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 "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)

References

[a1]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[a2]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. 360ff