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][1].

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!





  [1]: 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