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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