Evelyn Lamb in Scientific American:
For Halloween, I wrote about a very scary topic: higher homotopy groups.Homotopy is an idea in topology, the field of math concerned with properties of shapes that stay the same no matter how you squish or stretch them, as long as you don’t tear them or glue things together. Both homotopy groups and the somewhat related homology groups are different ways to describe the topology of shapes using algebra. In my post, I said that homology detects “holes” of different dimensions. But, as one commenter asked, what do I mean by holes of different dimensions?
Good question! I deliberately used “hole” as a wiggle word because there isn’t a real mathematical definition of hole. But here’s my short answer that is also the reason I’m not an algebraic topologist. If you can put it on a necklace, it has a one-dimensional hole. If you can fill it with toothpaste, it has a two-dimensional hole. For holes of higher dimensions, you’re on your own.
That answer isn’t very satisfying. Is there a better way to describe holes? I talked with some of my topologist friends and discovered two things: topologists don’t all agree on what a hole is, and it’s fun and interesting to think about different interpretations of a word whose mathematical definition isn’t completely settled. I think my larger conclusion, in the spirit of the season, is that holes are like Santa Claus: the true meaning is in your heart. So let’s look into our hearts and think about what holes are.