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.
Let be a surface and let be the one-skeleton of a triangulation of . In other words, is a graph embedded in so that its complement is a collection of triangles. If is a simple closed curve in then we can isotope to be transverse to (so that is a finite number of points in the edges of ). In fact, we can do even better: If ever intersects the same edge of twice in a row, then the arc of between these two points will be contained in a triangle, and moreover will cut off a disk in that triangle. (Such an arc is called a bent arc because it cannot be drawn by a straight line in the interior of the triangle.) If we isotope this arc of across this disk and into the adjacent triangle, we will strictly reduce the number of points of intersection . Note that we can isotope an arc with endpoints in different edges out of the triangle, but this will usually increase the number of points of intersection. If we each bent arc in this way, we will eventually isotope so that each arc of has its endpoints in distinct edges of . We will say that such an arc is normal.
If, after this isotopy, is disjoint from then is contained in a triangle and is trivial (bounds a disk by the Jordan Curve Theorem). Otherwise, we will say that is a normal loop. Note that in each triangle, there are exactly three types of normal arcs, defined by the pair of edges that contain their endpoints. Each edge of the triangulation contains the endpoints of four different classes of normal arcs, two in each of the adjacent disks and the total numbers of normal arcs on each side must be equal, since is a loop. So every normal loop determines a vector whose entries are the number of normal arcs in each class. (So, these are dimensional vectors where is the number of faces in the triangulation.) Because the arcs must match up along the edges, they all satisfy a collection of linear equations of the form where are the numbers of normal arcs on one side of an edge and are the numbers of arcs on the other side.
Conversely, if we choose an integer vector with positive entries that satisfies this set of linear equations, we can draw the corresponding normal arcs in the surface and there will be a unique way to match up the endpoints to form a closed one-manifold in . So this means that there is a one-to-one correspondence between normal loops and vectors in with all positive entries that lie in the subspace defined by the linear equations. (This is called the positive cone of the subspace.)
This vector structure on the set of normal loops turns out to be very nice. For example, we might translate a given normal loop into a vector, then multiply the vector by some integer , then translate the new vector back into a normal loop. In this case, the resulting one-dimensional manifold will (usually) correspond to copies of the original loop. (For one-sided loops in non-orientable surfaces, it’s a little more complicated.) But what happens if we add two of these vectors? If and are vectors corresponding to disjoint normal loops then the vector will correspond to the union of the two normal loops. However, if the original loops intersect non-trivially, their union is not an embedded one-manifold.
In particular, if we look at the normal arcs of these loops inside a given triangle in , the normal arcs will intersect. However, if we shuffle the endpoints of these arcs, we can make them pairwise disjoint. In the process of shuffling the endpoints, we will break the original loops then reattach them to form some new one-manifold. (It may or may not be a single loop.) There will always be exactly one way to do this and the result is called the Haken sum of the original loops.
Note that while every essential loop can be isotoped to a normal loop, not every normal loop is necessarily essential. In particular, a small trivial loop around any vertex of the triangulation will be normal. If the triangulation has two or more vertices, one can construct very complicated normal loops that are topologically trivial, as the boundary of a regular neighborhood of a complicated arc between the two vertices. Moreover, (as if this weren’t bad enough) an isotopy class of essential loops will generally have a lot of different normal representatives, for similar reasons. In order to correct these problems, we need to introduce train tracks, and this will be the subject of my next post.