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.^[1]

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 ${\displaystyle C=\{C_{i}\}}$ be a chain complex of abelian groups, and assume that the homology groups of ${\displaystyle C}$ are finitely generated. Assume that there exists a map ${\displaystyle \Delta \colon C\to C\otimes C}$, called a chain-diagonal, with the property that ${\displaystyle (\varepsilon \otimes 1)\Delta =(1\otimes \varepsilon )\Delta }$. Here the map ${\displaystyle \varepsilon \colon C_{0}\to \mathbb {Z} }$ denotes the ring homomorphism known as the augmentation map, which is defined as follows: if ${\displaystyle n_{1}\sigma _{1}+\cdots +n_{k}\sigma _{k}\in C_{0}}$, then ${\displaystyle \varepsilon (n_{1}\sigma _{1}+\cdots +n_{k}\sigma _{k})=n_{1}+\cdots +n_{k}\in \mathbb {Z} }$.^[2]

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

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

where ${\displaystyle \scriptstyle \frown }$ denotes the cap product.^[3]

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

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 ${\displaystyle \mu \in H_{n}(C)}$, such that the maps given by

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

are group isomorphisms for all ${\displaystyle 0\leq k\leq n}$. These isomorphisms are the isomorphisms of Poincaré duality.^[4][5]

Example

• The singular chain complex of an orientable, closed n-dimensional manifold ${\displaystyle M}$ is an example of a Poincaré complex, where the duality isomorphisms are given by capping with the fundamental class ${\displaystyle [M]\in H_{n}(M;\mathbb {Z} )}$.^[1]

See also

• Poincaré space