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.
[1] 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.
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
[2] Using the diagonal as defined above, we are able to form pairings, namely: where
denotes the cap product.
[3] A chain complex C is called geometric if a chain-homotopy exists between
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.