Corollary 2.3.15. A Serre fibration \(p\colon E \to B\) is a weak homotopy equivalence if and only if the fiber \(F_b\) is weakly contractible for every \(b \in B\).
Proof. By working separately over each path component of \(B\), we may assume that \(B\) is path-connected. Picking a preferred basepoint \(b_0\) of \(B\), it then suffices to show that \(p\) is a weak homotopy equivalence if and only if the fiber \(F := F_{b_0}\) is weakly contractible. This follows immediately from the long exact sequence from Theorem 2.3.14. □
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