Manifold: Difference between revisions

Latest revision as of 19:57, 22 June 2016

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

The article on this topic in the Differential Geometry Wiki can be found at: topological manifold

Definition

A topological space is said to be a manifold if it satisfies all of the following conditions:

It is Hausdorff: any two distinct points can be separated by disjoint open subsets.
It is second-countable: it has a countable basis.
It is locally Euclidean, viz., every point has a neighborhood that is homeomorphic to some open set in Euclidean space. If the topological space is connected, the dimension of the Euclidean space is the same for all points (we usually assume that the same Euclidean space is used for all points, viz., that the dimension is the same at all points)

If the dimension of the Euclidean space at each point is $m$ , then we call the manifold a $m$ -manifold.

Significance of three parts of the definition

Significance of local Euclideanness

Locally Euclidean is the most important property of manifolds, since this means that all the nice properties that we know about Euclidean spaces, are applicable locally. Thus, manifolds are locally contractible, locally path-connected, locally metrizable, and so on. Also, many properties of the manifold that are essentially local in nature can be proved using local Euclideanness.

Significance of Hausdorffness

If we do not assume Hausdorffness, we get pathologies like the line with two origins.

More pertinently, the important way in which we use Hausdorffness is as follows: in a Hausdorff space, any compact subset is closed. Thus, in particular, the images of closed discs of Euclidean space, inside the manifold, continue to remain closed in the whole manifold. This is crucial to applying the gluing lemma for closed subsets, for proofs like those of the fact that any connected manifold is homogeneous or that the inclusion of any point in a manifold is a cofibration.

Significance of second-countability

The assumption of second-countability can be dispensed with for a number of purposes, but is crucial for some applications. The standard example of something that is a manifold but for the second-countability assumption, is the long line.

Relation with other properties

Stronger properties

Property	Meaning	Proof of strictness (reverse implication failure)	Intermediate notions
connected manifold	manifold that is connected as a topological space; equivalently, it is path-connected		\|FULL LIST, MORE INFO
compact manifold	manifold that is compact as a topological space		\|FULL LIST, MORE INFO
compact connected manifold	manifold that is both compact and connected		Connected manifold\|FULL LIST, MORE INFO
differentiable manifold	manifold that occurs as the underlying topological manifold of a differential manifold	the E8 manifold is a 4-manifold that is not differentiable. For dimensions 1, 2, and 3, all manifolds are differentiable	Triangulable manifold\|FULL LIST, MORE INFO
triangulable manifold	manifold that admits a triangulation, i.e., is homeomorphic to the geometric realization of a simplicial complex		\|FULL LIST, MORE INFO
manifold admitting a PL structure	underlying topological manifold of a PL manifold		Triangulable manifold\|FULL LIST, MORE INFO
Lie group	manifold that occurs as the underlying topological manifold of a Lie group		\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
homology manifold	locally compact space whose homology groups with respect to the exclusion of any point look like those of a manifold			\|FULL LIST, MORE INFO
manifold with boundary	Hausdorff, second-countable, and every point is contained in an open subset that is homeomorphic to an open subset of Euclidean half-space			\|FULL LIST, MORE INFO
sub-Euclidean space	homeomorphic to a closed subset of Euclidean space		obvious counterexamples, such as a closed unit disk	\|FULL LIST, MORE INFO
metrizable space	underlying topological space of a metric space			\|FULL LIST, MORE INFO
paracompact Hausdorff space	paracompact and Hausdorff	(via metrizable)		Elastic space, Metrizable space\|FULL LIST, MORE INFO
normal space	any two disjoint closed subsets can be separated by disjoint open subsets			Elastic space, Metrizable space, Monotonically normal space, Paracompact Hausdorff space, Protometrizable space\|FULL LIST, MORE INFO
regular space	any point and closed subset not containing it can be separated by disjoint open subsets			Metrizable space, Monotonically normal space\|FULL LIST, MORE INFO
Hausdorff space	any two distinct points can be separated by disjoint open subsets			Metrizable space, Monotonically normal space, Submetrizable space\|FULL LIST, MORE INFO
locally Euclidean space	every point is contained in an open subset that is homeomorphic to an open subset of Euclidean space			\|FULL LIST, MORE INFO
locally contractible space	it has a basis of open subsets that are all contractible			Locally Euclidean space\|FULL LIST, MORE INFO
locally metrizable space	it has a basis of open subsets that are all metrizable			Locally Euclidean space\|FULL LIST, MORE INFO
locally compact space	every point is contained in an open subset whose closure is compact			Homology manifold\|FULL LIST, MORE INFO
nondegenerate space	the inclusion of any point in it is a cofibration	manifold implies nondegenerate		\|FULL LIST, MORE INFO
compactly nondegenerate space	every point is contained in an open subset whose closure is compact, and the inclusion of the point in the closure is a cofibration.			\|FULL LIST, MORE INFO

Potentially weaker properties

It is most likely true that every manifold is a CW-space, i.e., it can be given the structure of a CW-complex. However, the case of dimension four is still open. See for instance this Math Stack Exchange question.

Incomparable properties

A topological space that occurs as the geometric realization of a simplicial complex is termed a polyhedron. Not every manifold is a polyhedron. For instance, the E8 manifold in 4 dimensions is not a polyhedron. Conversely, not every polyhedron is a manifold. For instance, the geometric realization of any graph with a vertex of degree more than two is not a manifold.

A manifold that is homeomorphic to the geometric realization of a simplicial complex is termed a triangulable manifold.

Metaproperties

Products

This property of topological spaces is closed under taking finite products

A direct product of manifolds is again a manifold. Fill this in later

Covering spaces

This property of topological spaces is closed under passing to covering spaces; viz if a topological space has this property, so does any covering space of it

Any covering space of a manifold naturally gets the structure of a manifold.

Fiber bundles

This property of topological spaces is a fiber bundle-closed property of topological spaces: it is closed under taking fiber bundles, viz if the base space and fiber both satisfy the given property, so does the total space.
Manifold, Orientable manifold

If $E$ is a fiber bundle with base space $B$ and fiber space $F$ , and both $B$ and $F$ are manifolds, then $E$ is also a manifold. Note that this covers the particular cases of direct products and covering spaces.

References

Topology (2nd edition) by James R. Munkres^{More info}, Page 225, Chapter 4, Section 36 (formal definition, as definition of $m$ -manifold, where $m$ is the dimension)

@@ Line 1: / Line 1: @@
-{{topospace property}}
+{{pivotal topospace property}}
+{{diffgeom version at|topological manifold}}
 ==Definition==
-A [[topological space]] is said to be a '''manifold''' if it satisfies the following equivalent conditions:
+A [[topological space]] is said to be a '''manifold''' if it satisfies '''all of the following conditions''':
+* It is [[defining ingredient::Hausdorff space|Hausdorff]]: any two distinct points can be separated by disjoint open subsets.
+* It is [[defining ingredient::second-countable space|second-countable]]: it has a countable [[basis of a topology|basis]].
+* It is [[locally Euclidean space|locally Euclidean]], viz., every point has a neighborhood that is homeomorphic to some open set in Euclidean space. If the topological space is connected, the dimension of the Euclidean space is the same for all points (we usually assume that the same Euclidean space is used for all points, viz., that the dimension is the same at all points)
+If the dimension of the Euclidean space at each point is <math>m</math>, then we call the manifold a <math>m</math>-manifold.
+==Significance of three parts of the definition==
-* It is [[Hausdorff space|Hausdorff]]
+===Significance of local Euclideanness===
-* It is [[second-countable space|second-countable]]
-* It is [[locally Euclidean space|locally Euclidean]], viz every point has a neighbourhood that is homeomorphic to some open set in Euclidean space. If the topological space is connected, the dimension of the Euclidean space is the same for all points
+Locally Euclidean is the most important property of manifolds, since this means that all the nice properties that we know about Euclidean spaces, are applicable locally. Thus, manifolds are locally contractible, locally path-connected, locally metrizable, and so on. Also, many properties of the manifold that are essentially local in nature can be proved using local Euclideanness.
+===Significance of Hausdorffness===
+If we do not assume Hausdorffness, we get pathologies like the [[line with two origins]].
+More pertinently, the important way in which we use Hausdorffness is as follows: in a Hausdorff space, any compact subset is closed. Thus, in particular, the images of closed discs of Euclidean space, inside the manifold, continue to remain closed in the whole manifold. This is crucial to applying the [[gluing lemma for closed subsets]], for proofs like those of the fact that [[connected manifold implies homogeneous|any connected manifold is homogeneous]] or that [[manifold implies nondegenerate|the inclusion of any point in a manifold is a cofibration]].
+===Significance of second-countability===
+The assumption of second-countability can be dispensed with for a number of purposes, but is crucial for some applications. The standard example of something that is a manifold but for the second-countability assumption, is the [[long line]].
 ==Relation with other properties==
@@ Line 13: / Line 33: @@
 ===Stronger properties===
-* [[Complete manifold]]
+{| class="sortable" border="1"
-* [[Compact manifold]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Paracompact manifold]]
+|-
-* Underlying space of a [[Lie group]]
+| [[Weaker than::connected manifold]] || manifold that is [[connected space|connected]] as a topological space; equivalently, it is [[path-connected space|path-connected]] || || || {{intermediate notions short|manifold|connected manifold}}
+|-
+| [[Weaker than::compact manifold]] || manifold that is [[compact space|compact]] as a topological space || || || {{intermediate notions short|manifold|compact manifold}}
+|-
+| [[Weaker than::compact connected manifold]] || manifold that is both compact and connected || || || {{intermediate notions short|manifold|compact connected manifold}}
+|-
+| [[Weaker than::differentiable manifold]] || manifold that occurs as the underlying topological manifold of a [[differential manifold]] || || the [[E8 manifold]] is a 4-manifold that is not differentiable. For dimensions 1, 2, and 3, all manifolds are differentiable || {{intermediate notions short|manifold|differentiable manifold}}
+|-
+| [[Weaker than::triangulable manifold]] || manifold that admits a triangulation, i.e., is homeomorphic to the geometric realization of a simplicial complex || || || {{intermediate notions short|manifold|triangulable manifold}}
+|-
+| [[Weaker than::manifold admitting a PL structure]] || underlying topological manifold of a [[PL manifold]] || || || {{intermediate notions short|manifold|manifold admitting a PL structure}}
+|-
+| [[Weaker than::Lie group as a topological space|Lie group]] || manifold that occurs as the underlying topological manifold of a [[Lie group]] || || || {{intermediate notions short|manifold|Lie group as a topological space}}
+|}
 ===Weaker properties===
-* [[Manifold with boundary]]
+{| class="sortable" border="1"
-* [[Locally Euclidean space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Polyhedron]]
+|-
-* [[CW-space]]
+| [[Stronger than::homology manifold]] || [[locally compact space]] whose homology groups with respect to the exclusion of any point look like those of a manifold || || || {{intermediate notions short|homology manifold|manifold}}
-* [[Perfectly normal space]] (and all separation axioms sitting below it)
+|-
-* [[Locally contractible space]]
+| [[Stronger than::manifold with boundary]] || Hausdorff, second-countable, and every point is contained in an open subset that is homeomorphic to an open subset of Euclidean half-space || || || {{intermediate notions short|manifold with boundary|manifold}}
+|-
+| [[Stronger than::sub-Euclidean space]] || homeomorphic to a closed subset of Euclidean space || || obvious counterexamples, such as a closed unit disk || {{intermediate notions short|closed sub-Euclidean space|manifold}}
+|-
+| [[Stronger than::metrizable space]] || underlying topological space of a [[metric space]] || || || {{intermediate notions short|metrizable space|manifold}}
+|-
+|[[Stronger than::paracompact Hausdorff space]] || [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]] || (via metrizable) || || {{intermediate notions short|paracompact Hausdorff space|manifold}}
+|-
+| [[Stronger than::normal space]] || any two disjoint closed subsets can be separated by disjoint open subsets || || || {{intermediate notions short|normal space|manifold}}
+|-
+| [[Stronger than::regular space]] || any point and closed subset not containing it can be separated by disjoint open subsets || || || {{intermediate notions short|regular space|manifold}}
+|-
+| [[Stronger than::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets || || || {{intermediate notions short|Hausdorff space|manifold}}
+|-
+| [[Stronger than::locally Euclidean space]] || every point is contained in an open subset that is homeomorphic to an open subset of Euclidean space || || || {{intermediate notions short|locally Euclidean space|manifold}}
+|-
+| [[Stronger than::locally contractible space]] || it has a basis of open subsets that are all contractible || || || {{intermediate notions short|locally contractible space|manifold}}
+|-
+| [[Stronger than::locally metrizable space]] || it has a basis of open subsets that are all metrizable || || || {{intermediate notions short|locally metrizable space|manifold}}
+|-
+| [[Stronger than::locally compact space]] || every point is contained in an open subset whose closure is compact || || || {{intermediate notions short|locally compact space|manifold}}
+|-
+| [[Stronger than::nondegenerate space]] || the inclusion of any point in it is a [[cofibration]] || [[manifold implies nondegenerate]] || || {{intermediate notions short|nondegenerate space|manifold}}
+|-
+|[[Stronger than::compactly nondegenerate space]] || every point is contained in an open subset whose closure is compact, and the inclusion of the point in the closure is a cofibration. || || || {{intermediate notions short|compactly nondegenerate space|manifold}}
+|}
+=== Potentially weaker properties ===
+It is most likely true that every manifold is a [[CW-space]], i.e., it can be given the structure of a [[CW-complex]]. However, the case of dimension four is still open. See for instance [http://math.stackexchange.com/questions/593041/cw-complexes-and-manifolds this Math Stack Exchange question].
+=== Incomparable properties ===
+A topological space that occurs as the geometric realization of a simplicial complex is termed a [[polyhedron]]. Not every manifold is a polyhedron. For instance, the [[E8 manifold]] in 4 dimensions is not a polyhedron. Conversely, not every polyhedron is a manifold. For instance, the geometric realization of any graph with a vertex of degree more than two is not a manifold.
+A manifold that is homeomorphic to the geometric realization of a simplicial complex is termed a [[triangulable manifold]].
 ==Metaproperties==
@@ Line 32: / Line 100: @@
 A direct product of manifolds is again a manifold. {{fillin}}
+{{covering space-closed}}
+Any [[covering space]] of a manifold naturally gets the structure of a manifold.
+{{fiber-bundle-closed}}
+If <math>E</math> is a [[fiber bundle]] with base space <math>B</math> and fiber space <math>F</math>, and both <math>B</math> and <math>F</math> are manifolds, then <math>E</math> is also a manifold. Note that this covers the particular cases of direct products and covering spaces.
+==References==
+* {{booklink-defined|Munkres}}, Page 225, Chapter 4, Section 36 (formal definition, as definition of <math>m</math>-manifold, where <math>m</math> is the dimension)