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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.
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} .
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.
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 be a manifold with boundary, and be a submanifold of .
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.
A Seifert surface of a knot is however a manifold with boundary, the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary.
In order to describe a Dehn surgery, [1] one picks two oriented simple closed curves and on the corresponding boundary torus of the drilled 3-manifold, where is a meridian of (a curve staying in a small ball in and having linking number +1 with or, equivalently, a curve that bounds a disc that intersects once the component ) and is a longitude ...
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 ...