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  2. 3-manifold - Wikipedia

    en.wikipedia.org/wiki/3-manifold

    In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer ...

  3. Manifold - Wikipedia

    en.wikipedia.org/wiki/Manifold

    For two dimensional manifolds a key invariant property is the genus, or "number of handles" present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed, it is possible to fully characterize compact, two-dimensional manifolds on the basis of genus and orientability.

  4. 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.

  5. Classification of manifolds - Wikipedia

    en.wikipedia.org/wiki/Classification_of_manifolds

    There is a unique connected 0-dimensional manifold, namely the point, and disconnected 0-dimensional manifolds are just discrete sets, classified by cardinality. They have no geometry, and their study is combinatorics. A connected compact 1-dimensional manifold without boundary is homeomorphic (or diffeomorphic if it is smooth) to the circle.

  6. Spherical 3-manifold - Wikipedia

    en.wikipedia.org/wiki/Spherical_3-manifold

    A prism manifold is a closed 3-dimensional manifold M whose fundamental group is a central extension of a dihedral group. The fundamental group π 1 (M) of M is a product of a cyclic group of order m with a group having presentation , =, =

  7. Introduction to 3-Manifolds - Wikipedia

    en.wikipedia.org/wiki/Introduction_to_3-Manifolds

    Familiar examples of two-dimensional manifolds include the sphere, torus, and Klein bottle; this book concentrates on three-dimensional manifolds, and on two-dimensional surfaces within them. A particular focus is a Heegaard splitting, a two-dimensional surface that partitions a 3-manifold into two handlebodies. It aims to present the main ...

  8. List of manifolds - Wikipedia

    en.wikipedia.org/wiki/List_of_manifolds

    4.3 Infinite-dimensional manifolds. 5 See also. 6 References. ... For more examples see 3-manifold. 4-manifolds. Complex projective plane; Del Pezzo surface; E 8 ...

  9. Handlebody - Wikipedia

    en.wikipedia.org/wiki/Handlebody

    A manifold is called a "k-handlebody" if it is the union of r-handles, for r at most k. This is not the same as the dimension of the manifold. For instance, a 4-dimensional 2-handlebody is a union of 0-handles, 1-handles and 2-handles. Any manifold is an n-handlebody, that is, any manifold is the union of handles.

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