Difference between revisions of "Relative homology"
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| − | The homology groups (cf. [[Homology group|Homology group]]) | + | 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 | + | They are defined by the quotient complex of the chain complex $X$ |
| − | with coefficients in a group | + | with coefficients in a group $G$ |
| − | by the subcomplex consisting of all chains with support in | + | by the subcomplex consisting of all chains with support in $A$. |
| − | These groups are usually not altered by | + | 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 | + | where $U$ is an open subset of $X$ contained in $A$. |
| − | is an open subset of | + | The relative cohomology groups $H^{p} (X, A; G)$ are defined by the subcomplex of the chain complex $X$ |
| − | contained in | + | consisting of all cochains with support in $X \setminus A$, |
| − | The relative cohomology groups | + | while the quotient complex usually defines cohomology groups of the subset $A \subset X$. |
| − | are defined by the subcomplex of the chain complex | ||
| − | consisting of all cochains with support in | ||
| − | while the quotient complex usually defines cohomology groups of the subset | ||
====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> |
| − | + | <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> | |
| − | + | <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=48498
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