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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.
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.
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property .
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:
Spherical 3-manifolds; Euclidean 3-manifolds, Bieberbach Theorem, Flat manifolds, Crystallographic groups; Seifert fiber space; Heegaard splitting. Waldhausen conjecture; Compression body; Handlebody; Incompressible surface. Dehn's lemma; Loop theorem (aka the Disk theorem) Sphere theorem; Haken manifold; JSJ decomposition; Branched surface ...
Manifolds in contemporary mathematics come in a number of types. These include: smooth manifolds, which are basic in calculus in several variables, mathematical analysis and differential geometry; piecewise-linear manifolds; topological manifolds. There are also related classes, such as homology manifolds and orbifolds, that resemble manifolds.
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]