# 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.

How to Cite This Entry:
Poincaré complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_complex&oldid=53764
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article