gt.geometric topology – On the proof of the surgery step in Wall’s book Answer

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gt.geometric topology – On the proof of the surgery step in Wall’s book

This question regards a part of the proof of the so called surgery stepin Wall’s book “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 $f:\mathbb S^r\times \mathbb D^{mr}\to M$ is on embedding, $m\geq r+2$ other $f_0=f|_{\mathbb S^r\times\{0\}}$ makes this diagram commute:

$\require{AMScd}$
\begin{CD} \mathbb S^r @>>f_0> M\\ @VViV @VV{\phi}V \\ \mathbb D^{r+1} @>>Q> X \\ \end{CD }
for some $Q$, then we can perform the surgery stepie 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 trivializations coincide?

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