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 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.

Let M be an orientable compact manifold of dimension n, with boundary ${\displaystyle \partial (M)}$, and let ${\displaystyle z\in H_{n}(M,\partial (M);\mathbb {Z} )}$ 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 ${\displaystyle (M,\partial (M))}$. Furthermore, this gives rise to isomorphisms of ${\displaystyle H^{k}(M,\partial (M);\mathbb {Z} )}$ with ${\displaystyle H_{n-k}(M;\mathbb {Z} )}$, and of ${\displaystyle H_{k}(M,\partial (M);\mathbb {Z} )}$ with ${\displaystyle H^{n-k}(M;\mathbb {Z} )}$ for all ${\displaystyle k}$.^[2]

Here ${\displaystyle \partial (M)}$ can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality.

There is a version for triples. Let ${\displaystyle \partial (M)}$ 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 ${\displaystyle k}$, there is an isomorphism^[3]

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

• "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, 12 (12), National Academy of Sciences: 737–739, doi:10.1073/pnas.12.12.737, ISSN 0027-8424, JSTOR 84764, PMC 1084792, PMID 16587146