Singular Homology, Inclusion Inducing a Homology Isomorphism

Inclusion Inducing a Homology Isomorphism

If the inclusion of a space S into a space T induces a homology isomorphism, the relative homology h(T/S) = 0. We saw this in the last section, when S was a retract of T, and the homotopy kept S within S. After this theorem, we no longer need to keep S within S. A retract implies f(S) into T produces a homology isomorphism, and that's all we need.

Let C be the simplexes of S, let D be the simplexes of T, and let E be the simplexes of T mod the simplexes of S. This notation extends to all dimensions. Thus the i^th row, corresponding to the i^th dimension, builds a short exact sequence that looks like this.

0 → C_i → D_i → E_i → 0

These exact sequences stack up on top of each other, with boundary operators connecting each row to the row below. The homology is the kernal mod the image as usual. The homology of C corresponds to S, the homology of D corresponds to T, and the homology of E is the relative homology of T/S. This is the group we are interested in.

Construct the long exact sequence, and each arrow from h(C_i) to h(D_i) is, by assumption, an isomorphism. That means each h(E_i) = 0, and that completes the proof.

The above can be reversed. If the relative homology E is 0, then the embedding of C into D induces a homology isomorphism.