Train tracks

A few posts back, I defined normal loops in the triangulation of a surface and said I would use this idea to define train tracks on a surface. The key property of normal loops is that the normal arcs form parallel families and we can encode the topology of the curve by keeping track of how many parallel arcs are in each family. Train tracks encode loops in a surface in a very similar way. A train track is a union of bands in the surface (disks parameterized as ) with disjoint interiors, but that fit together along their horizontal sides. In other words, the top and bottom edges of each band are contained in the union of the horizontal edges of other bands. A picture of this is shown below the fold.

Normal loops in surfaces

I plan to write a post or two about normal surfaces and branched surfaces in three-dimensional manifolds, but I want to warm up first, with two posts about the two-dimensional analogues of these objects. Train tracks play a huge role in the approach to the topology of surfaces initiated by Nielsen and Thurston, for understanding mapping class groups, Teichmuller space, laminations, etc. They organize the set of isotopy classes of simple closed curves in a surface in a way that allows one to take limits of infinite sequences of loops. (The limits are called projective measured laminations.)  In this post and the next, I will discuss train tracks from a rather unusual perspective, via normal loops in a triangulation of the given surface.