Relative homology

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