In singular homology
Let be a topological space and a subspace. Write for the chain complex of singular homology on and for the chain map induced by the subspace inclusion.
The cokernel of this inclusion, hence the quotient of by the image of under the inclusion, is the chain complex of -relative singular chains.
A boundary in this quotient is called an -relative singular boundary,
a cycle is called an -relative singular cycle.
The chain homology of the quotient is the -relative singular homology of
Long exact sequences
Let . The corresponding relative homology sits in a long exact sequence of the form
The connecting homomorphism sends an element represented by an -relative cycle , to the class represented by the boundary .
This is the homology long exact sequence induced by the given short exact sequence of chain complexes.
Let be a sequence of two inclusions. Then there is a long exact sequence of relative homology groups of the form
Observe that we have a (degreewise) short exact sequence of chain complexes
The corresponding homology long exact sequence is the long exact sequence in question.
Let be a sequence of topological subspace inclusions such that the closure of is still contained in the interior of : .
In the above situation, the inclusion induces isomorphism in relative singular homology groups
for all .
Let be two topological subspaces such that their interior is a cover of .
In the above situation, the inclusion induces isomorphisms in relative singular homology groups
for all .
A proof is spelled out in (Hatcher, from p. 128 on).
Relative homology is homotopy invariant in both arguments.
Relation to reduced homology of quotient topological spaces
For a CW complex, the inclusion of any subcomplex is a good pair (called a CW-pair ).
This is discussed at CW complex – Subcomplexes.
For instance (Hatcher, prop. 2.22).
By assumption we can find a neighbourhood such that has a deformation retract and hence in particular is a homotopy equivalence and so induces also isomorphisms on all singular homology groups.
It follows in particular that for all the canonical morphism is an isomorphism, by prop. 2.
Given such we have an evident commuting diagram of pairs of topological spaces
Here the right vertical morphism is in fact a homeomorphism.
Applying relative singular homology to this diagram yields for each the commuting diagram of abelian groups
Here the left horizontal morphisms are the above isomorphims induced from the deformation retract. The right horizontal morphisms are isomorphisms by prop. 3 and the right vertical morphism is an isomorphism since it is induced by a homeomorphism. Hence the left vertical morphism is an isomorphism (2-out-of-3 for isomorphisms).
Relation to reduced homology
Let be a inhabited topological space and let any point. Then the relative singular homology is isomorphic to the absolute reduced singular homology of
This is the special case of prop. 5 for a point.
The reduced singular homology of the -sphere equals the -relative homology of the -disk with respect to the canonical boundary inclusion : for all
The -sphere is homeomorphic to the -disk with its entire boundary identified with a point:
Moreover the boundary inclusion is evidently a good pair in the sense of def. 3. Therefore the example follows with prop. 5.
Detecting homology isomorphisms
If an inclusion is such that all relative homology vanishes, , then the inclusion induces isomorphisms on all singular homology groups.
Under the given assumotion the long exact sequence in prop. 1 secomposes into short exact pieces of the form
Exactness says that the middle morphism here is an isomorphism.
Relative homology of CW-complexes
Let be a CW-complex and write
for its filtered topological space-structure with the topological space obtained from by gluing on -cells.
The relative singular homology of the filtering degrees is
where denotes the set of -cells of and denotes the free abelian group on this set.
For instance (Hatcher, lemma 2.34).
The inclusion is clearly a good pair in the sense of def. 3. The quotient is by definition of CW-complexes a wedge sum of -spheres, one for each element in . Therefore by prop. 5 we have an isomorphism with the reduced homology of this wedge sum. The statement then follows by the respect of reduced homology for wedge sums as discussed at Reduced homology - Respect for wedge sums.
A standard textbook account for relative singular homology is section 2.1 of