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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]]).
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An algebraic Poincaré complex is a chain complex with a formal Poincaré duality — the analogue of the preceding.
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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
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A generalization of the concept of a [[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]] isomorphism). A Poincaré complex is a space where this isomorphism is taken as an axiom (see also [[Poincaré space]]).
  
<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>
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An algebraic Poincaré complex is a chain complex with a formal Poincaré duality — the analogue of the preceding.
 
 
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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Let  $  C = \{ C _ {i} \} $
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be a chain complex, with  $  C _ {i} = 0 $
 +
when  $  i < 0 $,
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whose homology groups are finitely generated. In addition, let  $  C $
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be provided with a (chain) diagonal  $  \Delta :  C \rightarrow C \otimes C $
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such that  $  ( \epsilon \otimes 1 ) \Delta = ( 1 \otimes \epsilon ) \Delta $,
 +
where  $  \epsilon : C \rightarrow \mathbf Z $
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is the augmentation (and $  C $
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is identified with  $  C \otimes \mathbf Z $
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and  $  \mathbf Z \otimes C $).
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The presence of the diagonal enables one to define pairings
  
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$$
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H  ^ {k} ( C) \otimes H _ {n} ( C)  \rightarrow  H _ {n-} k ( C) ,\ \
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x \otimes y  \rightarrow  x \cap y .
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$$
  
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The complex  $  C $
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is called geometric if a chain homotopy is given between  $  \Delta $
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and  $  T \Delta $,
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where  $  T :  C \otimes C \rightarrow C \otimes C $
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is transposition of factors,  $  T ( a \otimes b ) = b \otimes a $.
  
====Comments====
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A geometric chain complex is called an algebraic Poincaré complex of formal dimension  $  n $
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if there exists an element of infinite order  $  \mu \in H _ {n} ( C) $
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such that for any  $  k $
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the homomorphism  $  \cap \mu :  H  ^ {k} ( C) \rightarrow H _ {n-} k ( C) $
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is an isomorphism.
  
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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  $  ( X , A ) $.
 +
One also considers Poincaré complexes (and Poincaré chain pairs) of modules over appropriate rings.
  
 
====References====
 
====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>
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<table>
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<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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.T.C. Wall,  "Surgery on compact manifolds" , Acad. Press  (1970)</TD></TR>
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</table>

Latest revision as of 17:53, 10 April 2023


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

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

Let $ C = \{ C _ {i} \} $ be a chain complex, with $ C _ {i} = 0 $ when $ i < 0 $, whose homology groups are finitely generated. In addition, let $ C $ be provided with a (chain) diagonal $ \Delta : C \rightarrow C \otimes C $ such that $ ( \epsilon \otimes 1 ) \Delta = ( 1 \otimes \epsilon ) \Delta $, where $ \epsilon : C \rightarrow \mathbf Z $ is the augmentation (and $ C $ is identified with $ C \otimes \mathbf Z $ and $ \mathbf Z \otimes C $). The presence of the diagonal enables one to define pairings

$$ H ^ {k} ( C) \otimes H _ {n} ( C) \rightarrow H _ {n-} k ( C) ,\ \ x \otimes y \rightarrow x \cap y . $$

The complex $ C $ is called geometric if a chain homotopy is given between $ \Delta $ and $ T \Delta $, where $ T : C \otimes C \rightarrow C \otimes C $ is transposition of factors, $ T ( a \otimes b ) = b \otimes a $.

A geometric chain complex is called an algebraic Poincaré complex of formal dimension $ n $ if there exists an element of infinite order $ \mu \in H _ {n} ( C) $ such that for any $ k $ the homomorphism $ \cap \mu : H ^ {k} ( C) \rightarrow H _ {n-} k ( C) $ 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 $ ( X , A ) $. One also considers Poincaré complexes (and Poincaré chain pairs) of modules over appropriate rings.

References

[a1] C.T.C. Wall, "Surgery of non-simply-connected manifolds" Ann. of Math. (2) , 84 (1966) pp. 217–276
[a2] C.T.C. Wall, "Surgery on compact manifolds" , Acad. Press (1970)
How to Cite This Entry:
Poincaré complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_complex&oldid=22915
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article