While reading GMTW - The homotopy type of the cobordism category (https://arxiv.org/abs/math/0605249, p. 16) I found the following passage:

Lef $f:W\to \mathbb{R}$ be the projection. Then $(\pi_2,f):W\to X\times \mathbb{R}$ is proper since we have assumed that $X$ is compact. For $n\gg d$ we get an embedding $W\subset X\times \mathbb R\times \mathbb R^{d-1+n}$ which lifts $(\pi_2,f)$.

In this context I think $(\pi_2,f):W\to X\times \mathbb{R}$ is a submersion so by Ehresmann's lemma it is actually a fiber bundle. I'm not sure where the embedding comes from though. Is this some version of Whitney's theorem that is good for fiber bundles?


I think it is simpler than that if $W$ is assumed to be compact. For $n$ enough large, you have an embedding $\iota:W\to\mathbb{R}^{d-1+n}$ by Whitney's theorem assuming that $W$ is compact. A map such as $((\pi_2,f),\iota):W\to X\times\mathbb{R}\times\mathbb{R}^{d-1+n}$ is an example of the sort claimed in the above assertion. If $W$ is not necessarily compact, but you know that the fibre of $W\to X\times\mathbb{R}$ is compact, then you can apply a similar construction in a fibre-wise manner, I think!

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  • $\begingroup$ You are right, thank you. Just to make sure, what's the problem with using Whitney's theorem if $W$ is not compact? $\endgroup$ – user109300 Sep 4 '17 at 7:23

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