Closed manifold

{{Short description|Topological concept in mathematics}} {{RefImprove|date=March 2023}}{{broader|Classification of manifolds#Point-set}}

In [[mathematics]], a '''closed manifold''' is a [[manifold]] [[Manifold with boundary|without boundary]] that is [[Compact space|compact]]. In comparison, an '''open manifold''' is a manifold without boundary that has only ''non-compact'' components.

== Examples ==

The only [[Connected space|connected]] one-dimensional example is a [[circle]]. The [[sphere]], [[torus]], and the [[Klein bottle]] are all closed two-dimensional manifolds. The [[real projective space]] '''RP'''<sup>''n''</sup> is a closed ''n''-dimensional manifold. The [[complex projective space]] '''CP'''<sup>''n''</sup> is a closed 2''n''-dimensional manifold.<ref>See Hatcher 2002, p.231</ref> A [[Real line|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 [[Orientability|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 "[[Shape of the universe|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 ==

* {{annotated link|Tame manifold}}

== 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]], [https://pi.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic Topology.''] Cambridge University Press, Cambridge, 2002.

[[Category:Differential geometry]] [[Category:Manifolds]] [[Category:Geometric topology]]