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
Poincaré spaces are used in surgery theory to analyze and classify manifolds.
Sometimes,[2] Poincaré space means a homology sphere with non-trivial fundamental group—for instance, the Poincaré dodecahedral space in 3 dimensions.