I want to prove that the homotopy groups of some topological space $B$ of interest to me (not a CW complex) are trivial. I have a strategy of proof that consists in introducing another space $E$ that is contractible, and easily comes with a continuous surjection $\pi :E\to B$. If I can prove that any continuous map $f:I^k\to B$ lifts to a continuous map $\tilde f : I^k\to E$, then I'm done.
If I am not mistaken, this lifting property is true as soons as $\pi$ is a Serre fibration. Here is my question: are there classical way to prove such a thing, and were can I learn them (or simply learn about Serre fibrations)? Of course, any reference for the initial problem, which seems slightly weaker, is welcome too.
I guess that I should be able to manage my case by hand, but I think it may be an opportunity to learn more mathematics.