Poincaré complex


In mathematics, and especially topology, a Poincaré complex 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 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.

Example