Namespaces
Variants
Actions

Difference between revisions of "Lefschetz duality"

From Encyclopedia of Mathematics
Jump to: navigation, search
(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057970/l0579701.png" /> is a pair of spaces such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057970/l0579702.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057970/l0579703.png" />-dimensional topological manifold, then for any Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057970/l0579704.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057970/l0579705.png" /> there is an isomorphism
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057970/l0579706.png" /></td> </tr></table>
+
$$H_i(X,A;G)\approx H_c^{n-i}(X\setminus A;G).$$
  
On the right-hand side one has cohomology with compact support. If the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057970/l0579707.png" /> is non-orientable, one must, as usual, take cohomology with local coefficients.
+
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
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article