Let $B$ be a connected pointed CW complex, let $E$ and $E'$ be two CW complexes and let $f\colon E\to B$ and $f'\colon E'\to B$ be two Serre fibrations. Let $g\colon E\to E'$ be a continuous map such that

\begin{eqnarray} E &\xrightarrow{f}& B\\ \small{g}\downarrow & &\|\\ E'&\xrightarrow{f'}& B \end{eqnarray}

commutes and suppose that $g$ induces a weak equivalence $g_x\colon E_x\xrightarrow{\cong} E'_x$ when restricted to the fibers $E_x$ and $E_x'$ of $E$ and $E′$ over the basepoint $x$ of $B$.

Is $g$ a weak equivalence?

By the 5-lemma, the only problem is the injectivity of $\pi_0g\colon\pi_0E\to \pi_0E'$. A related question on math.stackexchange does not provide an answer.

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    $\begingroup$ Doesn't it suffice to take into account the action of $\pi _1(B)$ on $\pi _0$ of everything? $\endgroup$ – user43326 Jun 26 '14 at 14:03

Note that $\pi _1(B)$ acts on $\pi _0(F)$. So the ending of the long exact sequence for the fibration $$\cdots\rightarrow \pi _1(B)\rightarrow \pi _0(F)\rightarrow \pi _0(E)$$ is not just an exact sequence of sets, but it also shows that $\pi _0(E)$ is the quotient of the set $\pi _0(F)$ by the group $\pi _1(B)$. So your hypotheses imply that $\pi _0(E)\cong \pi _0(E^{\prime})$. Thus combined with what you already know about $\pi _i$'s with $i>0$, you get the weak equivalence between $E$ and $E^{\prime}$.


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