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In [[algebraic topology]], a '''Poincaré space'''<ref>{{springer | |||
| title=Poincaré space | |||
| id=p/p073110 | |||
| last=Rudyak | |||
| first=Yu.B. | |||
}}</ref> is an ''n''-dimensional [[topological space]] with a distinguished element ''µ'' of its ''n''th [[homology group]] such that taking the [[cap product]] with an element of the ''k''th [[cohomology]] group yields an isomorphism to the (''n'' − ''k'')th homology group. The space is essentially one for which [[Poincaré duality]] is valid; more precisely, one whose singular chain complex forms a [[Poincaré complex]] with respect to the distinguished element ''µ''. | |||
For example, any closed, orientable, connected manifold ''M'' is a Poincaré space, where the distinguished element is the [[fundamental class]] <math>[M].</math> | |||
Poincaré spaces are used in [[surgery theory]] to analyze and classify manifolds. Not every Poincaré space is a manifold, but the difference can be studied, first by having a [[Normal invariant|normal map]] from a manifold, and then via [[obstruction theory]]. | |||
==Other uses== | |||
Sometimes,<ref name="EGB">{{Cite web|url=http://www.jstor.org/discover/10.2307/2371704?uid=3739656&uid=2134&uid=2&uid=70&uid=4&uid=3739256&sid=21101618075721|title=''Locally Connected Spaces and Generalized Manifolds''|author= Edward G. Begle|year=1942|accessdate=February 1, 2013}}</ref> ''Poincaré space'' means a [[homology sphere]] with non-trivial [[fundamental group]]—for instance, the Poincaré dodecahedral space in 3 dimensions. | |||
==See also== | |||
*[[Stable normal bundle]] | |||
==References== | |||
{{Reflist}} | |||
{{DEFAULTSORT:Poincare space}} | |||
[[Category:Algebraic topology]] | |||
[[Category:Abstract algebra]] | |||
{{Topology-stub}} | |||
Revision as of 21:10, 25 January 2014
In algebraic topology, a Poincaré space[1] is an n-dimensional topological space with a distinguished element µ of its nth homology group such that taking the cap product with an element of the kth cohomology group yields an isomorphism to the (n − k)th homology group. The space is essentially one for which Poincaré duality is valid; more precisely, one whose singular chain complex forms a Poincaré complex with respect to the distinguished element µ.
For example, any closed, orientable, connected manifold M is a Poincaré space, where the distinguished element is the fundamental class
Poincaré spaces are used in surgery theory to analyze and classify manifolds. Not every Poincaré space is a manifold, but the difference can be studied, first by having a normal map from a manifold, and then via obstruction theory.
Other uses
Sometimes,[2] Poincaré space means a homology sphere with non-trivial fundamental group—for instance, the Poincaré dodecahedral space in 3 dimensions.
See also
References
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