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Consider two fibrations $F\to E\to B$ and $F'\to E'\to B$ such that the following diagram commutes:

$\begin{array}{ccccc} F&\stackrel{}{\rightarrow}&E&\stackrel{}{\rightarrow}&B\\ \downarrow\scriptstyle{f}&&\downarrow\scriptstyle{g}&&\scriptstyle{=}\\ F'&\stackrel{}{\rightarrow}&E'&\stackrel{}{\rightarrow}&B \end{array}$

and $f$ is a homotopoy equivalence. Can we conclude that also $g$ is a homotopy equivalence?

Thank you!

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Hint: Assume that every space here are connected and are $CW$-complexes. The Serre homotopy sequence associated to the fibrations implies that $\pi_n(E)\simeq \pi_n(E'), n\geq 0$ via the morphism induced by $g$, since $E$ and $E'$ are $CW$-complexes and connected, the Whitehead theorem implies that $g$ is a homotopy equivalence.

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