This question regards a part of the proof of the so called *surgery step*, in Wall's [book](https://www.maths.ed.ac.uk/%7Ev1ranick/books/scm.pdf) "surgery on compact manifolds", Theorem 1.1.


**Setting**


$M^m$ smooth manifold, $X$ CW complex, $\phi :M\to X$ continuous map, $\nu\to X$ a rank-$v$ vector bundle and $F:TM\oplus \phi^\*\nu \oplus \varepsilon^q \to \varepsilon^{m+q+v}$ is a given stable trivialization of $TM\oplus \phi^\*\nu$.


The second part of Theorem 1.1. asserts that if $f:\mathbb S^r\times \mathbb D^{m-r}\to M$ is an *embedding*, $m\geq r+2$ and $f\_0=f|\_{\mathbb S^r\times\{0\}}$ makes this diagram


$\require{AMScd}$
\begin{CD}
 \mathbb S^r @>>f\_0> M\\
 @VViV @VV{\phi}V \\
 \mathbb D^{r+1} @>>Q> X \\
\end{CD}
commute for some $Q:\mathbb D^{r+1}\to X$, *then we can perform the surgery step*, i.e. denoting by $W^{m+1}$ the trace of the surgery along $f$, $\phi$ extends to $W$, yielding $ \phi\_W: W\to X$, and also the trivialization $F$ extends to a stable trivialization of $TW\oplus \phi\_W^\*\nu$.
In other words, we obtain a cobordism of the normal maps.


**Question**
The proof of this fact takes a few lines in the book (pg. 11, 3rd and 4th paragraph) and
relies on the fact that $TW\oplus \phi\_W^\*\nu$ restricted to the handle is trivial (the handle is contractible) and that this trivialization coincides with that induced by $F$ so that the two glue to a trivialization over $W$.



> 
> Why the two trivialization coincide?
> 
> 
> 


I don't see any good reason why the two trivialization should agree in general. Once a trivialization is chosen the other one defines an element in $\pi\_r(O(N))$ with $N>>r$, but these homotopy groups do not vanish in general if $r$ is congruent to $0,1,3,7$ modulo $ 8$ (stabilizing does not seem to help).
So we really need the two trivializations to be related in a special so that the element defined in $\pi\_r(O(N))$ is trivial. On the other hand, the choice of $F$ seems to be too arbitrary to me.
Does somebody see why the two coincide?



On the proof of the surgery step in Wall's book
The theorem has the hypothesis "$f$ is in this class", meaning that the embedding $f$ is in the regular homotopy class of immersions determined by $F$ together with the element of the relative homotopy group $\pi\_{r+1}(\phi)$ that is given by the maps $f\_0$ and $Q$. This hypothesis actually says that the two stable trivializations are the same.


I think of it like this. $F$ is a stable trivialization of $TM\oplus\phi^\ast\nu$. The bundle $f\_0^\ast TM$ gets a stable trivialization from the way in which $f\_0$ is extended to an embedding $f$ of the handle. The bundle $f\_0^\ast \phi^\ast\nu$ gets a trivialization from the way ($Q$) that $\phi\circ f\_0$ is extended to $D^{r+1}$. The resulting stable trivialization of $f\_0^\ast ( TM \oplus \phi^\ast \nu)$ must be assumed to coincide with $f\_0^\ast F$.


All this stuff about regular homotopy classes of immersions is just a slightly indirect way of saying that.


