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 above, but we still need a good criterion for knowing when something in , represented by , is zero. Or, when , when it represents the base element. The obvious reason is that if there is a homotopy starting with and ending at the constant map. Here is another that will be useful.

Theorem 1 (Compression criterion)A map represents zero in if and only if is homotopic relative to a map .

*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…)