Differential Topology

In the previous post we studied some “easy” cases of homotopy groups of spheres. We focused most on the group $\pi_3(S^2)$ and its computation from the Hopf fibration. All groups calculated last time were part of the so-called unstable range, meaning that they are not invariant under suspension. Due to the Freudenthal suspension theorem we know precicely the stable range for homotopy groups of spheres, and these are given by the stable homotopy groups. These groups are what we will look at today (and what we looked at during the second part of the talk these two blog-posts are based upon). We will compute some of the low stable homotopy groups of spheres using the socalled $J$-homomorphism. But, in order to do this calculation we must cover a plethora of interesting mathematics. ...

For those that don’t know I am a fifth year mathematics student at NTNU, meaning I am finishing my masters degree after this semester. During my time at NTNU I have had some wonderful classes, and some wonderful teachers. Since most I post about on this blog is related to topology, it is very safe to assume that some of my most memorable courses are exactly the topology courses. I very recently looked at my notes from my first topology course, or rather one of the two first, as I took two in parallel during my fourth semester. The course was focused on differential topology and the study of smooth manifolds. It was taught by my now supervisor, but on a couple of the last lectures we had some guest appearances from the other topology professors at NTNU. One of these guest lectures is the focus of todays blog post. ...