Topological manifold

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In mathematics, a topological manifold is a Hausdorff topological space which looks locally like Euclidean 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.

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[edit] Formal definition

A topological space X is locally Euclidean if every point in X has a neighborhood which is homeomorphic to an open subset of Euclidean space Rn. The integer n, called the dimension of X, must be the same for all points of X.

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 of dimension n.

[edit] Examples

See also: List of manifolds

[edit] Properties

The property of being locally Euclidean is preserved by local homeomorphisms. That is, if X is locally Euclidean of dimension n and f : X → Y 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.

A manifold need not be connected, but every manifold M is a disjoint union of connected manifolds (all of the same dimension). 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.

[edit] 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 not Hausdorff because the two origins cannot be separated.

[edit] 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 (see the Whitney embedding theorem). 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.

[edit] Dimensionality

The dimension of a manifold is a topological property, meaning that any manifold homeomorphic to an n-manifold also has dimension n. It follows from invariance of domain that an n-manifold cannot be homeomorphic to an m-manifold for n ≠ m.

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.

[edit] 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 on 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 a diffeomorphisms.

[edit] Classification of manifolds

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.

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

Every 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.

The 3-dimensional case may be solved. Thurston's geometrization conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman sketched a proof of this conjecture in 2003 which (as of 2006) appears to be essentially correct.

The classification of n-manifolds for n greater than three is known to be impossible; it is equivalent to the so-called word problem in group theory, which has been shown to be undecidable. In other words, 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.

[edit] 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):

\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.

Let M be a manifold with boundary. The interior of M, denoted Int M, is the set of points in M which have neighborhoods homeomorphic to an open subset of Rn. The boundary of M, denoted ∂M, is the complement of Int M in M. The boundary points can be characterized as those points which land on the boundary hyperplane (xn = 0) of Rn+ under some coordinate chart.

If M is a manifold with boundary of dimension n, then Int M is a manifold (without boundary) of dimension n and ∂M is a manifold (without boundary) of dimension n − 1.

[edit] See also

[edit] References

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