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 N decompose into subspaces A and B, themselves compact orientable manifolds with common boundary Z, which is the intersection of A and B. Then there is an isomorphism[3]

${\displaystyle D_{M}:H^{p}(M,A;\mathbb {Z} )\to H_{n-p}(M,B;\mathbb {Z} ).}$

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

• "Lefschetz_duality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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, doi:10.1073/pnas.12.12.737, ISSN 0027-8424, JSTOR 84764, PMC 1084792, PMID 16587146