Compact Manifold

In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.

Examples

The only connected one-dimensional example is a circle.
The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP^''n'' is a closed ''n''-dimensional manifold. The complex projective space CP^''n'' is a closed 2''n''-dimensional manifold.
A line is not closed because it is not compact.
A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary.

Properties

Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups.

If $M$ is a closed connected n-manifold, the n-th homology group $H_(M;\mathbb)$ is $\mathbb$ or 0 depending on whether $M$ is orientable or not. Moreover, the torsion subgroup of the (n-1)-th homology group $H_(M;\mathbb)$ is 0 or $\mathbb_2$ depending on whether $M$ is orientable or not. This follows from an application of the universal coefficient theorem.

Let $R$ be a commutative ring. For $R$-orientable $M$ with
fundamental class in H_(M;R) , the map $D: H^k(M;R) \to H_(M;R)$ defined by D(\alpha)= cap\alpha is an isomorphism for all k. This is the Poincaré duality.See Hatcher 2002, p.250 In particular, every closed manifold is $\mathbb_2$-orientable. So there is always an isomorphism $H^k(M;\mathbb_2) \cong H_(M;\mathbb_2)$.

Open manifolds

For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.

Abuse of language

Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical conditions), thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesn’t cover some basic objects such as a closed disk, so authors sometimes define a manifold with boundary and abusively say ''manifold'' without reference to the boundary. But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used.

The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and it is a manifold, but not a closed manifold.

Use in physics

The notion of a " closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive Ricci curvature.

See also

*

References

* Michael Spivak: ''A Comprehensive Introduction to Differential Geometry.'' Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, .
* Allen Hatcher
''Algebraic Topology.''
Cambridge University Press, Cambridge, 2002.

{{Manifolds

Differential geometry
Manifolds
Geometric topology