Fibration with contractible total space

Question

Let $F \to E \to B$ be a fibration, and suppose $E$ is contractible. Then the long exact sequence of homotopy groups shows that $F$ has the same homotopy groups as the loop space $\Omega B$. Is it true that $F$ and $\Omega B$ are homotopy equivalent?

Answer 1

Yes if they are CW complexes, this is a corollary of this theorem of Whitehead.

https://en.wikipedia.org/wiki/Whitehead_theorem