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 groupfor 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.


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