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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, 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 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 differential topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold. To define this more precisely, first let M {\displaystyle M} be a manifold with boundary, and
The case of dimension 4 is somehow a boundary case, as it manifests "low dimensional" behaviour smoothly (but not topologically); see discussion of "low" versus "high" dimension. Different categories of manifolds yield different classifications; these are related by the notion of "structure", and more general categories have neater theories.
Manifold decomposition works in two directions: one can start with the smaller pieces and build up a manifold, or start with a large manifold and decompose it. The latter has proven a very useful way to study manifolds: without tools like decomposition, it is sometimes very hard to understand a manifold.
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.