Let $p:E\to B$ be a Serre fibration, $b_0\in B, x_0\in p^{-1}(b_0)=F.$ In Hatcher page 376 theorem 4.41, the long exact sequence for homotopy groups associated to this fibration is deduced from the long exact sequence for relative homotopy group, by proving that $p_*:\pi_n(E,F,x_0)\to\pi_n(B,b_0)$ is an isomorphism for all $n\ge 1$. As shown below:
Near the end, he stated "after permuting the last two coordinates on $I^n\times I$". Can anyone tell me what he meant by this and how is it used? I have no idea about it. Thanks.
He means mapping $(t_1, \ldots, t_n, t_{n + 1}) \in I^n$ to $(t_1, \ldots, t_{n + 1}, t_n)$. As for why this is necessary, recall that the relative homotopy lifting property says that any map $I^n \times I \to B$ can be lifted to extend an initial lift of the form $(I^n \times \{0\}) \cup (A \times I) \to E$ where $A$ is a subcomplex of $I^n$. Now the conditions in the proof give you a lift $(I^n \times \{0\}) \cup (J^{n - 1} \times I) \cup (I^n \times \{1\}) \to E$ which is on the nose not what you want, so you need to take a different perspective, and doing the coordinate interchange in question reveals that this space is in fact of the form $(I^n \times \{0\}) \cup (\partial I^n \times I)$ (if I have computed this correctly :) which allows you to go ahead.