- Source: 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.
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.
If
M
{\displaystyle M}
is a closed connected n-manifold, the n-th homology group
H
n
(
M
;
Z
)
{\displaystyle H_{n}(M;\mathbb {Z} )}
is
Z
{\displaystyle \mathbb {Z} }
or 0 depending on whether
M
{\displaystyle M}
is orientable or not. Moreover, the torsion subgroup of the (n-1)-th homology group
H
n
−
1
(
M
;
Z
)
{\displaystyle H_{n-1}(M;\mathbb {Z} )}
is 0 or
Z
2
{\displaystyle \mathbb {Z} _{2}}
depending on whether
M
{\displaystyle M}
is orientable or not. This follows from an application of the universal coefficient theorem.
Let
R
{\displaystyle R}
be a commutative ring. For
R
{\displaystyle R}
-orientable
M
{\displaystyle M}
with
fundamental class
[
M
]
∈
H
n
(
M
;
R
)
{\displaystyle [M]\in H_{n}(M;R)}
, the map
D
:
H
k
(
M
;
R
)
→
H
n
−
k
(
M
;
R
)
{\displaystyle D:H^{k}(M;R)\to H_{n-k}(M;R)}
defined by
D
(
α
)
=
[
M
]
∩
α
{\displaystyle D(\alpha )=[M]\cap \alpha }
is an isomorphism for all k. This is the Poincaré duality. In particular, every closed manifold is
Z
2
{\displaystyle \mathbb {Z} _{2}}
-orientable. So there is always an isomorphism
H
k
(
M
;
Z
2
)
≅
H
n
−
k
(
M
;
Z
2
)
{\displaystyle H^{k}(M;\mathbb {Z} _{2})\cong H_{n-k}(M;\mathbb {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.
See also
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, Algebraic Topology. Cambridge University Press, Cambridge, 2002.
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