Massey Exercise 2.7.6

Theorem: Assume in addition to the hypotheses of 7.5 that there is a continuous map such that for all , . Then for any , .

Proof: For , we have

Now, we examine what this path inside actually is. We have:

This path in performs in the first coordinate over the first half of time, and then performs in the second coordinate over the second half of time. This is homotopic to the map that performs in the first coordinate and in the second, both over the whole time frame. That is, we have:

By the hypothesis, for all , so this is actually the identity map. Then we have

Where the last equivalence comes from applying the first property of in the special case of both arguments being . A symmetric argument obtains , so .