# How can I endow a “locally product” CW structure on a vector bundle over a CW complex?

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.

• 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? – Naruki Masuda Jun 19 '17 at 5:45
• @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. – John Klein Jun 19 '17 at 12:04