This one will be a quick post. In effect, we continue with last time, where we defined the relative homotopy groups, and now describe a practical means of determining when something in one of these relative groups is zero or not. This will become useful in the future.

The compression criterion

We have defined the group ${\pi_n(X, A)}$ above, but we still need a good criterion for knowing when something in ${\pi_n(X, A)}$, represented by ${f: (D^n, S^{n-1}) \rightarrow (X, A)}$ , is zero. Or, when ${n = 1}$, when it represents the base element. The obvious reason is that if there is a homotopy ${H: (D^n, S^{n-1}) \times I \rightarrow (X, A)}$ starting with ${f}$ and ending at the constant map. Here is another that will be useful.

Theorem 1 (Compression criterion) A map ${f: (D^n, S^{n-1}) \rightarrow (X, A)}$ represents zero in ${\pi_n(X, A)}$ if and only if ${f}$ is homotopic relative ${S^{n-1}}$ to a map ${g: D^n \rightarrow A}$.

Proof: This is one of those things which is not really all that hard to prove, but for which pictures help significantly. So I will try to draw pictures. (more…)