In algebraic topology, a Poincaré space 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.[1] 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 usesedit
Sometimes,[2]Poincaré space means a homology sphere with non-trivial fundamental group—for instance, the Poincaré dodecahedral space in 3 dimensions.
^Edward G. Begle (1942). "Locally Connected Spaces and Generalized Manifolds". American Journal of Mathematics. 64 (1): 553–574. doi:10.2307/2371704. JSTOR 2371704.
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poincaré, space, algebraic, topology, dimensional, topological, space, with, distinguished, element, homology, group, such, that, taking, product, with, element, cohomology, group, yields, isomorphism, homology, group, space, essentially, which, poincaré, dual. In algebraic topology a Poincare space 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 1 The space is essentially one for which Poincare duality is valid more precisely one whose singular chain complex forms a Poincare complex with respect to the distinguished element µ For example any closed orientable connected manifold M is a Poincare space where the distinguished element is the fundamental class M displaystyle M Poincare spaces are used in surgery theory to analyze and classify manifolds Not every Poincare 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 editSometimes 2 Poincare space means a homology sphere with non trivial fundamental group for instance the Poincare dodecahedral space in 3 dimensions See also editStable normal bundleReferences edit Rudyak Yu B 2001 1994 Poincare space Encyclopedia of Mathematics EMS Press Edward G Begle 1942 Locally Connected Spaces and Generalized Manifolds American Journal of Mathematics 64 1 553 574 doi 10 2307 2371704 JSTOR 2371704 nbsp This topology related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Poincare space amp oldid 896273742, wikipedia, wiki, book, books, library,
