Poincaré complex
# Poincaré complex

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, 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 groups. A chain complex is called a Poincaré complex if its homology groups and cohomology groups have the abstract properties of Poincaré duality.

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 $C=\{C_{i}\}$ be a chain complex of abelian groups, and assume that the homology groups of $C$ are finitely generated. Assume that there exists a map $\Delta \colon C\to C\otimes C$, called a chain-diagonal, with the property that $(\varepsilon \otimes 1)\Delta =(1\otimes \varepsilon )\Delta$. Here the map $\varepsilon \colon C_{0}\to \mathbb {Z}$ denotes the ring homomorphism known as the augmentation map, which is defined as follows: if $n_{1}\sigma _{1}+\cdots +n_{k}\sigma _{k}\in C_{0}$, then $\varepsilon (n_{1}\sigma _{1}+\cdots +n_{k}\sigma _{k})=n_{1}+\cdots +n_{k}\in \mathbb {Z}$.

Using the diagonal as defined above, we are able to form pairings, namely:

    $\rho \colon H^{k}(C)\otimes H_{n}(C)\to H_{n-k}(C),\ {\text{where}}\ \ \rho (x\otimes y)=x\frown y$,

where $\scriptstyle \frown$ denotes the cap product.

A chain complex C is called geometric if a chain-homotopy exists between $\Delta$ and $\tau \Delta$, where $\tau \colon C\otimes C\to C\otimes C$ is the transposition/flip given by $\tau (a\otimes b)=b\otimes a$.

A geometric chain complex is called an algebraic Poincaré complex, of dimension n, if there exists an infinite-ordered element of the n-dimensional homology group, say $\mu \in H_{n}(C)$, such that the maps given by

    $(\frown \mu )\colon H^{k}(C)\to H_{n-k}(C)$

are group isomorphisms for all $0\leq k\leq n$. These isomorphisms are the isomorphisms of Poincaré duality.

## Example

* The singular chain complex of an orientable, closed n-dimensional manifold $M$ is an example of a Poincaré complex, where the duality isomorphisms are given by capping with the fundamental class $[M]\in H_{n}(M;\mathbb {Z} )$.
