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!

anyfactorization, and worry about whether the homotopy can be made nice later. – HJRW Nov 18 '09 at 23:45