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 C = { C_i } be a chain complex, and assume that the homology groups of C are finitely generated. Assume that there exists a map Δ : C → C⊗C, called a chain-diagonal, with the property that (ε⊗1)Δ = (1⊗ε)Δ; where the map ε : C₀ → Z denotes the ring homomorphism known as the augmentation map. It is defined as follows: if n₁σ₁ + … + n_kσ_k ∈ C₀ then ε(n₁σ₁ + … + n_kσ_k) = n₁ + … + n_k ∈ Z.^[2]

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

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

where ${\displaystyle \scriptstyle \frown }$ denotes the cap product.^[3] A chain complex C is called geometric if a chain-homotopy exists between Δ and τΔ, where τ : C⊗C → C⊗C is given by τ(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 μ ∈ H_n(C), such that the maps given by

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

are group isomorphisms for all 0 ≤ k ≤ n. These isomorphisms are the isomorphisms of Poincaré duality.^[4][5]

Example

• The singular chain complex of an orientable, closed manifold is an example of a Poincaré complex,^[1]

See also

• Poincaré space

References

1. Yu. B. Rudyak. "Poincaré complex". Retrieved August 6, 2010.
2. Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, p. 110, ISBN 978-0-521-79540-1
3. Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, pp. 239–241, ISBN 978-0-521-79540-1
4. Wall, C. T. C. (1966). "Surgery of non-simply-connected manifolds". Ann. of Math. 84 (2): 217 − 276. doi:10.2307/1970519.
5. Wall, C. T. C. (1970). Surgery on compact manifolds. Academic Press.

• Wall, C. T. C. (1999) [1970], Ranicki, Andrew, ed., Surgery on compact manifolds (PDF), Mathematical Surveys and Monographs, 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388 – especially Chapter 2

External links

• Classifying Poincaré complexes via fundamental triples on the Manifold Atlas