Difference between revisions of "Lefschetz duality"
From Encyclopedia of Mathematics
(Importing text file) |
(→References: + ZBL link) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
''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. |
| Line 13: | Line 14: | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Lefschetz, "Manifolds with a boundary and their transformations" ''Trans. Amer. Math. Soc.'' , '''29''' (1927) pp. 429–462</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C.R.F. Maunder, "Algebraic topology" , Cambridge Univ. Press, reprint (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> B. Iversen, "Cohomology of sheaves" , Springer (1986)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Lefschetz, "Manifolds with a boundary and their transformations" ''Trans. Amer. Math. Soc.'' , '''29''' (1927) pp. 429–462 {{ZBL|53.0552.04}}</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> C.R.F. Maunder, "Algebraic topology" , Cambridge Univ. Press, reprint (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> B. Iversen, "Cohomology of sheaves" , Springer (1986)</TD></TR></table> | ||
Latest revision as of 13:08, 17 April 2023
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 Zbl 53.0552.04 |
| [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=11981
Lefschetz duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lefschetz_duality&oldid=11981
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article