I am currently stuck on Problem 8-18 (b) in Lee's Introduction to Smooth Manifolds.
We have a smooth submersion $F: M \to N$ and a smooth vector field $Y$ on $N$. The problem in part (b) asks to show that if $\dim M \neq \dim N$, then $Y$ has a lift, but the lift is not unique. I am struggling to show the existence of a lift globally.
If we assume that $\dim M = \dim N$ instead (this is part (a) of the problem) then it is clear that $dF_p: T_pM \to T_pN$ is an isomorphism for all $p \in M$ so we can define the lift of $Y$ by $X_p = dF_p^{-1}\left(Y_{F(p)}\right)$ for all $p$ in $M$. By the rank theorem, there are smooth charts $(U, \varphi)$ for $M$ containing $p$ and $(V, \psi)$ for $N$ such that $F(U) \subseteq V$ and $\psi \circ F \circ \phi^{-1} = \text{id}$. Then working in the coordinate basis $\frac{\partial}{\partial x^i}$ associated to $(U, \varphi)$ and $\frac{\partial}{\partial y^i}$ associated to $(V, \psi)$ and using the Jacobian of $dF_p$ which is just $I_n$ in this case, we see that $X^i = Y^i \circ F$ so the components of $X$ are smooth, and so $X$ is a smooth lift of $Y$.
This argument fails when $\dim M \gt \dim N$ because there is no canonical way to choose a lift of $Y_{F(p)}$. We now have more than one choice for picking an element in $dF_p^{-1}\left(Y_{F(p)}\right)$ since $dF_p$ is not injective. And there is no clear way to make that arbitrary choice behave smoothly over all $p \in M$.
One idea I had was that by the local section theorem, for each $p \in M$, we can choose a smooth local section $\sigma_p: V_p \to M$ of $F$ defined on a neighborhood $V_p$ of $F(p)$ in $N$ whose image contains $p$, and define $X_p = d\left(\sigma_p\right)_{F(p)}\left(Y_{F(p)}\right)$. However it is not clear that this defines a smooth vector field because if we want to apply the gluing lemma to stitch together the smooth vector fields $d\sigma_p \circ Y$ defined on $V_p$, we need to know that $d\sigma_p \circ Y$ agrees with $d\sigma_{p'} \circ Y$ on their overlaps $V_p \cap V_{p'}$.
Another approach is to work in coordinates as in part (a) and try to define $X$ explicitly by $X = \sum\limits_{i=1}^n (Y^i \circ F) \frac{\partial}{\partial x^i}$ on each chart. But again it is not clear how to show that definitions agree on overlaps between different charts. For example, you could have a situation where $X_{p'} = \sum\limits_{i=1}^n (Y^i \circ F) \frac{\partial}{\partial x^i}\bigg\vert_{p'} + \sum\limits_{i=n+1}^mc_i\frac{\partial}{\partial x^i}\bigg\vert_{p'}$ for some $p'$ arbitrarily close to $p$, where the nonzero terms $c_i$ for $i \gt n$ arise from the change of coordinates when writing $\frac{\partial}{\partial {x'}^i}\bigg\vert_{p'}$ in terms of $\frac{\partial}{\partial x^i}\bigg\vert_{p'}$.