Volume 12: The Poincaré Conjecture


The Klein Bottle

A Tale of Rubber Bands and the Shape of the Universe

“Marty, it’s perfect! You’re just not thinking fourth-dimensionally!” – Emmett Brown

 

What is the shape of the universe? Are we even allowed to ask?

We know a lot about our universe. Painstaking work has taught us much about what’s in it, how it began, how it is changing and even how it might end. A century of study of our existence on the vastest scales – Relativity – and the smallest scales – Quantum Mechanics – have given us a glimpse of what’s out there. But we have more questions today than ever before.

But our story doesn’t begin in the present. It begins in France at the end of the 19th century. It begins with Henri Poincaré, often considered the last universal mathematician. It begins with a piece of pure mathematics from the obscure discipline of topology.

  • What is topology?

  • What is the Poincaré Conjecture?

  • What is the shape of the universe?

 

What is topology?

The last 150 years could be broadly described as an attempt to make mathematics more general – or to use the mathematical term, abstract. Number theory is the abstract version of arithmetic. Analysis is the abstract version of calculus. Abstract algebra is the abstract version of…well…algebra. Topology is the abstract version of geometry.

Topology is geometry on steroids.[1] You remember geometry – 9th grade, compass, straightedge, step-by-step proofs. In geometry, you were concerned with sizes, shapes and distances. A circle was the collection of all points equidistant from its center. An equilateral triangle had three sides of equal length.

Unhelpfully, Wikipedia defines topology as “the study of…topological spaces.” More helpful is to think of topology as geometry, but where you are allowed to bend or stretch objects as long as you never tear or glue. It’s like rubber or Play-Doh. In topology, a square can be smoothed (carefully) to make a circle. They are therefore the same thing in topology.[2] But they are both different from a figure “8,” which you can only get to by pinching the center and gluing. Similarly, a donut and coffee mug are equivalent to each other, but both are different from a sphere or a cube.

Another good example to describe topological equivalence is the alphabet. In the below table, each “class” of letters is equivalent to all of its class’s members (and not equivalent to members of any other class). Each class can be described by the number of “holes” and “tails” of its members. Of course, the classes could be a bit different depending on your font.[3]

Shapes that have different numbers of holes or tails can never be equivalent. Shapes with the same number of holes and tails can be different, “K” and “X”. In the latter, the four tails meet at a central point, where in the former there is a small connector. Equivalence is transitive. In other words, if A is equivalent to B, and B is equivalent to C, then we know that A is equivalent to C. I’ll leave the proof of this to the reader.

Topologists love to think and talk about properties of objects. In topology, we only care about those properties that are preserved among all objects that are equivalent. We saw some examples of this above – number of holes and tails are examples of topological properties. However, “has corners” is not a valid property in topology, because a square meets the criteria and a circle does not. Because it is not conserved between these equivalent shapes it is not a property in topology.[4]

We are going to work with some of these properties, so it makes sense to define them and give some examples. Note: these definitions are all non-technical – feel free to look them up on your own if you want the real meaning.[5]

Manifold: If an object “looks flat” when you zoom in on any small portion, it is a manifold. The surface of the Earth is a 2-dimensional manifold; when you stand on the street it appears flat, the curvature is too far away to see. The letter “X” is not a manifold; if you are at the intersection, there is no way to make it look flat (without tearing or gluing, which are prohibited in topology). A manifold with n dimensions is called a n-Manifold.

Boundary: The points you can get to both from inside and outside of an object are its boundary. In the below example, if the light blue area is the object, the dark blue is the boundary.