Factoring maps of handlebodies

2009-11-18T20:07:47Z

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!

Answer by Sam Nead for Factoring maps of handlebodies

2009-11-19T01:21:42Z

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 $V$.

Factoring maps of handlebodies - MathOverflow

Factoring maps of handlebodies

Answer by Sam Nead for Factoring maps of handlebodies