Factoring maps of handlebodies - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:56:19Z http://mathoverflow.net/feeds/question/6006 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6006/factoring-maps-of-handlebodies Factoring maps of handlebodies HW 2009-11-18T20:07:47Z 2009-11-19T17:48:46Z <p>Any map of finite graphs (1-dimensional CW-complexes) factors as a composition of</p> <ol> <li>a finite sequence of folds;</li> <li>an inclusion; and</li> <li>a finite-to-one covering map.</li> </ol> <p>There should be a corresponding result for handlebodies, which presumably should say that, after a homotopy, a continuous map of handlebodies factors as:</p> <ol> <li>a compression (by which I mean a map of a handle into the complement of its interior);</li> <li>an inclusion; and</li> <li>a finite-to-one covering map.</li> </ol> <p>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?</p> <p><strong>Notes</strong></p> <p>A <em>fold</em> 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?</p> <p>The graph-theoretic result is due to <a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=TI&amp;pg6=RT&amp;pg7=ALLF&amp;pg8=ET&amp;review%5Fformat=html&amp;s4=stallings&amp;s5=finite%20graphs&amp;s6=&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=2&amp;mx-pid=695906" rel="nofollow">Stallings</a>.</p> <p>By an <em>inclusion</em> of handlebodies, I mean that the new one should be obtained from the old by attaching 1-handles.</p> <p><strong>EDIT</strong> (prompted by Sam's comments below) I'm not quite sure what "respect the boundary" should mean, at this point. Suggestions welcome!</p> http://mathoverflow.net/questions/6006/factoring-maps-of-handlebodies/6047#6047 Answer by Sam Nead for Factoring maps of handlebodies Sam Nead 2009-11-19T01:21:42Z 2009-11-19T17:48:46Z <p>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 $f(W) \cap D$.)</p> <p>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. </p> <p>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). </p> <p>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.) </p> <p>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 $V$.</p>