Difference between revisions of "Lefschetz duality"
From Encyclopedia of Mathematics
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''Lefschetz–Poincaré duality'' | ''Lefschetz–Poincaré duality'' | ||
| − | An assertion about the duality between homology and cohomology, established by S. Lefschetz. More precisely, if | + | An assertion about the duality between homology and cohomology, established by S. Lefschetz. More precisely, if $(X,A)$ is a pair of spaces such that $X\setminus A$ is an $n$-dimensional topological manifold, then for any Abelian group $G$ and any $i$ there is an isomorphism |
| − | + | $$H_i(X,A;G)\approx H_c^{n-i}(X\setminus A;G).$$ | |
| − | On the right-hand side one has cohomology with compact support. If the manifold | + | On the right-hand side one has cohomology with compact support. If the manifold $X\setminus A$ is non-orientable, one must, as usual, take cohomology with local coefficients. |
Revision as of 09:12, 27 July 2014
Lefschetz–Poincaré duality
An assertion about the duality between homology and cohomology, established by S. Lefschetz. More precisely, if $(X,A)$ is a pair of spaces such that $X\setminus A$ is an $n$-dimensional topological manifold, then for any Abelian group $G$ and any $i$ there is an isomorphism
$$H_i(X,A;G)\approx H_c^{n-i}(X\setminus A;G).$$
On the right-hand side one has cohomology with compact support. If the manifold $X\setminus A$ is non-orientable, one must, as usual, take cohomology with local coefficients.
Comments
The original reference is [a1]. Good modern accounts of Lefschetz duality can be found in [a2] and (from the point of view of sheaf cohomology) in [a3].
References
| [a1] | S. Lefschetz, "Manifolds with a boundary and their transformations" Trans. Amer. Math. Soc. , 29 (1927) pp. 429–462 |
| [a2] | C.R.F. Maunder, "Algebraic topology" , Cambridge Univ. Press, reprint (1980) |
| [a3] | B. Iversen, "Cohomology of sheaves" , Springer (1986) |
How to Cite This Entry:
Lefschetz duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lefschetz_duality&oldid=32557
Lefschetz duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lefschetz_duality&oldid=32557
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article