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A manifold is a topological space which is locally homeomorphic to Euclidean space. Manifolds provide a setting for other structures important in many fields of mathematics, so they are often referred to as topological manifolds when being studied in their own right, rather than in the presence of other structures.
Charts and transition maps
A K-chart at p is a homeomorphism from an open neighbourhood of p to K. Instead of saying "there is a K-chart at p", you can say "at p there is a K-chart". In manifold theory, K is generally taken to be an subset of a Euclidean space, or just the space itself. Now suppose that there are K-charts at points p and q, which may or may not be distinct. The intersection of the p-neighborhood and the q-neighborhood is mapped to two subsets of K, one for each chart. Composing one chart with the inverse of the other produces a transition map between the two subsets of K. This map is also a homeomorphism; thus there is no concept of compatibility in the topological context; all atlases are compatible.
Topological manifolds
Topological manifold without boundary
The prototypical example of a topological manifold without boundary is Euclidean space. A general manifold without boundary looks locally, as a topological space, like Euclidean space. This is formalized by requiring that a manifold without boundary is a non-empty topological space in which every point has an open neighbourhood homeomorphic to (an open subset of) Rn (Euclidean n-space). Another way of saying this, using charts, is that a manifold without boundary is a non-empty topological space in which at every point there is an Rn-chart. If the dimension of the manifold is constant, then it is sometimes called a pure manifold.
Topological manifold with boundary
More generally it is possible to allow a topological manifold to have a boundary. The prototypical example of a topological manifold with boundary is the Euclidean closed half-space. Most points in Euclidean closed half-space, those not on the boundary, have a neighbourhood homeomorphic to Euclidean space in addition to having a neighbourhood homeomorphic to Euclidean closed half-space, but the points on the boundary only have neighbourhoods homeomorphic to Euclidean closed half-space and not to Euclidean space. Thus we need to allow for two kinds of points in our topological manifold with boundary: points in the interior and points in the boundary. Points in the interior will, as before, have neighbourhoods homeomorphic to Euclidean space, but may also have neighbourhoods homeomorphic to Euclidean closed half-space. Points in the boundary will have neighbourhoods homeomorphic to Euclidean closed half-space. Thus a topological manifold with boundary is a non-empty topological space in which at each point there is an Rn-chart or an [0,∞)×Rn−1-chart. The set of points at which there are only [0,∞)×Rn−1-charts is called the boundary and its complement is called the interior. The interior is always non-empty and is a topological n-manifold without boundary. If the boundary is non-empty then it is a topological (n−1)-manifold without boundary. If the boundary is empty, then we regain the definition of a topological manifold without boundary.
Sheaf of continuous functions
The continuous real-valued functions on a topological space form a sheaf. An alternative definition of a topological manifold is a topological space with a sheaf of continuous functions locally isomorphic to Euclidean space with its sheaf of continuous functions.Properties
A manifold with empty boundary is said to be closed if it is compact, and open if it is not compact.A 1-manifold is called a curve and a 2-manifold is called a surface. Examples of curves include circles, hyperbolas, and the trefoil knot. The torus, projective plane, and Klein bottle are examples of surfaces. All compact surfaces are homeomorphic to exactly one of the 2-sphere, a connected sum of tori, or a connected sum of projective planes: see classification theorems of surfaces.
Manifolds inherit many of the local properties of Euclidean space. In particular, they are locally path-connected, locally compact and locally metrizable. Being locally compact Hausdorff spaces they are necessarily Tychonoff spaces.
The line with two origins
Requiring a manifold to be Hausdorff may seem strange; it is tempting to think that being locally homeomorphic to Euclidean space (a Hausdorff space) implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the real line with two origins is not Hausdorff, because the two origins cannot be separated.
More formally, take two copies of R, write them as
- and ,
- if .
A generalization of the topological manifold allows one to omit the requirement that a manifold be Hausdorff. It still must be locally Euclidean, however. Such spaces are called non-Hausdorff manifolds and are used in the study of codimension-1 foliations.
Additional properties of manifolds
A topological space is said to be homogeneous if its homeomorphism group acts transitively on it. Every connected manifold without boundary is homogeneous, but manifolds with nonempty boundary are not homogeneous.
It can be shown that a manifold is metrizable if and only if it is paracompact. Non-paracompact manifolds (such as the long line) are generally regarded as pathological, so it's common to add paracompactness to the definition of an n-manifold. Sometimes n-manifolds are defined to be second-countable, which is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space. Note that every compact manifold is second-countable, and every second-countable manifold is paracompact. A manifold consisting of an uncountable number of disconnected circles is paracompact, but not second-countable, while a manifold consisting of a countably infinite number of circles will be second-countable but not compact.
Every Hausdorff, second countable manifold of dimension n admits an atlas consisting of at most n + 1 charts.
Subtypes
Piecewise linear manifold
The idea of a piecewise linear (PL) structure on a topological manifold M is used in geometric topology. Smooth manifolds have PL structures, but not conversely, in general. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. A more slick definition is to use a sheaf, locally isomorphic to the sheaf of piecewise linear functions on Euclidean space.
Differentiable manifold
A differentiable manifold is a topological manifold with a globally defined differentiable structure. Any topological manifold can be given a differentiable structure locally by using the homeomorphisms in its atlas, combined with the standard differerentiable structure on the Euclidean space. In other words, the homeomorphism can be used to give a local coordinate system. To induce a global differentiable structure, one can show that the natural compositions of the homeomorphisms on overlaps between charts in the atlas produce differentiable functions on Euclidean space. In other words, the coordinates defined by each homeomorphism are differentiable with respect to the coordinates defined by each other homeomorphism, provided the two charts have overlapping domains. This idea is often presented formally using transition maps.
"Topological manifold"