Poincaré space (original) (raw)
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In algebraic topology, a Poincaré space 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.[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 [ M ] . {\displaystyle [M].} ![{\displaystyle [M].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43458fccbecd7e748851a81f669fa398ae1b3c54)
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.
Sometimes,[2] Poincaré space means a homology sphere with non-trivial fundamental group—for instance, the Poincaré dodecahedral space in 3 dimensions.
- ^ Rudyak, Yu.B. (2001) [1994], "Poincaré 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.