On the proof of the surgery step in Wall's book

Question

This question regards a part of the proof of the so called surgery step, in 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 $f:\mathbb S^r\times \mathbb D^{m-r}\to M$ 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?

Answer 1

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.