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Homotopy invariance of numerable fiber bundle: let $\xi = (E,π,B)$ (i.e. $\pi : E \to B$) a numerable topological fiber bundle, $X$ a topological space, and $f,\,g : X \to B$ two continuous applications. If $f$ and $g$ are homotopic, then the pullback fiber bundles $f^* \xi$ and $g^* \xi$ are $X$-isomorphic.

NB : A space $B$ is "numerable" if it has an open covering with a (locally finite) partition of unity (weaker property than $B$ paracompact), and the fiber bundle $\xi$ is "numerable" if $B$ has a trivializing open covering with also has a subordinate partition of unity (see Dold in "Partitions of unity in the theory of fibration").

The usual theorem of homotopy invariance of fiber bundle assumes that the space $X$ is paracompact but no restriction on the fiber bundle which is pulled back. Here there is no restriction on $X$ but the bundle is numerable. This theorem is the corollary 7.10 of Dold's paper, but I can't fill what looks like a gap for me in the proof.

I admit that numerable fiber bundles possess the CHP (Homotopy lifting property ) i.e. if $\xi =(E,π,B)$, $\xi' =(E',π',B')$ are two numerable topological fiber bundles, $(\tilde{f},\,f):\xi \to \xi'$ a morphism of fiber bundles, and $H$ a homotopy of $f$ i.e. $H(\cdot,\,0) = f$, then there is a lifting $\tilde{H}:E \times I \to E'$ of $H$ such as $\tilde{H}(\cdot,\,0) = \tilde{f}$ and $\pi' \circ \tilde{H} = H \circ (\pi \times Id_I)$.(I think that the proof in Dold's paper for principal fiber bundles can be adapted for plain fiber bundles).

First it is clear that if $\xi$ is numerable, the pullback $f^*\xi$ is also numerable. Then I can apply the preceding theorem to the canonical morphism $(pr_2^f,f):f^*\xi \to \xi$ and the homotopy $H : X \times I \to B$ from $f$ to $g$, giving a lifting $\tilde{H}:f^{*}E \times I \to E$ such as $\pi \circ \tilde{H} = H \circ (pr_{1} \times Id_{I})$.

I can also pullback $\xi$ with $H$ to get $H^*\xi = (H^*E,pr_1^H \times Id_I,X \times I)$ and the isomorphisms $(H^{*}E) \mathclose{}|\mathopen{}_{X \times \{0\}}\, \cong f^{*}\xi$ and $(H^{*}E) \mathclose{}|\mathopen{}_{X \times \{1\}}\, \cong g^{*}\xi$.

By the universal property of fibered product, there is a continuous application $\theta \colon f^{*}E \times I \to H^{*}E$, such that $\tilde{H} = pr_{2}^H \circ \theta$ and $pr_{1}^f \times Id_I = pr_{1}^H \circ \theta$. Then for $(x,\,e) \in f^{*}E$ we have \begin{equation*} \theta((x,\,e),\,t) = (\,(x,\,t),\,\tilde{H}((x,\,e),\,t)\,) \in H^{*}E \end{equation*}

If I can prove that $\theta$ is also an isomorphism, it's done because \begin{equation*} f^*\xi \cong (f^*E \times I) \mathclose{}|\mathopen{}_{X \times \{1\}} \cong (H^* E) \mathclose{}|\mathopen{}_{X \times \{1\}}\cong g^*\xi \end{equation*}

But how can I prove that $\theta$ really an isomorphism ??

To be clear, I know there are other kinds of proof, like in Husemoller's "Fibre Bundles" or in Tammo Tom Dieck "algebraic topology" (which are for numerable principal bundles but seem to work also for plain fiber bundles). But here I am interested by proofs relying on the Covering Homotopy Property (CHP) for fiber bundle. Once you have this property, it should be relatively easy to prove the homotopy invariance (all the "dirty work" has already been done).

My understanding of Dold's proof (Corollary 7.10) is that since there is a lift $\tilde{H}:f^{*}E \times I \to E$ such as $\pi \circ \tilde{H} = H \circ (pr_{1} \times Id_{I})$,... it is "obvious that $f^* \xi$ and $g^* \xi$ are $X$-isomorphic. Not at all obvious for me ! Ralph Cohen uses the same kinf of proof in its "Topology of fiber bundles", with a little bit more details. In fact he says exactly what I have written above, but for the crucial part he just writes that $\theta$ is obviously an isomorphism "since it induces identity on the base and the fibers$... and once again this doesn't seem obvious at all ! (and even if it is true, being an homeo restricted to each fiber doesn't imply that the morphism is an homeo).

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  • $\begingroup$ I think Dold proves this in his book Algebraic Topology, no? There's a lemma about something called a stacked cover that he uses, which seems to me to be key. $\endgroup$ – David Roberts Sep 23 '17 at 22:18
  • $\begingroup$ @Roberts, Dold's book does define "stacked cover", but doesn't even mention fiber bundles in its index and so doesn't prove what I am asking for. In fact there is a very short proof of this in "Dold/Partition of Unity in the theory of fibrations" (Corollary 7.10) as an "easy" consequence of the Homotopy lifting of numerable fiber bundles, but I don't understand it... $\endgroup$ – ychemama Sep 26 '17 at 8:29
  • $\begingroup$ I will look at it and see if I can answer this tomorrow (it is evening here now) $\endgroup$ – David Roberts Sep 26 '17 at 11:38
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    $\begingroup$ This answer circumvents that conclusion, but in doing so appeals to the proof of the covering homotopy theorem. I also find the answer @Ben linked to unsatisfactory, but for a different reason: I don't think it proves that $\theta$ is an iso on fibers, as $\tilde H\colon f_0^* E\times I\to H^*E \to E$ is essentially different from $f_0^* E\times I\to f_0^*E\to E$ (iso on fibers by construction). If $\theta$ were known to be an iso on fibers, then the continuity of the map of sets $\theta^{-1}$ could be checked locally on a trivialization. $\endgroup$ – Fabian Henneke Jan 4 '18 at 16:59
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    $\begingroup$ @Fabian Henneke , I have checked [this answer]. It explains Ralph Cohen's affirmation by refering to his proof of the 1st covering homotopy theorem. But this one is for compact base space, and refers to Steenrod (thm 11.3 p50) for the general case. But Steenrod's setting assumes that the base space is what he calls $C^{\sigma}$ (ie Normal, locally compact, any covering reducible to countable covering). It's different than mine : $\xi$ numerable locally trivial fibration, no restriction on space $X$, and my proof of the CHP is an adaptation of Dold's thm 7.8. $\endgroup$ – ychemama Jan 5 '18 at 12:05

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