Relative homology

Definition

Given a subspace A ⊂ X , one may form the short exact sequence

where C ∙ ( X ) denotes the singular chains on the space X. The boundary map on C ∙ ( X ) leaves C ∙ ( A ) invariant and therefore descends to a boundary map on the quotient. The corresponding homology is called relative homology:

One says that relative homology is given by the relative cycles, chains whose boundaries are chains on A, modulo the relative boundaries (chains that are homologous to a chain on A, i.e. chains that would be boundaries, modulo A again).

Properties

The above short exact sequences specifying the relative chain groups gives rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence

The connecting map δ takes a relative cycle, representing a homology class in H_n(X, A), to its boundary (which is a cycle in A).

It follows that H_n(X, x₀), where x₀ is a point in X, is the n-th reduced homology group of X. In other words, H_i(X, x₀) = H_i(X) for all i > 0. When i = 0, H₀(X, x₀) is the free module of one rank less than H₀(X). The connected component containing x₀ becomes trivial in relative homology.

The excision theorem says that removing a sufficiently nice subset Z ⊂ A leaves the relative homology groups H_n(X, A) unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that H_n(X, A) is the same as the n-th reduced homology groups of the quotient space X/A.

The n-th local homology group of a space X at a point x₀ is defined to be H_n(X, X - {x₀}). Informally, this is the "local" homology of X close to x₀.

Relative homology readily extends to the triple (X, Y, Z) for Z ⊂ Y ⊂ X.

One can define the Euler characteristic for a pair Y ⊂ X by

The exactness of the sequence implies that the Euler characteristic is additive, i.e. if Z ⊂ Y ⊂ X, one has

Functoriality

The map C ∙ can be considered to be a functor

where Top² is the category of pairs of topological spaces and C C is the category chain complexes of abelian groups.

Examples

One important use of relative homology is the computation of the homology groups of quotient spaces X / A . In the case that A is a subspace of X fulfilling the mild regularity condition that there exists a neighborhood of A that has A as a deformation retract, then the group H ~ n ( X / A ) is isomorphic to H n ( X , A ) . We can immediately use this fact to compute the homology of a sphere. We can realize S n as the quotient of an n-disk by its boundary, i.e. S n = D n / S n − 1 . Applying the exact sequence of relative homology gives the following:
⋯ → H n ( D n ) → H n ( D n , S n − 1 ) → H n − 1 ( S n − 1 ) → H n − 1 ( D n ) → ⋯ .

Because the disk is contractible, we know its homology groups vanish in all dimensions, so the above sequence collapses to the short exact sequence:

0 → H n ( D n , S n − 1 ) → H n − 1 ( S n − 1 ) → 0.

Therefore, we get isomorphisms H n ( D n , S n − 1 ) ≅ H n − 1 ( S n − 1 ) . We can now proceed by induction to show that H n ( D n , S n − 1 ) ≅ Z . Now because S n − 1 is the deformation retract of a suitable neighborhood of itself in D n , we get that H n ( D n , S n − 1 ) ≅ H n ( S n ) ≅ Z .