Namespaces
Variants
Actions

Difference between revisions of "Relative homology"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
The homology groups (cf. [[Homology group|Homology group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r0810101.png" /> of a pair of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r0810102.png" />. They are defined by the quotient complex of the chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r0810103.png" /> with coefficients in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r0810104.png" /> by the subcomplex consisting of all chains with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r0810105.png" />. These groups are usually not altered by  "excisionexcision" , i.e. by the replacement of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r0810106.png" /> by a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r0810107.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r0810108.png" /> is an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r0810109.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r08101010.png" />. The relative cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r08101011.png" /> are defined by the subcomplex of the chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r08101012.png" /> consisting of all cochains with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r08101013.png" />, while the quotient complex usually defines cohomology groups of the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081010/r08101014.png" />.
+
<!--
 +
r0810101.png
 +
$#A+1 = 14 n = 0
 +
$#C+1 = 14 : ~/encyclopedia/old_files/data/R081/R.0801010 Relative homology
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
The homology groups (cf. [[Homology group|Homology group]]) $  H _ {p}  ^ {c} ( X, A;  G) $
 +
of a pair of spaces $  ( X, A) $.  
 +
They are defined by the quotient complex of the chain complex $  X $
 +
with coefficients in a group $  G $
 +
by the subcomplex consisting of all chains with support in $  A $.  
 +
These groups are usually not altered by  "excisionexcision" , i.e. by the replacement of the pair $  ( X, A) $
 +
by a pair $  ( X \setminus  U, A \setminus  U) $,  
 +
where $  U $
 +
is an open subset of $  X $
 +
contained in $  A $.  
 +
The relative cohomology groups $  H  ^ {p} ( X, A;  G) $
 +
are defined by the subcomplex of the chain complex $  X $
 +
consisting of all cochains with support in $  X \setminus  A $,  
 +
while the quotient complex usually defines cohomology groups of the subset $  A \subset  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.G. Sklyarenko,  "Homology and cohomology of general spaces" , Springer  (Forthcoming)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.G. Sklyarenko,  "Homology and cohomology of general spaces" , Springer  (Forthcoming)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. 360ff</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. 360ff</TD></TR></table>

Revision as of 08:10, 6 June 2020


The homology groups (cf. Homology group) $ H _ {p} ^ {c} ( X, A; G) $ of a pair of spaces $ ( X, A) $. They are defined by the quotient complex of the chain complex $ X $ with coefficients in a group $ G $ by the subcomplex consisting of all chains with support in $ A $. These groups are usually not altered by "excisionexcision" , i.e. by the replacement of the pair $ ( X, A) $ by a pair $ ( X \setminus U, A \setminus U) $, where $ U $ is an open subset of $ X $ contained in $ A $. The relative cohomology groups $ H ^ {p} ( X, A; G) $ are defined by the subcomplex of the chain complex $ X $ consisting of all cochains with support in $ X \setminus A $, while the quotient complex usually defines cohomology groups of the subset $ A \subset X $.

References

[1] E.G. Sklyarenko, "Homology and cohomology of general spaces" , Springer (Forthcoming) (Translated from Russian)

Comments

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[a2] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. 360ff
How to Cite This Entry:
Relative homology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relative_homology&oldid=48498
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article