Poincaré space

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

  1. ↑ Rudyak, Yu.B. (2001), "Poincaré space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
  2. ↑ Edward G. Begle (1942). "Locally Connected Spaces and Generalized Manifolds". Retrieved February 1, 2013.


This article is issued from Wikipedia - version of the 3/7/2013. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.