So, it’s been a little while since I last wrote and many of you are wondering what has happened about my surgery illustrations. I am wondering about this myself, not having heard from Erica (the journalist) in nearly two weeks now. The latest news was that the editor was considering hiring a professional animator (they’re willing to pay him, but not poor Haggis!) but they wanted my pictures to give him an idea of what was needed. I duly sent them off, all annotated and everything, but have no idea what the animator is going to do with them or when this article is going to appear.
Seeing as I’ve made these pictures I may as well give you (my loving readers) a brief description of what surgery is to show them off again.
So, you start off with your favourite manifold. A manifold is a mathematical space or shape which looks flat if you zoom in on any particular bit of it. For example, a circle looks curved from a distance but if you zoom in far enough then it’ll look like a straight line. Similarly, we live on a sphere but walking about in our everyday lives we think we’re on a flat surface. (Some people actually still think we live on a flat earth!) My favourite 2-dimensional manifold is a torus (or doughnut):
To start off the surgery procedure we have to cut out a “disc times a sphere”. In this case we will cut out a 1-dimensional disc (which is a line) times a 1-dimensional sphere (which is a circle).
This gives us the following picture:
The other half of the surgery procedure is to glue back in a different “disc times a sphere” along the edge where the other one was removed. In this case we will glue in a 2-dimensional disc (which is a filled-in circle) times a 0-dimensional sphere (which is two points – the boundary of the 1-dimensional disc).
Gluing these discs in gives us:
Finally, because I am a topologist I can pretend that this shape is made of plasticine and I can squeeze and push it about however I like (so long as I don’t make any holes in it). Hopefully you will agree with me that the simplest way to display this shape (after a bit of squeezing) is a sphere:
So what surgery has done is to take a shape with a hole in it (the torus) and change it into a shape with no holes in it (the sphere). This is the general idea of surgery theory: to try and make shapes simpler. We can do it with high-dimensional shapes too (in fact, this is where it is most useful!) but this is very hard to visualise.
I hope you have enjoyed my few small pictures, and I will post again when I know more about the Kervaire article. Comments, as always, are welcome.