Poincaré complex

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.

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