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.
Formulations
Let M be an orientable compact manifold of dimension n, with boundary [math]\displaystyle{ \partial(M) }[/math], and let [math]\displaystyle{ z\in H_n(M,\partial(M); \Z) }[/math] 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 [math]\displaystyle{ (M,\partial(M)) }[/math]. Furthermore, this gives rise to isomorphisms of [math]\displaystyle{ H^k(M,\partial(M); \Z) }[/math] with [math]\displaystyle{ H_{n-k}(M; \Z) }[/math], and of [math]\displaystyle{ H_k(M,\partial(M); \Z) }[/math] with [math]\displaystyle{ H^{n-k}(M; \Z) }[/math] for all [math]\displaystyle{ k }[/math].[2]
Here [math]\displaystyle{ \partial(M) }[/math] can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality.
There is a version for triples. Let [math]\displaystyle{ \partial(M) }[/math] 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 [math]\displaystyle{ k }[/math], there is an isomorphism[3]
- [math]\displaystyle{ D_M\colon H^k(M,A; \Z)\to H_{n-k}(M,B; \Z). }[/math]
Notes
- ↑ Biographical Memoirs By National Research Council Staff (1992), p. 297.
- ↑ Vick, James W. (1994). Homology Theory: An Introduction to Algebraic Topology. p. 171.
- ↑ Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. p. 254. ISBN 0-521-79160-X. https://pi.math.cornell.edu/~hatcher/AT/ATpage.html.
References
- Hazewinkel, Michiel, ed. (2001), "Lefschetz_duality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Lefschetz_duality
- 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, PMID 16587146
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