In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold. The singular homology and cohomology groups of a closed, orientable manifold are related by Poincaré duality. Poincaré duality is an isomorphism between homology and cohomology groups. A chain complex is called a Poincaré complex if its homology groups and cohomology groups have the abstract properties of Poincaré duality. A Poincaré space is a topological space whose singular chain complex is a Poincaré complex. These are used in surgery theory to analyze manifold algebraically.

Definition

Let C = {C_i} be a chain complex of abelian groups, and assume that the homology groups of C are finitely generated. Assume that there exists a map, called a chain-diagonal, with the property that. Here the map denotes the ring homomorphism known as the augmentation map, which is defined as follows: if, then. Using the diagonal as defined above, we are able to form pairings, namely: where denotes the cap product. A chain complex C is called geometric if a chain-homotopy exists between \Delta and \tau\Delta, where is the transposition/flip given by. A geometric chain complex is called an algebraic Poincaré complex, of dimension n, if there exists an infinite-ordered element of the n-dimensional homology group, say, such that the maps given by are group isomorphisms for all. These isomorphisms are the isomorphisms of Poincaré duality.

Example