I'm self studying from Lee's Introduction to smooth manifolds, and I'm unsure about my solution to problem 10.10.
Given a rank $k$ vector bundle $E\to M$, with $M$ compact, I'm searching for a smooth global section which is transverse to the embedded submanifold $\zeta(M)\subseteq E$, where $\zeta:M\to E$ is the zero section. The transversality homotopy theorem gets me a smooth map $\sigma$ homotopic to $\zeta$ which is transverse, but it need not be a section.
Could you take a look at this proof which attempts to construct such a section from $\sigma$? The gist of it is to take $\sigma(p)$ and leave its vector components alone, while forcing its manifold components to be $p$.
Consider $d\sigma_p(T_pM)$. By transversality, its direct sum with $T_{\sigma(p)}\zeta(M)$ is $T_{\sigma(p)}E$. For this to remain true, we don't need any components of $d\sigma_p(T_pM)$ which are already represented in $T_{\sigma(p)}\zeta(M)$. Moreover, there exist coordinates for $T_{\sigma(p)}E$ such that $T_{\sigma(p)}\zeta(M)$ lies in the first $m=\dim M$ coordinates.
So, let $\hat\sigma$ be the map such that $(d\hat\sigma_p(T_pM))^i=0$ if $1\leq i\leq m$ and $(d\hat\sigma_p(T_pM))^i = (d\sigma_p(T_pM))^i$ if $m<i\leq m+k$, in some coordinates. This corresponds to $\hat\sigma(p) = (p, v)$, with $v$ as in $\sigma$. This is smooth because each component is smooth, transverse to $\zeta(M)$, and it's clearly a global section.
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My hesitation is simply whether I can go ahead and define $\hat\sigma$ component-wise without worrying about whether the definition is coordinate-independent. At the same time, the definition certainly looks coordinate independent. I'm not sure how I'd go about verifying that this step is ok? Intuitively, I'm not doing anything with $v$, so the modification to $\sigma$ is coordinate independent. But formally, I can't seem to avoid relying on a choice of slice chart.
My other thought on an approach would be that we have a homotopy $F$ from $\zeta$ to $\sigma$, almost all maps $F_t$ are also transverse to $\zeta(M)$. But it may be that for all $t\neq 0$, $F_t$ is not a section, so searching for a transverse section that way doesn't work out, unless I somehow dig into the proof of the transversality homotopy theorem and use the $F_s$ which has some more dimensions to work with. That feels like overcomplicating something though. It's also possible that $\pi\circ\sigma(p)\neq p$ for any $p$, so I can't get anything out of the closed set where $\sigma$ is a section. I found a use for compactness of $M$ in a different part of the problem, I don't think it's relevant to this part.