So two questions initially probably popped out to you when you read that.
- What does it mean for something to be contractible?
- The fuck is an infinite sphere?
Let's tackle contractability first. Roughly speaking, for something to be contractible you need to be able to “deform” is down to a point. Deformation includes things like stretching, shrinking, and moving, but not cutting or pasting. For example, a filled-in circle is contractible, since I can just shrink everything down to a single point. A circle that isn’t filled in is not contractible since there’s no way to get rid of that “hole” in the center. Mathematicians joke that they can’t tell the difference between a doughnut and a cup of coffee because one can deform one into the other. But one cannot turn a doughnut into a point. Most of the time. We’ll get to that “most” in a moment.
Now onto the infinite sphere. It’s important to note that when mathematiciants talk about spheres we typically mean the following thing. “An n-1 dimensional sphere consists of all points at distance 1 away from the origin in n dimensional space.” Take a circle, or a 1-sphere as a mathematician might call it. It’s in 2 dimensional space, and it’s all points at a distance 1 away from the origin of that graph.
The 0-sphere is just the two points at 1 and -1, given that the first dimension is just a line. Notice how you can get the 1-sphere from the 0 sphere by rotating it 180 degrees about the new dimension we add? And then try rotating the 1 sphere in three dimensions. You get the 2-sphere, something that looks like a hollow ball. If you continue to do this forever and ever you get the infinite sphere.
But wait! I hear you say. The 1 sphere certainly can’t be squished down to a point, and the 2-sphere is hollow! How could the infinite sphere be contractible? Here’s where we use a clever trick. Notice how if we overlay the 1-sphere on the 2-sphere (like a rubberband on a ball) we can shrink it, moving it closer to the pole of the 2-sphere until it all meets at a single point? So while the 1-sphere isn’t contractible, it is equivalent to a single point on the surface of the 2-sphere. While you can’t shrink the 2-sphere down, you can move it along the 3-sphere in 4-dimensions to shrink it down to a single point. So for any n-sphere in the infinite sphere we can have it be contractible in the n+1 sphere which exists in the infinite sphere. So every part of the infinite sphere is contractible. So the infinite sphere isn’t contractible. This is a very rough proof, but I hope you get the idea behind it. The idea that infinity is fucking weird and makes everything act bananas and I love it.
