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 = \$ 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 $\varepsilon\colon C_0\to \mathbb$ 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 $\varepsilon(n_1\sigma_1 + \cdots + n_k\sigma_k) = n_1+ \cdots + n_k\in \mathbb$.

Using the diagonal as defined above, we are able to form pairings, namely:
:$\rho \colon H^k(C)\otimes H_n(C) \to H_(C), \ \text \ \ \rho(x\otimes y) = x \frown y$,
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
:$(\frown\mu) \colon H^k(C) \to H_(C)$
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 \in H_(M; \mathbb).

See also

*Poincaré space

References

* – especially Chapter 2

External links

Classifying Poincaré complexes via fundamental triples
on the Manifold Atlas

{{DEFAULTSORT:Poincare Space
Algebraic topology
Homology theory
Duality theories