# Poincaré complex

A generalization of the concept of a manifold; a space with homology groups having, in a certain sense, the same structure as the homology groups of a closed orientable manifold. H. Poincaré showed that the homology groups of a manifold satisfy a certain relation (the Poincaré duality isomorphism). A Poincaré complex is a space where this isomorphism is taken as an axiom (see also Poincaré space).

An algebraic Poincaré complex is a chain complex with a formal Poincaré duality — the analogue of the preceding.

Let be a chain complex, with when , whose homology groups are finitely generated. In addition, let be provided with a (chain) diagonal such that , where is the augmentation (and is identified with and ). The presence of the diagonal enables one to define pairings

The complex is called geometric if a chain homotopy is given between and , where is transposition of factors, .

A geometric chain complex is called an algebraic Poincaré complex of formal dimension if there exists an element of infinite order such that for any the homomorphism is an isomorphism.

Examples of algebraic Poincaré complexes are: the singular chain complex of an orientable closed manifold or, more generally, a Poincaré complex with suitable finiteness conditions. One can also define Poincaré chain pairs — algebraic analogues of the Poincaré pairs . One also considers Poincaré complexes (and Poincaré chain pairs) of modules over appropriate rings.