# Preservation of fiberwise normal bundles under fiberwise homotopy equivalences

I am looking to replicate, in the fiberwise setting, the result of Spivak/Wall that the fiber homotopy type of the normal sphere bundle of a manifold is preserved under homotopy equivalences.

A particularly slick way to prove this result, is to notice that the mapping cylinder of a homotopy equivalence $$M \rightarrow M'$$ is a relative Poincare duality space, and as such we can talk about its normal fibration (obtained from a relative embedding into $$\mathbb{R}^N \times (0,1]$$), which restricts to the normal fibrations of $$M$$ and $$M'$$ on either side. Thus, because a mapping cylinder of a homotopy equivalence has both end inclusions homotopy equivalences, we can conclude there is some fiberwise homotopy equivalence of normal spherical fibrations.

So here is my question: is there is enough fiberwise embedding theory in the literature to be able to conclude that the fiberwise mapping cylinder of a fiber homotopy equivalence of manifold bundles over $$B$$ embeds into $$B \times (\mathbb{R}^{N} \times (0,1])$$, such that $$M,M'$$ land in the boundary?

After looking at the actual construction of the fiberwise embedding of a bundle $$M \rightarrow E \rightarrow B$$ into $$B \times \mathbb{R}^N$$, it became clear how to adjust it to embed the mapping cylinder. For convenience, let's assume that $$B$$ is a finite CW complex.
To obtain a fiberwise embedding of the bundle $$p:E \rightarrow B$$ into $$B \times \mathbb{R}^N$$, simply pick an embedding $$i$$ of $$E$$ into $$\mathbb{R}^N$$ and define $$E \rightarrow B \times \mathbb{R}^N$$ by $$x \rightarrow (p(x),i(x))$$.
Suppose now we have a fiberwise homotopy equivalence between $$E \rightarrow B$$ and $$E' \rightarrow B'$$. Let $$I$$ be a relative embedding of the mapping cylinder into $$\mathbb{R}^L \times (0,1]$$. Now we can define a fiberwise embedding of the mapping cylinder of the bundle equivalence by $$x \rightarrow (p(x),I(x))$$.