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  2. Topological manifold - Wikipedia

    en.wikipedia.org/wiki/Topological_manifold

    A topological manifold with boundary is a Hausdorff space in which every point has a neighborhood homeomorphic to an open subset of Euclidean half-space (for a fixed n):

  3. Boundary (topology) - Wikipedia

    en.wikipedia.org/wiki/Boundary_(topology)

    A boundary point of a set is any element of that set's boundary. The boundary defined above is sometimes called the set's topological boundary to distinguish it from other similarly named notions such as the boundary of a manifold with boundary or the boundary of a manifold with corners, to name just a few examples.

  4. Manifold - Wikipedia

    en.wikipedia.org/wiki/Manifold

    A manifold with boundary is a manifold with an edge. For example, ... Lee, John M. (2000) Introduction to Topological Manifolds. Springer-Verlag.

  5. Surface (topology) - Wikipedia

    en.wikipedia.org/wiki/Surface_(topology)

    More generally, a (topological) surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H 2 in C. These homeomorphisms are also known as (coordinate) charts. The boundary of the upper half-plane is the x-axis.

  6. Classification of manifolds - Wikipedia

    en.wikipedia.org/wiki/Classification_of_manifolds

    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:

  7. Solid torus - Wikipedia

    en.wikipedia.org/wiki/Solid_torus

    The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to S 1 × S 1 {\displaystyle S^{1}\times S^{1}} , the ordinary torus . Since the disk D 2 {\displaystyle D^{2}} is contractible , the solid torus has the homotopy type of a circle, S 1 {\displaystyle S^{1}} . [ 3 ]

  8. Cobordism - Wikipedia

    en.wikipedia.org/wiki/Cobordism

    In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds.

  9. Lefschetz duality - Wikipedia

    en.wikipedia.org/wiki/Lefschetz_duality

    In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary.Such a formulation was introduced by Solomon Lefschetz (), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. [1]