Today's rabbit hole: I've been reading about homotopy groups of spheres, the Hopf fibration, etc. Homotopy groups are a way of describing how spheres of one dimension can be mapped into spheres of other dimensions. The first homotopy group, the fundamental group, is easy to 1/7

calculate for all spheres, but computing higher homotopy groups is notoriously challenging. Here's a chart showing some of them. Note that those below the diagonal are zero because a sphere of lower dimension inside one of higher dimension can always be shrunken to a point. 2/7

Those along the diagonal are Z because the homotopy group measures how many times S^n is wrapped around S^n. (Imagine wrapping a rubber band around your finger a certain number of times.) Notice that other than these and the few yellow ones, the homotopy groups are finite. 3/7

The first interesting example is π_3(S^2) b/c there's a way to map the 3-sphere into the 2-sphere in a nontrivial way—the so-called Hopf fibration. Inverse images of points in S^2 are circles in S^3. This image gives an idea of how some of these circles are linked. 4/7

In particular, if you take the inverse image of a circle of latitude on S^2, you get a torus in S^3. The preimage of each point on the circle is a Villarceau circle on the torus like one of the metal rings in the previous photo. 5/7

Here's a beautiful video by @NilesTopologist helping to visualize the Hopf fibration: . And here he is giving a very clear lecture on the topic: . 6/7

I don't know much about this topic, but when I was a grad student at Northwestern in the 90s, a lot of the older topologists in the department worked in this area and appear in the references on the Wikipedia page (Mahowald, Barratt, Kahn, Priddy): https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres 7/7

Credit: all photos were from Wikipedia. 8/7

Also, I now want to make one of those Hopf fibration tori out of key rings! 9/7