Closed manifold explained

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; Z ) is 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; Z ) is 0 or 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 withfundamental 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 Z _{2} -orientable. So there is always an isomorphism H ^{k(M; Z} _{2 )} \cong H _{n-k} (M; 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

• 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.

Notes and 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