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 [math]\displaystyle{ M }[/math] is a closed connected n-manifold, the n-th homology group [math]\displaystyle{ H_{n}(M;\mathbb{Z}) }[/math] is [math]\displaystyle{ \mathbb{Z} }[/math] or 0 depending on whether [math]\displaystyle{ M }[/math] is orientable or not.[3] Moreover, the torsion subgroup of the (n-1)-th homology group [math]\displaystyle{ H_{n-1}(M;\mathbb{Z}) }[/math] is 0 or [math]\displaystyle{ \mathbb{Z}_2 }[/math] depending on whether [math]\displaystyle{ M }[/math] is orientable or not. This follows from an application of the universal coefficient theorem.[4]
Let [math]\displaystyle{ R }[/math] be a commutative ring. For [math]\displaystyle{ R }[/math]-orientable [math]\displaystyle{ M }[/math] with fundamental class [math]\displaystyle{ [M]\in H_{n}(M;R) }[/math], the map [math]\displaystyle{ D: H^k(M;R) \to H_{n-k}(M;R) }[/math] defined by [math]\displaystyle{ D(\alpha)=[M]\cap\alpha }[/math] is an isomorphism for all k. This is the Poincaré duality.[5] In particular, every closed manifold is [math]\displaystyle{ \mathbb{Z}_2 }[/math]-orientable. So there is always an isomorphism [math]\displaystyle{ 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 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
References
- 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.
Original source: https://en.wikipedia.org/wiki/Closed manifold.
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