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 {{harvs|txt|author-link=Solomon Lefschetz|first=Solomon|last=Lefschetz|year=1926}}, at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem.Biographical Memoirs By National Research Council Staff (1992), p. 297. 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 $\partial(M)$ , and let $z\in H_n(M,\partial(M); \Z)$ 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 $(M,\partial(M))$ . Furthermore, this gives rise to isomorphisms of $H^k(M,\partial(M); \Z)$ with $H_{n-k}(M; \Z)$ , and of $H_k(M,\partial(M); \Z)$ with $H^{n-k}(M; \Z)$ for all $k$ .{{cite book|first=James W.|last= Vick|title=Homology Theory: An Introduction to Algebraic Topology|year=1994|page=171}}

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

There is a version for triples. Let $\partial(M)$ 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 $k$ , there is an isomorphism{{cite book|last=Hatcher|first= Allen|author-link=Allen Hatcher|url= https://pi.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic topology|publisher= Cambridge University Press|location=Cambridge| year=2002|isbn=0-521-79160-X|page=254}}

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

Notes

References

{{springer|id=Lefschetz_duality}}
{{Citation | last=Lefschetz | first=Solomon | author-link=Solomon Lefschetz|title=Transformations of Manifolds with a Boundary | jstor=84764 | publisher=National Academy of Sciences | year=1926 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=12 | issue=12 | pages=737–739 | doi=10.1073/pnas.12.12.737| pmc=1084792 | pmid=16587146| doi-access=free | bibcode=1926PNAS...12..737L }}

Category:Duality theories

Category:Manifolds