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. 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].}
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, 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
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