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.<ref>See Hatcher 2002, p.231</ref> 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.<ref>See Hatcher 2002, p.536</ref>

If <math>M</math> is a closed connected n-manifold, the n-th homology group <math>H_{n}(M;\mathbb{Z})</math> is <math>\mathbb{Z}</math> or 0 depending on whether <math>M</math> is orientable or not.<ref>See Hatcher 2002, p.236</ref> Moreover, the torsion subgroup of the (n-1)-th homology group <math>H_{n-1}(M;\mathbb{Z}) </math> is 0 or <math>\mathbb{Z}_2</math> depending on whether <math>M</math> is orientable or not. This follows from an application of the universal coefficient theorem.<ref>See Hatcher 2002, p.238</ref>

Let <math>R</math> be a commutative ring. For <math>R</math>-orientable <math>M</math> with fundamental class <math>[M]\in H_{n}(M;R) </math>, the map <math>D: H^k(M;R) \to H_{n-k}(M;R)</math> defined by <math>D(\alpha)=[M]\cap\alpha</math> is an isomorphism for all k. This is the Poincaré duality.<ref>See Hatcher 2002, p.250</ref> In particular, every closed manifold is <math>\mathbb{Z}_2</math>-orientable. So there is always an isomorphism <math>H^k(M;\mathbb{Z}_2) \cong H_{n-k}(M;\mathbb{Z}_2)</math>.

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

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References

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