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.

For broader coverage of this topic, see Classification of manifolds § Point-set.

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

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

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.

See also

• Tame manifold

References

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

• 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.