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 SolomonLefschetz(1926), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. 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 ∂ ( M ) {\displaystyle \partial (M)}, and let z ∈ H n ( M , ∂ ( M ) ; Z ) {\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 ( M , ∂ ( M ) ) {\displaystyle (M,\partial (M))}. Furthermore, this gives rise to isomorphisms of H k ( M , ∂ ( M ) ; Z ) {\displaystyle H^{k}(M,\partial (M);\mathbb {Z} )} with H n − k ( M ; Z ) {\displaystyle H_{n-k}(M;\mathbb {Z} )}, and of H k ( M , ∂ ( M ) ; Z ) {\displaystyle H_{k}(M,\partial (M);\mathbb {Z} )} with H n − k ( M ; Z ) {\displaystyle H^{n-k}(M;\mathbb {Z} )} for all k {\displaystyle k}.

Here ∂ ( M ) {\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 ∂ ( M ) {\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 k {\displaystyle k}, there is an isomorphism

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

Notes