Description

An immersion of a circle into the plane is a C^1-map with no-where vanishing first derivative. Such a map does not need to be injective, that is, its image is a circle which can have self-intersections.

It is an interesting question whether, given two such immersions, one can be deformed into the other through a one-parameter family of immersions. Such a deformation is called a regular homotopy. There is the notion of winding number for an in immersion of a circle into the plane. It measures, algebraically, how often the tangent vector in the sense of travel winds counterclockwise around itself, as one follows the immersed circle.

The winding number remains constant under a regular homotopy. The Whitney-Graustein theorem states that the converse also holds: If two immersed circles in the plane have the same winding number, then they are regular homotopic.

One can restate the Whitney-Graustein theorem in that the winding number is the only obstruction to the existence of a regular homotopy. For the same problem, but one dimension up - the immersion of 2-dimensional spheres in the 3-dimensional space - Stephen Smale proved that there is no such obstruction - any two immersed 2-spheres are regularly homotopic. In fact, there exists sphere eversions, meaning that one can turn a 2-sphere inside out.

This is a beautiful area of mathematics where one uses differentiable methods to prove results about topological spaces and their immersions in other topological spaces. The aim of the project is to get familiar with methods from differential topology, and to understand a proof of the Whitney-Graustein theorem, and maybe understand aspects of the sphere eversion. While these two topics are now more than half a century old, the methods of differential topology are standard tools in contemporary research in topology and therefore a worthwhile topic to study.

The problem of sphere eversion and the Whitney-Graustein theorem are nicely illustrated to a not-yet-topologist in the following video from 1994.

• Regular homotopies in the plane (part I)
• Regular homotopies in the plane (part II)

Prerequisites

Resources

• The Whitney-Graustein (Wikipedia) gives a rough idea of the topic, but it considers more general cases.
• Sphere eversion (Wikipedia). This discusses Smale’s proof of the existence of sphere eversion.
• A contact geometric proof of the Whitney-Graustein theorem by Hansjoerg Geiges. This is a modern proof of the Whitney-Graustein theorem using the theory of contact geometry from 2008.