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Difference between revisions of "Relative homology"

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The homology groups (cf. [[Homology group|Homology group]]) $ H _ {p}  ^ {c} ( X, A;  G) $
+
The homology groups (cf. [[Homology group|Homology group]]) $H _ {p}  ^ {c} ( X, A;  G) $
 
of a pair of spaces  $  ( X, A) $.  
 
of a pair of spaces  $  ( X, A) $.  
They are defined by the quotient complex of the chain complex $ X $
+
They are defined by the quotient complex of the chain complex $X$
with coefficients in a group $ G $
+
with coefficients in a group $G$
by the subcomplex consisting of all chains with support in $ A $.  
+
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) $
+
These groups are usually not altered by "excision", i.e. by the replacement of the pair $(X, A)$
by a pair  $  ( X \setminus  U, A \setminus  U) $,  
+
by a pair  $  ( X \setminus  U, A \setminus  U)$,  
where $ U $
+
where $U$ is an open subset of $X$ contained in $A$.  
is an open subset of $ X $
+
The relative cohomology groups $H^{p} (X, A; G)$ are defined by the subcomplex of the chain complex $X$
contained in $ A $.  
+
consisting of all cochains with support in $X \setminus A$,  
The relative cohomology groups $ H ^ {p} ( X, A; G) $
+
while the quotient complex usually defines cohomology groups of the subset $A \subset X$.
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><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR>
====Comments====
+
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)  pp. 360ff</TD></TR>
 
+
</table>
====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>
 

Latest revision as of 16:43, 9 April 2023


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 "excision", 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)
[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=53716
This article was adapted from an original article by E.G. Sklyarenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article