# Probing a manifold with closed curves

Since fedja's excellent comment on Joseph's question on probing a manifold with geodesics remained uncommented (especially by topologists), I'd like to make a question out of it:

Conjecture: Given an orientable 2-dimensional manifold and two closed curves on it which intersect transversally in exactly one point. Then the two curves cannot be homotopic.

(An immediate consequence of this would be that living on a surface with two such curves, one would know, that it is not homeomorphic to the sphere.)

How to proof this conjecture (if it's true)?

-
Counting signed intersections of curves gives a symplectic form on the first homology group of a closed surface. In particular, in such a surface two homologous curves never intersect transversally in exactly one point. –  Mariano Suárez-Alvarez Mar 14 at 17:42
Did you omit the word "orientable" from your conjecture? It's false as it stands. –  maproom Mar 14 at 17:43
(My comment refers to orientable surfaces, as well!, as otherwise there are no signed intersection numbers to get us started) –  Mariano Suárez-Alvarez Mar 14 at 17:50
@maproom: Indeed, I had only orientable surface in mind. So I added it. Thanks for the correction! –  Hans Stricker Mar 14 at 17:51
If you really want to learn about curves on surfaces, I recommend "Travaux de Thurston" by Fathi, Laudenbach and Poeneru. If you are averse to reading French, Magalit and Kim have translated it into English ( Work of Thurston). Exposes 3 and 4 are the real stuff. –  Charlie Frohman Mar 15 at 2:38

I wanted to give a somewhat "low-tech" answer to this, here's the best I came up with:

Call the curves $C_1$ and $C_2$ and the surface that they lie in $\Sigma$. Take a small open neighborhood $N$ of $C_1 \cup C_2$. What does $N$ look like? Well, it's a neighborhood of $C_1$ stuck to a neighborhood of $C_2$ in an operation called a "plumbing" (which is the same operation as sticking together two strips of paper to make a cross).

The neighborhood of $C_i$ is a cylinder that has been plumbed to itself a few times (one plumbing for each self-intersection of $C_i$).

Now if $C_1$ can be homotoped to coincide with $C_2$ inside $\Sigma$ then it can be homotoped to $C_2$ inside the space $S$ that I make by crushing $\Sigma \setminus N$ to a point.

The idea is to convince yourself that $S$ is a torus with a finite number of points identified (one point for each boundary circle of $N$), and that $C_1$ wraps once around one direction of the torus while $C_2$ wraps once around the transverse direction.

-
Thanks for all the upvotes, but I've realized that my argument is not correct. The space space $S$ is not necessarily a torus with some points identified but just some genus $g \geq 1$ surface with a finite number of points identified. It's fixable though. Orient $C_i$ and choose an oriented loop $P_i$ on $C_i$ with no self-intersections such that $P_1$ and $P_2$ intersect once. Then crushing the complement of a neighborhood of $P_1 \cup P_2$ gives an honest torus such that $C_1$ wraps once in one direction and $C_2$ wraps once in the other direction. –  Andrew Lobb Mar 15 at 15:34

There are exactly two one plane bundles over the circle, the open annulus, and open Moebius band, where we take projection to be onto the central circle. We can now ask what the mod 2 self intersection of the zero section is. In the case of an annulus it is zero, in the case of the Moebius band it is one.

The tubular neighborhood theorem says that any smooth submanifold of a smooth manifold has a neighborhood homeomorphic to a k-plane bundle over the submanifold, where k is it's codimension.

You can't embed a Moebius band in an orientable surface, so the self intersection of any simple closed curve in an orientable surface is zero. The self intersection number is invariant under homotopy, so if gamma and gamma' are homotopic simple closed curves in a surface that intersect each other transversely then they intersect in an even number of points.

-
It took me a few seconds to realize that "one plane bundle" is the same as a "Line bundle" in your first line :-) –  Mariano Suárez-Alvarez Mar 14 at 19:03
I only begin to learn that and why orientability is crucial for my question. So excuse me for not having mentioned orientability in my original question.) –  Hans Stricker Mar 14 at 19:05
@Mariano: I grinned over your "few seconds" - for others it takes a few years! –  Hans Stricker Mar 14 at 19:07