Directed arcs on a surface - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T12:20:22Z http://mathoverflow.net/feeds/question/68765 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68765/directed-arcs-on-a-surface Directed arcs on a surface Greg Muller 2011-06-24T20:39:45Z 2011-06-24T20:39:45Z <p>This question is a little odd. I have specific class of structures on a surface, which satisfy several nice properties, and I want to know if they are more natural geometric structures in disguise (my guess is a spin structure with some extra information).</p> <p>Let $S$ be a connected, oriented surface (possibly with boundary $\partial S$). Let $M$ be a finite collection of marked points, such that every component of $\partial S$ has at least one marked point. Define an <strong>arc</strong> in $S$ be an smooth, unoriented curve in $S$ with endpoints in $M$; considered up to endpoint-fixing isotopy (we prohibit contractible paths).</p> <p>I now introduce an odd-seeming structure on $S$, which I believe corresponds to some intrinsic geometric information (but I don't know what). A <strong>directing</strong> of $S$ assigns an orientation to each arc in $S$; equivalently, it chooses an endpoint to be the 'source'. A directing is <strong>clockwise</strong> if, for any three arcs $a,b,c$ which bound a triangle in $S$, an <em>odd</em> number of $a,b,c$ are oriented clockwise around the interior.</p> <p><strong>Example.</strong> Consider the disc with $n$ marked points, all on the boundary. Starting with any marked point and moving counter-clockwise, number the marked points $1$ through $n$. Then there is a clockwise directing such that an arc goes from $p$ to $q$ whenever $p &lt; q$.</p> <p>Clockwise directings satisfy several remarkably nice properties.</p> <ul> <li>If $\partial S\neq \emptyset$, clockwise directings always exist. (The condition on the boundary cannot be removed)</li> <li>A clockwise directing is completely determined by its restriction to the arcs in a triangulation of $S$. In fact, any choice of orientation of the arcs in a triangulation which satisfies the 'clockwise' condition determines a unique clockwise directing.</li> <li>There are finitely many clockwise directings.</li> <li>Any two clockwise directings may be related by a sequence of elementary moves. (1) For a given marked point $p$, reversing the orientation of each arc which has one endpoint at $p$. (2) for a given loop $l$ in $S$, reversing the orientation of each arc which intersects $l$ an odd number of times.</li> </ul> <p>For these reasons and others, I believe that this contrived structure is equivalent to a more natural geometric structure. Thus, my question is, <Blockquote> Is there a class of geometric structures on $S$ such that a choice of clockwise directing is equivalent to a choice of such a structure?</Blockquote></p> <p>Specifically, I suspect that this corresponds to an extended version of a spin-structure. That is, I conjecture that a clockwise directing on $S$ is equivalent to a choice of spin-structure on $S$, together with some extra information at the marked points. However, I know very little about spin structures, and so I have no idea if there is a natural way to add extra information at a marked point.</p>