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  2. Topological manifold - Wikipedia

    en.wikipedia.org/wiki/Topological_manifold

    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.

  3. Low-dimensional topology - Wikipedia

    en.wikipedia.org/wiki/Low-dimensional_topology

    A topological space X is a 3-manifold if every point in X has a neighbourhood that is homeomorphic to Euclidean 3-space. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.

  4. Topology - Wikipedia

    en.wikipedia.org/wiki/Topology

    A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...

  5. Introduction to 3-Manifolds - Wikipedia

    en.wikipedia.org/wiki/Introduction_to_3-Manifolds

    A manifold is a space whose topology, near any of its points, is the same as the topology near a point of a Euclidean space; however, its global structure may be non-Euclidean. Familiar examples of two-dimensional manifolds include the sphere , torus , and Klein bottle ; this book concentrates on three-dimensional manifolds, and on two ...

  6. Geometric topology - Wikipedia

    en.wikipedia.org/wiki/Geometric_topology

    Local flatness is a property of a submanifold in a topological manifold of larger dimension. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n).

  7. 3-manifold - Wikipedia

    en.wikipedia.org/wiki/3-manifold

    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.

  8. Solid Klein bottle - Wikipedia

    en.wikipedia.org/wiki/Solid_Klein_bottle

    In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. [ 1 ] It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder D 2 × I {\displaystyle \scriptstyle D^{2}\times I} to the bottom disk by a reflection across a diameter of the disk.

  9. Generalized Poincaré conjecture - Wikipedia

    en.wikipedia.org/wiki/Generalized_Poincaré...

    The generalized Poincaré conjecture is true topologically, but false smoothly in some dimensions. This results from the construction of the exotic spheres, manifolds that are homeomorphic, but not diffeomorphic, to the standard sphere, which can be interpreted as non-standard smooth structures on the standard (topological) sphere.