“Poincaré space”的意思、由来-开放百科全书

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Poincaré space

释义

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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.

References

1. ^{{springer | title=Poincaré space | id=p/p073110 | last=Rudyak | first=Yu.B.}}
2. ^{{Cite journal|title=Locally Connected Spaces and Generalized Manifolds|journal = American Journal of Mathematics|volume = 64|issue = 1|pages = 553–574|author= Edward G. Begle|year=1942|jstor = 2371704|doi = 10.2307/2371704}}

2 : Algebraic topology|Abstract algebra

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