Homology, Relative Homology

Relative Homology

Let A be a subchain of B. At each level, there is a quotient module Bi/Ai. With A6 mapping into A7, B6/A6 maps into B7/A7. Furthermore, any x in B6 maps to 0 in B8, hence the same is true of any coset represented by x. Therefore the sequence of quotient modules, denoted B/A, is a chain complex.

Like any chain complex, B/A has homology groups at every level. This is called a relative homology - the homology of B relative to A.

Here is a technical description of the 7th homology group of B/A. The element x in B7 represents a member of the homology group if its image lies in A8. Thus x maps to 0 in B8/A8. If x is spanned by A7 and the image of B6, then x is part of the trivial coset, which becomes 0 in the homology group. Since both A7 and the image of B6 wind up in A8, the span is contained entirely in the preimage of A8. The quotient group, i.e. the homology group, is well defined.

Relative Homology Homomorphism

Let A be a subchain of B and let C be a subchain of D. Let f be a chain map from B into D, and assume f carries A into C. Build a new map f′ from B/A into D/C. The image of the coset A+x is the coset C+f(x). Since A winds up in C, this is well defined.

Since f is compatible with the internal homomorphisms of the graded modules B and D, f′ is compatible with the internal homomorphisms of B/A and D/C. Everything follows the coset representative, where the diagram commutes. Therefore f′ is a chain map, and as described in the previous section, it induces a homomorphism from the homology of B/A into the homology of D/C.

The follow-on theorems carry over as well. For instance, let g be a second chain map from D into another chain complex R, such that g carries C into the subchain Q. The composition of f and g maps the relative homology of B/A into the relative homology of Q/R, and this is the same as the composition of the homomorphisms induced by f and g separately. Either way the result is g(f(x)), where x and g(f(x)) are both coset representatives in their respective homology groups.

The sum of two chain maps induces the sum of the individual homomorphisms on the relative homology. The restriction of a chain complex restricts the relative homology, and any homomorphism therefrom. The join or direct sum of two restrictions implies the join or direct sum of the corresponding relative homologies, and any homomorphisms therefrom. I'll leave the details to you.