By Megan MacGregor Math 10 H Topology: Homeomorphism, Euler’s Characteristic, and the Hairy Ball Theorem By Megan MacGregor Math 10 H

Topology For my inquiry project, I wanted to find a field of mathematics that I really didn’t know anything about. Through my searching, I found the topic of (click) Topology. Topology is the study of (click) objects that are unchanging through transformation. (click)

Topology was first written about by mathematician (click) Euler Topology was first written about by mathematician (click) Euler. His paper published in 1736 on (click) The Seven Bridges of Koonis-bare-g revealed the answer-or lack thereof- to the problem. His work in this newly discovered field led to advances in mathematics that had never been seen before. Euler understood that this new field of mathematics had (click) nothing to do with measurement, but more so with position. The first use of the word topology came in 1847 from German mathematician Johann Benedict Listing who got many of his ideas from famed mathematician Johann Gauss.

Two objects that can be morphed from one to the other and back without creating any holes or tears in the surface are referred to as homeomorphic. As shown in this (click) gif, the infamous ‘coffee cup or donut?’ . (click)

Basic Topology Topology is a very large field of mathematics. It stretches from algebra to applied mathematics to geometry and beyond. The basic ideas of topology lie in the bending and stretching of topological spaces, or as we say in regular English, objects. In this diagram, we see two loops, which are topologically similar because they can both be stretched out to form the same red ring. This is known as homeomorphism. (click)

Homotopic equivalence Homeomorphic Homotopic equivalence Homeomorphism is a fundamental part of topology. Formally, homeomorphism is the one to one ratio between points of two different objects. This means having the same relative size and number of holes. Homeomorphism also includes topological properties in order to be true. Topological properties are properties that are preserved via homeomorphism. If we have set A and B and set A has topological property C but set B does not, then sets A and B are not homeomorphic. When talking about topology, Homotopy is the functional counterpart of homeomorphism. Where two objects may be homeomorphic, the function of the transformation between them is called homotopy equivalence. (click) Keep in mind, a homeomorphic object does not necessarily have homotopy equivalence. For instance, the Mobius strip has homotopy equivalence to a circle. It can be squished into a line and the twist disappears and it becomes a circle. But the Mobius strip is not homeomorphic to a circle because of the twist in the strip. (click)

Mobius Strip The Mobius strip is one of the more well-known objects from Topology, for its simplicity and fascinating characteristics. It has only one side, one edge, and is completely un-orientable in that there is no determinable front or back. The Mobius strip has a Euler characteristic of zero. Meaning that using Euler’s formula for polyhedrons: (click)

Euler’s Characteristics Where V is number of vertices, e is the number of edges, and f is the number of faces, equalling X, Euler’s characteristic. (click)

The Hairy Ball Theorem The hairy ball theorem explores another popular idea in the field of topology. This particular theorem,(click)

invented by Henri Poincare in the 19th century and proved by L E J Brouwer in 1912, dictates that any continuous tangent vector field on the sphere must have a point where the vector is zero. This translates into somewhat real life scenarios as well. On the earth, there is constant wind flow, however, all this wind must flow around some point. (click)

These points are the spots where the vector is zero, and there is no wind. In this diagram this sticking up point would be that no wind spot. As the hairy ball theorem goes, ‘you cannot comb a hairy ball flat without creating a cowlick’ thus, you cannot comb all the hairs on a ball down without creating a part that is sticking up. (click)

Everything I’ve talked about relates to the vast field of topology Everything I’ve talked about relates to the vast field of topology. Something that, throughout this project, I have become more interested in, and would like to research further in the future.