Relative homology

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In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. It is somewhat analogous to forming quotient objects in other categories, a general way to gain insight into structures.

Homology groups of subspaces are not in a direct way subgroups of the homology groups of the whole space. We cannot form the quotient groups from the homology alone.

On the other hand, the singular chains on a subspace A (the free abelian groups C_p(A) that are used to define the singular homology groups) actually are subgroups of the singular chains on the whole space X--we can then consider the quotient group C_p(X) / C_p(A). The boundary map sends q-chains on A to (q-1)-chains on A and hence, by algebra, induces unique homomorphisms on the quotient groups. In these groups we can define the relative homology to be H_q(X, A) to be the quotient of the relative cycles (chains whose boundaries are chains on A, i.e. chains that would be cycles modulo the fact that they lie in A) by the relative boundaries (chains that are homologous to a chain on A, i.e. chains that would be boundaries, modulo A again).

[edit] The long exact sequence for relative homology

A very important feature of the relative homology is that it gives us a long exact sequence of the homology groups. Using the snake lemma we can define a connecting homomorphism which takes the sequence down one dimension for it to continue onward.