TITLE: How to "globalize" the inverse function theorem?
QUESTION [7 upvotes]: Let $F: V \times W\rightarrow Z$, where $V,W,Z$ are finite-dimensional smooth (or analytic) manifolds and $F$ is smooth (or analytic). Assume that $\dim W=\dim Z$ and the usual inverse function theorem applies when we fix the first argument of $F$, so locally we have an inverse (w.r.t. the second argument of $F$) mapping $f: V \times Z \rightarrow W$ such that for any $v,w,z$ from suitably chosen small vicinities we have $F(v,f(v,z))\equiv z$ and $f$  is smooth (or analytic). Here comes the question:

Is there any sensible way to glue together these $f$'s so that we can speak of them globally, and under which conditions?
More precisely, is there a way to treat this situation "as if" $f$ were defined globally, without doing the "restrict ourselves to suitably chosen small vicinities" thing all the time, even if we allow the Jacobian of $F$ w.r.t. $w$ to vanish at some points or even on some submanifolds of $V\times W$ of nonzero codimension?

I do not quite expect that (under reasonably mild conditions) one can construct
a global $f$ defined on the whole $Q=Z\times V/Sing$ where $Sing=\lbrace(v,z)\in V\times Z|f \quad \mbox{can't be reasonably defined}\rbrace$ but perhaps one can have a sheaf of such $f$'s on $Q$ or something of the sort? I just started with learning the sheaf theory, so I don't quite (yet) know how to make things work on my own.
While in my particular application I would like $V$ to be a manifold, if this makes things easier, I can quite well make do with $W$ and $Z$ being just open domains in $\mathbb{R}^n$ or $\mathbb{C}^n$ (or whole $\mathbb{R}^n$ or $\mathbb{C}^n$ for that matter); that's the main reason why I wrote the domain of $F$ as $V\times W$ rather than just some general manifold $M$.
In fact, I am quite convinced that things like that should have been treated in the literature but I wasn't able to google up anything reasonable so far (maybe I just don't know the right keywords), so pointing out any suitable references is most welcome.

REPLY [8 votes]: There exist several known global implicit function theorems. Those results tend to be tailored for specific applications. It seems to be rather difficult to state a universally useful one-size-fits-all version.
One result that I find particularly helpful goes back to Hadamard (see, e.g., Chapt. 6 of "The Implicit Function Theorem" by Krantz and Parks):  

Theorem. Let $M$ and $N$ be smooth, connected manifolds of dimension $d$ and let $f:M\to N$ be a $C^1$ mapping. If

$f$ is proper (i.e. $f^{-1}(K)\subset M$ is compact whenever $K\subset N$ is compact),
the Jacobian of $f$ vanishes nowhere on $M$, and
$N$ is simply connected,

then $f$ is a homeomorphism.

You might be interested also in this paper by Rheinboldt. It contains some topological conditions on when the local solvability of the equation
$$F(x,f(x,z))=z$$
leads to the global solvability.