Factoring maps of handlebodies Any map of finite graphs (1-dimensional CW-complexes) factors as a composition of


*

*a finite sequence of folds;

*an inclusion; and

*a finite-to-one covering map.


There should be a corresponding result for handlebodies, which presumably should say that, after a homotopy, a continuous map of handlebodies factors as:


*

*a compression (by which I mean a map of a handle into the complement of its interior);

*an inclusion; and

*a finite-to-one covering map.


Is my intuition correct, and does anyone have a reference?  I'm specifically interested in how well-behaved the homotopy can be taken to be.  For instance, can it be made to respect the boundary?
Notes
A fold is a map that identifies two edges with a common endpoint.  Many folds don't change the homotopy type of a graph, and one would expect not to need these in the handlebody setting.  The important folds are the ones that kill a loop.  In handlebody terms, you can think of this as gluing in a two-handle, or as cutting a one-handle - hence my use of the word "compression".  Is this word acceptable in this context?
The graph-theoretic result is due to Stallings.
By an inclusion of handlebodies, I mean that the new one should be obtained from the old by attaching 1-handles.
EDIT (prompted by Sam's comments below) I'm not quite sure what "respect the boundary" should mean, at this point.  Suggestions welcome!
 A: Suppose that we are given a PL map from a handlebody $W$ to a handlebody
$V$.  Choose a spine for $W$.  Homotope the map until the image is a
regular neighborhood of the image of the spine.  By general position,
our map is now an embedding.  Fix a pants decomposition of disks $D =
(D_i)$, for $V$.  Suppose that $P$ is a component of $V - D$ (so $P$ is a
three-ball with three distinguished disks on its boundary).  Consider a 
component $X$ of $f(W)
\cap P$.  This is essentially a knotted graph.  Via a homotopy (keeping
$X \cap D$ fixed) unknot $X$.  If the rank of $X$ is positive then another 
homotopy produces small lollipops which we shall compress a bit later.
If $X$ meets any disk $D_i \subset \partial P$ more than once 
then we may homotope a leg of
$X$ through $D_i$.  Let $Y$ be the resulting component of $f(W) \cap
P$. (Note that this reduces $f(W) \cap D$.)
Homotoping in this fashion we eventually arrive at an embedding of $W$
so that every component in every three-ball of $V - D$ is either a
tripod or an interval, possibly with lollipops attached.  The feet of
the tripod/interval lie in distinct disks in the boundary of the
containing solid pants. 
Now compress all of the lollipops to get $f'(W')$ (a new handlebody,
because we compressed and a new map because we have to extend it over
the two-handles we added).  
EDIT:
This reproduces, in our context, part of Stallings paper (eg sliding 
the leg is a fold, arriving at only tripods and intervals produces 
an immersion.)  
Since $f'$ is an immersion, it follows from Stallings paper that $f'$ 
is $\pi_1$ injective and that $W'$ embeds into a finite cover of $V$.
