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 $$0 \le k \le n$$. 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 $$[M] \in H_{n}(M; \mathbb{Z})$$.