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, an isomorphism between its homology and cohomology groups. A chain complex is called a Poincaré complex if its homology 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 manifolds 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 $\mathrm{\Delta }:C\to C\otimes C$, called a chain-diagonal, with the property that $(\epsilon \otimes 1)\mathrm{\Delta }=(1\otimes \epsilon )\mathrm{\Delta }$. Here the map $\epsilon :C_0\to \mathbb{Z}$ 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 $\epsilon (n_1{\sigma }_1+\cdots +n_k{\sigma }_k)=n_1+\cdots +n_k\in \mathbb{Z}$.

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

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

where $⌢$ denotes the cap product.

A chain complex C is called geometric if a chain homotopy exists between $\mathrm{\Delta }$ and $\tau \mathrm{\Delta }$, where $\tau :C\otimes C\to C\otimes C$ 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

$(⌢\mu ):H^k(C)\to H_{n-k}(C)$

are group isomorphisms for all $0\le k\le n$. 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 $[M]\in H_n(M;\mathbb{Z})$.

References

• Template:Citation – especially Chapter 2

External links

• Classifying Poincaré complexes via fundamental triples on the Manifold Atlas