Fiber-preserving homotopy equivalences are fiber homotopy equivalences

Let \[ p_1:E_1\to B, \qquad p_2:E_2\to B \] be fibrations, and let \[ f:E_1\to E_2, \qquad p_2f=p_1, \] be a homotopy equivalence. We write \(\simeq_B\) for a homotopy through fiber-preserving maps.

Choose an ordinary homotopy inverse \[ g:E_2\to E_1 \] with \[ fg\simeq\operatorname{id}_{E_2}, \qquad gf\simeq\operatorname{id}_{E_1}. \] Since \[ p_1g=p_2fg\simeq p_2, \] the homotopy lifting property for \(p_1\) deforms \(g\) to a map \[ g':E_2\to E_1 \] with \[ p_1g'=p_2. \] Thus we may replace \(g\) by \(g'\) and assume from now on that \(g\) itself is fiber-preserving. It remains to improve the two ordinary homotopies to fiber homotopies.

We use the following lemma.

Lemma. If \(p:E\to B\) is a fibration and \(u:E\to E\) is fiber-preserving and ordinarily homotopic to \(\operatorname{id}_E\), then \(u\) has a left homotopy inverse over \(B\).

Choose an ordinary homotopy \[ h_t:u\simeq\operatorname{id}_E. \] Since \(ph_0=p\), lift the base homotopy \(ph_t\), starting at \(\operatorname{id}_E\), to a homotopy \[ k_t:\operatorname{id}_E\simeq k_1 \] with \[ p k_t=p h_t. \] In particular \(k_1\) is fiber-preserving because \(ph_1=p\).

Now concatenate the homotopies \[ k_1u\simeq u=h_0\simeq h_1=\operatorname{id}_E \] to obtain a homotopy \(J\) from \(k_1u\) to \(\operatorname{id}_E\). Its projection \(pJ\) is a loop of maps \(E\to B\) based at \(p\). The square of homotopies obtained from \(h_t\) and the lifted homotopy \(k_t\) gives a homotopy of \(pJ\), rel endpoints, to the constant homotopy at \(p\). Lift this two-parameter homotopy using the fibration property of \(p\). Reading along the opposite boundary gives a homotopy over \(B\) \[ k_1u\simeq_B\operatorname{id}_E. \] This proves the lemma.

Apply the lemma to \[ u=gf:E_1\to E_1. \] Then there is a fiber-preserving \(a:E_1\to E_1\) with \[ a\,gf\simeq_B\operatorname{id}_{E_1}. \] Set \[ \ell=ag:E_2\to E_1. \] Then \[ \ell f\simeq_B\operatorname{id}_{E_1}. \tag{1} \] Since \(a\simeq\operatorname{id}_{E_1}\) ordinarily and \(g\) is an ordinary homotopy equivalence, \(\ell\) is also an ordinary homotopy equivalence. Repeating the preceding construction with \(\ell\) in place of \(f\), we obtain a fiber-preserving map \[ m:E_1\to E_2 \] with \[ m\ell\simeq_B\operatorname{id}_{E_2}. \tag{2} \] Compose (1) on the left with \(m\). Using (2), \[ m\simeq_Bm\ell f\simeq_B f. \] Therefore \[ f\ell\simeq_Bm\ell\simeq_B\operatorname{id}_{E_2}. \] Together with (1), this shows that \(\ell\) is a fiber-preserving homotopy inverse to \(f\). Hence \[ \boxed{f\text{ is a fiber homotopy equivalence}.} \]