Poincaré space
In algebraic topology, a Poincaré space is an "n"-dimensional topological space with a distinguished element "µ" of its "n"th homology group such that taking the cap product with an element of the "k"th 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 formula_1 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.