Preservation of fiberwise normal bundles under fiberwise homotopy equivalences

Question

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?

Answer 1

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 $x \rightarrow (p(x),I(x))$.