Manifold
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 , then we call the manifold a -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 implication | 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 is a fiber bundle with base space and fiber space , and both and are manifolds, then is also a manifold. Note that this covers the particular cases of direct products and covering spaces.
References
- Topology (2nd edition) by James R. MunkresMore info, Page 225, Chapter 4, Section 36 (formal definition, as definition of -manifold, where is the dimension)