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


1. a finite sequence of folds;
2. an inclusion; and
3. 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:


1. a compression (by which I mean a map of a handle into the complement of its interior);
2. an inclusion; and
3. 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](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=RT&pg7=ALLF&pg8=ET&review_format=html&s4=stallings&s5=finite%20graphs&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=695906).


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!



Factoring maps of handlebodies
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 $V$.


