Poincaré space

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 [M].

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

* Stable normal bundle

## References

1. ↑ Hazewinkel, Michiel, ed. (2001), "Poincaré space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p073110
2. ↑ Edward G. Begle (1942). "Locally Connected Spaces and Generalized Manifolds". American Journal of Mathematics 64 (1): 553–574. doi:10.2307/2371704.

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