystyle \pi =\pi _{1}(M)} be its fundamental group, then a lot of information can be derived from them.
In all dimensions, the fundamental group of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in dimension 4 and above every finitely presented group is the fundamental group of a manifold (note that it is sufficient to show this for 4- and 5-dimensional manifolds, and then to take ...
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 ...
The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of Michael Freedman () implies that the homeomorphism type of the manifold only depends on this intersection form, and on a / invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kirby ...