Poincaré complex

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. Let be a chain complex of abelian groups, and assume that the homology groups of 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 and , 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. The singular chain complex of an orientable, closed n-dimensional manifold is an example of a Poincaré complex, where the duality isomorphisms are given by capping with the fundamental class .