Difference between revisions of "Poincaré complex"
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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]]). | 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. | An algebraic Poincaré complex is a chain complex with a formal Poincaré duality — the analogue of the preceding. | ||
| − | Let | + | 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 | + | 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. | ||
====Comments==== | ====Comments==== | ||
| − | |||
====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> | <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> | ||
Revision as of 08:06, 6 June 2020
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.
Comments
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) |
Poincaré complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_complex&oldid=23464