Lefschetz duality

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 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

Let M be an orientable compact manifold of dimension n, with boundary N, and let z be the fundamental class of M. Then cap product with z induces a pairing of the (co)homology groups of M and the relative (co)homology of the pair (M, N); and this gives rise to isomorphisms of H^k(M, N) with H_{n - k}(M), and of H_k(M, N) with H^{n - k}(M).^[2]

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

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

Notes

1. ↑ Biographical Memoirs By National Research Council Staff (1992), p. 297.
2. ↑ James W. Vick, Homology Theory: An Introduction to Algebraic Topology (1994), p. 171.
3. ↑ Allen Hatcher, "Algebraic Topology" (2002), p. 254.

References

• Hazewinkel, Michiel, ed. (2001) [1994], "Lefschetz_duality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 
• Lefschetz, Solomon (1926), "Transformations of Manifolds with a Boundary", Proceedings of the National Academy of Sciences of the United States of America, National Academy of Sciences, 12 (12): 737–739, ISSN 0027-8424, JSTOR 84764, doi:10.1073/pnas.12.12.737