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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 manifold|closed]], [[orientable manifold|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 group]]s. A chain complex is called a Poincaré complex if its [[homology group]]s and cohomology groups have the abstract properties of Poincaré duality.<ref name="YUB">{{Cite web|url=http://eom.springer.de/p/p072990.htm|title=''Poincaré complex''|author=Yu. B. Rudyak|accessdate=August 6, 2010}}</ref>

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 {{nowrap|1=''C'' = { ''Ci'' }}} be a [[chain complex]], and assume that the homology groups of ''C'' are [[Generating set of a group|finitely generated]]. Assume that there exists a map {{nowrap|1=Δ : ''C'' → ''C''⊗''C''}}, called a chain-diagonal, with the property that {{nowrap|1=(ε⊗1)Δ = (1⊗ε)Δ}}; where the map {{nowrap|1=ε : ''C''0 → '''Z'''}} denotes the [[ring homomorphism]] known as the [[augmentation ideal|augmentation map]]. It is defined as follows: if {{nowrap|1=''n''1σ1 + … + ''n''''k''σ''k'' ∈ ''C''0}} then {{nowrap|1=ε(''n''1σ1 + … + ''n''''k''σ''k'') = ''n''1 + … + ''n''''k'' ∈ '''Z'''.}}<ref>{{Citation|first=Allen|last=Hatcher|title=Algebraic Topology|publisher=Cambridge University Press|year=2001|ISBN=978-0-521-79540-1|page=110|url=http://www.math.cornell.edu/~hatcher/AT/ATchapters.html}}</ref>

Using the diagonal as defined above, we are able to form pairings, namely:
:<math>\rho : H^k(C)\otimes H_n(C) \to H_{n-k}(C), \ \text{where} \ \ \rho(x\otimes y) = x \frown y ,</math>
where <math>\scriptstyle \frown</math> denotes the [[cap product]].<ref>{{Citation|first=Allen|last=Hatcher|title=Algebraic Topology|publisher=Cambridge University Press|year=2001|ISBN=978-0-521-79540-1|pages=239–241|url=http://www.math.cornell.edu/~hatcher/AT/ATchapters.html}}</ref> A chain complex ''C'' is called '''geometric''' if a chain-[[homotopy]] exists between Δ and τΔ, where {{nowrap|1=τ : ''C''⊗''C'' → ''C''⊗''C''}} is given by {{nowrap|1=τ(''a''⊗''b'') = ''b''⊗''a''.}}

A geometric chain complex is called an algebraic '''Poincaré complex''', of dimension ''n'', if there exists an infinite-[[Order (group theory)|order]]ed element of the ''n''-dimensional homology group, say {{nowrap|1=μ ∈ H''n''(''C'')}}, such that the maps given by
:<math> (\frown\mu) : H^k(C) \to H_{n-k}(C) </math>
are group [[isomorphism]]s for all {{nowrap|1=0 ≤ ''k'' ≤ ''n''}}. These isomorphisms are the isomorphisms of Poincaré duality.<ref>{{Cite journal|last=Wall|first=C. T. C.|year=1966|title=Surgery of non-simply-connected manifolds|journal=Ann. of Math.|volume=84|issue=2|pages=217 − 276}}</ref><ref>{{Cite journal|last=Wall|first=C. T. C.|year=1970|title=Surgery on compact manifolds|journal=Acad. Press}}</ref>

==Example==
*The [[singular homology|singular]] [[chain complex]] of an orientable, closed manifold is an example of a Poincaré complex,<ref name="YUB"/>

==See also==
*[[Poincaré space]]

==References==
{{Reflist}}
{{refbegin}}
*{{Citation | last1=Wall | first1=C. T. C. | editor1-last=Ranicki | editor1-first=Andrew | title=Surgery on compact manifolds | origyear=1970 | url=http://www.maths.ed.ac.uk/~aar/books/scm.pdf | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=Mathematical Surveys and Monographs | isbn=978-0-8218-0942-6 | mr=1687388 | year=1999 | volume=69}} – especially Chapter 2
{{refend}}

{{DEFAULTSORT:Poincare Space}}
[[Category:Algebraic topology]]
[[Category:Homology theory]]
[[Category:Duality theories]]

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