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 be a chain complex of abelian groups, and assume that the homology groups of are finitely generated. Assume that there exists a map , called a chain-diagonal, with the property that . Here the map denotes the ring homomorphism known as the augmentation map, which is defined as follows: if , then .[2]

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

,

where denotes the cap product.[3]

A chain complex C is called geometric if a chain-homotopy exists between and , where is the transposition/flip given by .

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 , such that the maps given by

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

Example

  • The singular chain complex of an orientable, closed n-dimensional manifold is an example of a Poincaré complex, where the duality isomorphisms are given by capping with the fundamental class .[1]

See also

References

  1. Rudyak, Yuli B. "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". Annals of Mathematics. 84 (2): 217–276. doi:10.2307/1970519.
  5. Wall, C. T. C. (1970). Surgery on compact manifolds. Academic Press.
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