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The boundary of an -manifold with boundary is an ()-manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A square with interior is also a 2-manifold with boundary. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold.
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.
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]
The boundary of a manifold is a manifold , which has dimension . An orientation on M {\displaystyle M} induces an orientation on ∂ M {\displaystyle \partial M} . We usually denote a submanifold by Σ ⊂ M {\displaystyle \Sigma \subset M} .
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
Union along a subset of the boundaries. Note that the handles must generally be added in a specific order. Haken hierarchy: Any Haken manifold: Cut along a sequence of incompressible surfaces 3-balls: Disk decomposition Certain compact, orientable 3-manifolds: Suture the manifold, then cut along special surfaces (condition on boundary curves ...
In low-dimensional topology, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional manifold whose topology cannot be made simpler by a certain type of operation known as boundary compression. Suppose M is a 3-manifold with boundary.
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.