Topological manifold
In
Formal definition
A
A topological manifold is a locally Euclidean
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 an 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 torusTn (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 flag manifolds, and Stiefel manifolds.
Other manifolds
- Differentiable manifolds are a class of topological manifolds equipped with a differential structure.
- Lens spaces are a class of differentiable manifolds that are quotients of odd-dimensional spheres.
- Lie groups are a class of differentiable manifolds equipped with a compatible group structure.
- The E8 manifold is a topological manifold which cannot be given a differentiable structure.
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
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
A manifold need not be connected, but every manifold M is a
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
Compactness and countability axioms
A manifold is
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
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.[6] 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.[7]
Coordinate charts
By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of . 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 . 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 ballin .
- every point of M has a neighborhood homeomorphic to itself.
A Euclidean neighborhood homeomorphic to an open ball in is called a Euclidean ball. Euclidean balls form a
For any Euclidean neighborhood U, a homeomorphism 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 and with overlapping domains U and V, there is a transition function
Such a map is a homeomorphism between open subsets of . 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
Classification of manifolds
Discrete spaces (0-Manifold)
A 0-manifold is just a discrete space. A discrete space is second-countable if and only if it is countable.[7]
Curves (1-Manifold)
Every nonempty, paracompact, connected 1-manifold is homeomorphic either to R or the circle.[7]
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.[8]
Volumes (3-Manifold)
A classification of 3-manifolds results from
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.[10]
In fact, there is no
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):
Every topological manifold is a topological manifold with boundary, but not vice versa.[7]
Constructions
There are several methods of creating manifolds from other manifolds.
Product manifolds
If M is an m-manifold and N is an n-manifold, the Cartesian product M×N is a (m+n)-manifold when given the product topology.[13]
Disjoint union
The disjoint union of a countable family of n-manifolds is a n-manifold (the pieces must all have the same dimension).[7]
Connected sum
The
Submanifold
Any open subset of an n-manifold is an n-manifold with the subspace topology.[13]
Footnotes
- ISBN 978-981-4324-35-9.
- ^ ISBN 978-0-387-22727-6.
- ISBN 978-0-8218-7214-7.
- ^ Topospaces subwiki, Locally compact Hausdorff implies completely regular
- ^ Stack Exchange, Hausdorff locally compact and second countable is sigma-compact
- ISBN 978-3-03719-048-7.
- ^ ISBN 978-1-4419-7940-7.
- ISBN 978-3-642-34364-3.
- ISBN 978-3-03719-082-1.
- ISBN 978-1-4757-2284-0.
- ^ Žubr A.V. (1988) Classification of simply-connected topological 6-manifolds. In: Viro O.Y., Vershik A.M. (eds) Topology and Geometry — Rohlin Seminar. Lecture Notes in Mathematics, vol 1346. Springer, Berlin, Heidelberg
- ^ Barden, D. "Simply Connected Five-Manifolds." Annals of Mathematics, vol. 82, no. 3, 1965, pp. 365–385. JSTOR, www.jstor.org/stable/1970702.
- ^ ISBN 978-0-8218-4815-9.
References
- Gauld, D. B. (1974). "Topological Properties of Manifolds". The American Mathematical Monthly. 81 (6). Mathematical Association of America: 633–636. JSTOR 2319220.
- 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.
External links
- Media related to Mathematical manifolds at Wikimedia Commons