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 = {Ci} be a chain complex, and assume that the homology groups of C are finitely generated. Assume that there exists a map Δ : CCC, called a chain-diagonal, with the property that (ε⊗1)Δ = (1⊗ε)Δ; where the map ε : C0Z denotes the ring homomorphism known as the augmentation map. It is defined as follows: if n1σ1 + + nkσkC0 then ε(n1σ1 + + nkσk) = n1 + + nkZ.[2]

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 \frown y ,

where \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 : CCCC is given by τ(ab) = ba.

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 ∈ Hn(C), such that the maps given by

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

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

Example

See also

References

  1. 1 2 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.

External links

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