Poincaré complex explained

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 ⊗ C , called a chain-diagonal, with the property that (\varepsilon ⊗ 1 )\Delta = ( 1 ⊗ \varepsilon)\Delta . Here the map \varepsilon\colon C _{0\to} Z denotes the ring homomorphism known as the augmentation map, which is defined as follows: if n _{1\sigma} _{1} + … + n _{k\sigma} _{k\in} C _{0} , then \varepsilon(n _{1\sigma} _{1} + … + n _{k\sigma} _{k)} = n _{1+} … + n _{k\in} Z .

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

\rho \colon H ^{k(C) ⊗} H _{n(C)} \to H _{n-k} (C), where \rho(x ⊗ 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 ⊗ C\to C ⊗ C is the transposition/flip given by \tau (a ⊗ b) = b ⊗ 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.^{[1] [2]}

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; Z ) .

See also

• Poincaré space

References

• – especially Chapter 2

External links

• Classifying Poincaré complexes via fundamental triples on the Manifold Atlas

Notes and References

1. Wall. C. T. C.. C. T. C. Wall. 1966. Surgery of non-simply-connected manifolds. Annals of Mathematics. 84. 2. 217–276. 10.2307/1970519. 1970519.
2. Book: Wall, C. T. C.. 1970. Surgery on compact manifolds. Academic Press.