Closed 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 RPⁿ is a closed n-dimensional manifold. The complex projective space CPⁿ is a closed 2n-dimensional manifold.^[1] 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.^[2]

If $M$ is a closed connected n-manifold, the n-th homology group $H_{n}(M;\mathbb {Z} )$ is $\mathbb {Z}$ or 0 depending on whether $M$ is orientable or not.^[3] Moreover, the torsion subgroup of the (n-1)-th homology group $H_{n-1}(M;\mathbb {Z} )$ is 0 or $\mathbb {Z} _{2}$ depending on whether $M$ is orientable or not. This follows from an application of the universal coefficient theorem.^[4]

Let $R$ be a commutative ring. For $R$ -orientable $M$ with fundamental class $[M]\in H_{n}(M;R)$ , the map $D:H^{k}(M;R)\to H_{n-k}(M;R)$ defined by $D(\alpha )=[M]\cap \alpha$ is an isomorphism for all k. This is the Poincaré duality.^[5] In particular, every closed manifold is $\mathbb {Z} _{2}$ -orientable. So there is always an isomorphism $H^{k}(M;\mathbb {Z} _{2})\cong H_{n-k}(M;\mathbb {Z} _{2})$ .

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 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.

References

^ See Hatcher 2002, p.231
^ See Hatcher 2002, p.536
^ See Hatcher 2002, p.236
^ See Hatcher 2002, p.238
^ See Hatcher 2002, p.250

Michael Spivak: A Comprehensive Introduction to Differential Geometry. Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, ISBN 0-914098-70-5.
Allen Hatcher, Algebraic Topology. Cambridge University Press, Cambridge, 2002.

[1] See Hatcher 2002, p.231

[2] See Hatcher 2002, p.536

[3] See Hatcher 2002, p.236

[4] See Hatcher 2002, p.238

[5] See Hatcher 2002, p.250

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