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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|authorlink=Solomon Lefschetz|last=Lefschetz|year=1926}}, at the same time introducing [[relative homology]], for application to the [[Lefschetz fixed-point theorem]].<ref>Biographical Memoirs By National Research Council Staff (1992), p. 297.</ref> There are now numerous formulations of Lefschetz duality or '''Poincaré-Lefschetz duality''', or '''Alexander-Lefschetz duality'''. | |||
==Formulations== | |||
Let ''M'' be an [[orientable]] [[compact space|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 group]]s of ''M'' and the relative (co)homology of the pair (''M'', ''N''); and this gives rise to isomorphisms of ''H''<sup>''k''</sup>(''M'', ''N'') with ''H''<sub>''n - k''</sub>(''M''), and of ''H''<sub>''k''</sub>(''M'', ''N'') with ''H''<sup>''n - k''</sup>(''M'').<ref>James W. Vick, ''Homology Theory: An Introduction to Algebraic Topology'' (1994), p. 171.</ref> | |||
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 | |||
:<math>D_M: H^p(M,A,Z)\to H_{n-p}(M,B,Z).</math> | |||
==Notes== | |||
<references/> | |||
==References== | |||
*{{springer|id=Lefschetz_duality}} | |||
*{{Citation | last1=Lefschetz | first1=Solomon | title=Transformations of Manifolds with a Boundary | jstor=84764 | publisher=National Academy of Sciences | year=1926 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=12 | issue=12 | pages=737–739}} | |||
[[Category:Duality theories]] | |||
[[Category:Manifolds]] | |||
Revision as of 12:23, 28 December 2013
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 Template:Harvs, 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 Hk(M, N) with Hn - k(M), and of Hk(M, N) with Hn - 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
Notes
References
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