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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 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.
In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold-with-boundary which is not diffeomorphic to the standard 4-ball. Usually these manifolds are further required to have a handle decomposition with a single 1 {\displaystyle 1} -handle, and a single 2 {\displaystyle 2 ...
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 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.