Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics.

A manifold can mean a topological manifold, or more frequently, a topological manifold together with some additional structure. Differentiable manifolds, for example, are topological manifolds equipped with a differential structure. Every manifold has an underlying topological manifold, obtained simply by forgetting the additional structure. An overview of the manifold concept is given in that article. This article focuses purely on the topological aspects of manifolds.

Formal definition

A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to the Euclidean space Eⁿ (or, equivalently, to the real n-space Rⁿ, or to some connected open subset of either of the two).^[1]

A topological manifold is a locally Euclidean Hausdorff space. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact or second-countable. The reasons, and some equivalent conditions, are discussed below.

In the remainder of this article a manifold will mean a topological manifold. An n-manifold will mean a topological manifold such that every point has a neighborhood homeomorphic to Rⁿ.

Examples

n-Manifolds

• The real coordinate space Rⁿ is the prototypical n-manifold.
• Any discrete space is a 0-dimensional manifold.
• A circle is a compact 1-manifold.
• A torus and a Klein bottle are compact 2-manifolds (or surfaces).
• The n-dimensional sphere Sⁿ is a compact n-manifold.
• The n-dimensional torus Tⁿ (the product of n circles) is a compact n-manifold.

Projective manifolds

• Projective spaces over the reals, complexes, or quaternions are compact manifolds.
  • Real projective space RPⁿ is a n-dimensional manifold.
  • Complex projective space CPⁿ is a 2n-dimensional manifold.
  • Quaternionic projective space HPⁿ is a 4n-dimensional manifold.
• Manifolds related to projective space include Grassmannians, flag manifolds, and Stiefel manifolds.

Other manifolds

• Lens spaces are a class of manifolds that are quotients of odd-dimensional spheres.
• Lie groups are manifolds endowed with a group structure.
• Any open subset of an n-manifold is a n-manifold with the subspace topology.

Constructions

• If M is an m-manifold and N is an n-manifold, the product M × N is a (m+n)-manifold.
• The disjoint union of a family of n-manifolds is a n-manifold (the pieces must all have the same dimension).
• The connected sum of two n-manifolds results in another n-manifold.

Properties

The property of being locally Euclidean is preserved by local homeomorphisms. That is, if X is locally Euclidean of dimension n and f : Y → X is a local homeomorphism, then Y is locally Euclidean of dimension n. In particular, being locally Euclidean is a topological property.

Manifolds inherit many of the local properties of Euclidean space. In particular, they are locally compact, locally connected, first countable, locally contractible, and locally metrizable. Being locally compact Hausdorff spaces, manifolds are necessarily Tychonoff spaces.

Adding the Hausdorff condition can make several properties become equivalent for a manifold. As an example, we can show that for a Hausdorff manifold, the notions of σ-compactness and second-countability are the same. Indeed, a Hausdorff manifold is a locally compact Hausdorff space, hence it is (completely) regular.^[2] Assume such a space X is σ-compact. Then it is Lindelöf, and because Lindelöf + regular implies paracompact, X is metrizable. But in a metrizable space, second-countability coincides with being Lindelöf, so X is second-countable. Conversely, if X is a Hausdorff second-countable manifold, it must be σ-compact.^[3]

A manifold need not be connected, but every manifold M is a disjoint union of connected manifolds. These are just the connected components of M, which are open sets since manifolds are locally-connected. Being locally path connected, a manifold is path-connected if and only if it is connected. It follows that the path-components are the same as the components.

The Hausdorff axiom

The Hausdorff property is not a local one; so even though Euclidean space is Hausdorff, a locally Euclidean space need not be. It is true, however, that every locally Euclidean space is T₁.

An example of a non-Hausdorff locally Euclidean space is the line with two origins. This space is created by replacing the origin of the real line with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This space is not Hausdorff because the two origins cannot be separated.

Compactness and countability axioms

A manifold is metrizable if and only if it is paracompact. Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold. In any case, non-paracompact manifolds are generally regarded as pathological. An example of a non-paracompact manifold is given by the long line. Paracompact manifolds have all the topological properties of metric spaces. In particular, they are perfectly normal Hausdorff spaces.

Manifolds are also commonly required to be second-countable. This is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space. For any manifold the properties of being second-countable, Lindelöf, and σ-compact are all equivalent.

Every second-countable manifold is paracompact, but not vice versa. However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a countable number of connected components. In particular, a connected manifold is paracompact if and only if it is second-countable. Every second-countable manifold is separable and paracompact. Moreover, if a manifold is separable and paracompact then it is also second-countable.

Every compact manifold is second-countable and paracompact.

Dimensionality

By invariance of domain, a non-empty n-manifold cannot be an m-manifold for n ≠ m. The dimension of a non-empty n-manifold is n. Being an n-manifold is a topological property, meaning that any topological space homeomorphic to an n-manifold is also an n-manifold.

A 1-dimensional manifold is often called a curve while a 2-dimensional manifold is called a surface. Higher-dimensional manifolds are usually just called n-manifolds. For n = 3, 4, or 5 see 3-manifold, 4-manifold, and 5-manifold.

Coordinate charts

By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of Rⁿ. Such neighborhoods are called Euclidean neighborhoods. It follows from invariance of domain that Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in Rⁿ. Indeed, a space M is locally Euclidean if and only if either of the following equivalent conditions holds:

• every point of M has a neighborhood homeomorphic to an open ball in Rⁿ.
• every point of M has a neighborhood homeomorphic to Rⁿ itself.

A Euclidean neighborhood homeomorphic to an open ball in Rⁿ is called a Euclidean ball. Euclidean balls form a basis for the topology of a locally Euclidean space.

For any Euclidean neighborhood U a homeomorphism φ : U → φ(U) ⊂ Rⁿ is called a coordinate chart on U (although the word chart is frequently used to refer to the domain or range of such a map). A space M is locally Euclidean if and only if it can be covered by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover M, together with their coordinate charts, is called an atlas on M. (The terminology comes from an analogy with cartography whereby a spherical globe can be described by an atlas of flat maps or charts).

Given two charts φ and ψ with overlapping domains U and V there is a transition function

ψφ⁻¹ : φ(U ∩ V) → ψ(U ∩ V).

Such a map is a homeomorphism between open subsets of Rⁿ. That is, coordinate charts agree on overlaps up to homeomorphism. Different types of manifolds can be defined by placing restrictions on types of transition maps allowed. For example, for differentiable manifolds the transition maps are required to be diffeomorphisms.

Classification of manifolds

Discrete Spaces (0-Manifold)

A 0-manifold is just a discrete space. Such spaces are classified by their cardinality. Every discrete space is paracompact. A discrete space is second-countable if and only if it is countable.

Curves (1-Manifold)

Every nonempty, paracompact, connected 1-manifold is homeomorphic either to R or the circle. The unconnected ones are just disjoint unions of these.

Surfaces (2-Manifold)

Every nonempty, compact, connected 2-manifold (or surface) is homeomorphic to the sphere, a connected sum of tori, or a connected sum of projective planes. See the classification theorem for surfaces for more details.

Volumes (3-Manifold)

A classification of 3-manifolds results from Thurston's geometrization conjecture, proven by Grigori Perelman.

General n-Manifold

The full classification of n-manifolds for n greater than three is known to be impossible; it is at least as hard as the word problem in group theory, which is known to be algorithmically undecidable. In fact, there is no algorithm for deciding whether a given manifold is simply connected. There is, however, a classification of simply connected manifolds of dimension ≥ 5.

Manifolds with boundary

A slightly more general concept is sometimes useful. A topological manifold with boundary is a Hausdorff space in which every point has a neighborhood homeomorphic to an open subset of Euclidean half-space (for a fixed n):

${\displaystyle \mathbb R^n_{+} = \{(x_1,\ldots,x_n) \in \mathbb R^n : x_n \ge 0\}.}$

The terminology is somewhat confusing: every topological manifold is a topological manifold with boundary, but not vice versa.

Footnotes

1. The topology of Eⁿ is identical to the standard topology of Rⁿ, so these two spaces are not distinguished in topology. Also, any non-empty open subset of Eⁿ contains an Euclidean open ball, which is homeomorphic to the entire space.
2. Topospaces subwiki, Locally compact Hausdorff implies completely regular
3. Stack Exchange, Hausdorff locally compact and second countable is sigma-compact

References

• Gauld, D. B. (1974). "Topological Properties of Manifolds". The American Mathematical Monthly. Mathematical Association of America. 81 (6): 633–636. doi:10.2307/2319220. JSTOR 2319220.
• Kirby, Robion C.; Siebenmann, Laurence C. (1977). Foundational Essays on Topological Manifolds. Smoothings, and Triangulations (PDF). Princeton: Princeton University Press. ISBN 0-691-08191-3.
• Lee, John M. (2000). Introduction to Topological Manifolds. Graduate Texts in Mathematics 202. New York: Springer. ISBN 0-387-98759-2.