I'd like to understand a part of the Theorem on Tammo Tom Dieck p.$130$. The theorem is the following:
I do understand that $T$ is homeomorphic to $J^n$ with the exchange of the last two coordinates, but I don't understand how to prove rigorously using $(3.2.4)$ that the bottom map with a diagram where $T$ is involved lifts. Any help would be appreciated in order to understand in detail why we can lift such a map. Here's a picture of $(3.2.4)$
