In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz (1926), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem.[1] There are now numerous formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz duality.

Formulations

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Let M be an orientable compact manifold of dimension n, with boundary  , and let   be the fundamental class of the manifold M. Then cap product with z (or its dual class in cohomology) induces a pairing of the (co)homology groups of M and the relative (co)homology of the pair  . Furthermore, this gives rise to isomorphisms of   with  , and of   with   for all  .[2]

Here   can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality.

There is a version for triples. Let   decompose into subspaces A and B, themselves compact orientable manifolds with common boundary Z, which is the intersection of A and B. Then, for each  , there is an isomorphism[3]

 

Notes

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  1. ^ Biographical Memoirs By National Research Council Staff (1992), p. 297.
  2. ^ Vick, James W. (1994). Homology Theory: An Introduction to Algebraic Topology. p. 171.
  3. ^ Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. p. 254. ISBN 0-521-79160-X.

References

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