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.
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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 = { Ci } be a chain complex, and assume that the homology groups of C are finitely generated. Assume that there exists a map Δ : C → C⊗C, called a chain-diagonal, with the property that (ε⊗1)Δ = (1⊗ε)Δ; where the map ε : C0 → Z denotes the ring homomorphism known as the augmentation map. It is defined as follows: if n1σ1 + … + nkσk ∈ C0 then ε(n1σ1 + … + nkσk) = n1 + … + nk ∈ Z.
Using the diagonal as defined above, we are able to form pairings, namely:
where
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 μ ∈ Hn(C), such that the maps given by
are group isomorphisms for all 0 ≤ k ≤ n. These isomorphisms are the isomorphisms of Poincaré duality.