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I asked the same question in math stackexchange: https://math.stackexchange.com/questions/2322883/how-can-i-endow-a-locally-product-cw-structure-on-a-vector-bundle-over-a-cw-co but it seems that it's harder than I thought, so I ask here:

I'm now learning characteristic classes, and I need a CW structure on the total space of a vector bundle $E\to B$ where $B$ is a CW complex such that the associated sphere bundle and rectriction over any subcomplex of $B$ are both subcomplexes (this is required in "Algebraic Topology from a Homotopical Viewpoint", page 364). I think this should be something like a locally product structure, but I couldn't figure out how to glue them together. I even doubt that this can be done.

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The authors of this book are attempting to use CW structures to justify certain cohomology isomorphisms, but this seems to be the wrong approach since some of their claims about CW structures are just not true. For example, they say a vector bundle over a CW complex base space has a CW structure such that the complement of the zero section is a subcomplex, but this cannot be true since a subcomplex is always a closed subspace. It seems best not to talk about CW structures on vector bundles and instead prove the cohomology isomorphisms using standard tools such as excision for general topological spaces.

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    $\begingroup$ Thank you for your answer. I understand it as there is no widely accepted CW structure on the total space. But I need at least that if $C\hookrightarrow B$ is a cofibration, then $E|_C \hookrightarrow E$ is a cofibration, in order to prove the Thom isomorphism theorem for general CW complexes from the finite dimensional case. Is this true? $\endgroup$ – Naruki Masuda Jun 19 '17 at 5:45
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    $\begingroup$ @NarukiMasuda: Yes, this is true. See the answers to mathoverflow.net/questions/178509/… $\endgroup$ – Mark Grant Jun 19 '17 at 10:43
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    $\begingroup$ @NarukiMasuda It depends on what how you define a cofibration. If $C\to B$ is a Hurewicz cofibration (the HEP holds) then the same holds for $E_{|C} \to E$ (this is a result of Strøm). If $C \to B$ is a Serre cofibration (i.e, $B$ is obtained from $C$ by attaching cells), then $E_{|C} \to E$ needn't be a Serre cofibration. $\endgroup$ – John Klein Jun 19 '17 at 12:04

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