Closed manifold

 ( Differential Geometry ) 

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

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Examples The only Connected space 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.See Hatcher 2002, p.231 A Real 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.

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Properties Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups.See Hatcher 2002, p.536

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

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

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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.

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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.

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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.

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See also

• 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.

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