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A generalization of the concept of a [[Manifold|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|Poincaré duality]] isomorphism). A Poincaré complex is a space where this isomorphism is taken as an axiom (see also [[Poincaré space|Poincaré space]]).

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

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p0729901.png" /> be a chain complex, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p0729902.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p0729903.png" />, whose homology groups are finitely generated. In addition, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p0729904.png" /> be provided with a (chain) diagonal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p0729905.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p0729906.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p0729907.png" /> is the augmentation (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p0729908.png" /> is identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p0729909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299010.png" />). The presence of the diagonal enables one to define pairings

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299011.png" /></td> </tr></table>

The complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299012.png" /> is called geometric if a chain homotopy is given between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299015.png" /> is transposition of factors, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299016.png" />.

A geometric chain complex is called an algebraic Poincaré complex of formal dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299017.png" /> if there exists an element of infinite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299018.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299019.png" /> the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299020.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072990/p07299021.png" />. One also considers Poincaré complexes (and Poincaré chain pairs) of modules over appropriate rings.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.T.C. Wall,   "Surgery of non-simply-connected manifolds"  ''Ann. of Math. (2)'' , '''84'''  (1966)  pp. 217–276</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.T.C. Wall,   "Surgery on compact manifolds" , Acad. Press  (1970)</TD></TR></table>