Description

Did you know that a theorem of topology implies that on the surface of the earth at any time there exist two points exactly opposite of each other at which reign the same temperature and the same air pressure? The conclusion holds under reasonable modelling assumptions - that both temperature and air pressure at ground level depend continously on the point, and that the surface of the earth is a 2-dimensional sphere. Did you know that a continuous map from an n-dimensional ball to itself always admits a fixed point? Did you know that you can’t comb a hairy ball so that all hairs are tangent to a ball? There will always be some hair standing up away from the surface? The first statement is a consequence of the Borsuk-Ulam theorem, the second is known as the Brouwer fixed point theorem, and the third is called the hairy ball theorem.

The theorems are typically proved using algebraic topology which you can learn in a 4th year course at Durham. This is a mathematical theory making extensive use of what is called homological algebra. It is less known that these theorems can also be proved using a different method, namely that of topology from the differentiable viewpoint. Here one studies maps which are not just continuous, but differentiable in a suitable sense, and which have domain and target what is called a differentiable manifold.

In this project you will learn how to obtain topological information by such differentiable methods. You will see how to prove the above mentioned theorems, and as we move further on we will study the Pontryagin-Thom construction with applications to the homotopy groups of spheres. These are abelian groups which measure whether maps can be continuously deformed into another.

This is a beautiful area of mathematics where one uses differentiable methods to prove results that otherwise one can only prove using the machinery of algebraic topology. While that theory has its justification and is in some sense more powerful, and at least more general, many aspects of topology become simpler and more intuitive in the differentiable context. While one of our main references, Milnor’s book ``Topology from the differentiable viewpoint’’ (linked below) is 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 project can be beneficial to be taken in parallel to the course Algebraic Topology IV in order to learn two different methods with similar applications, but it is at the same time completely independent from it.

Prerequisites

Resources

• Topology from the differentiable viewpoint by John Milnor, 1965.
• The Borsuk-Ulam theorem on Wikipedia.
• The Brouwer fixed point theorem on Wikipedia.
• The hairy ball theorem on Wikipedia.
• Homotopy groups of spheres on Wikipedia.
• Differentiable manifold on Wikipedia.