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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.

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!

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# Factoring maps of handlebodies

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.

By an inclusion of handlebodies, I mean that the new one should be obtained from the old by attaching 1-handles.