Homotopy groups

Recently my friend Elias started his own math blog adventure, and his first post gave a nice introduction to spectral sequences. Reading it I remembered that I should really understand some of the parts better myself, because a lot of the arguments one makes in chromatic homotopy theory are based on spectral sequences. There is a framework for constructing spectral sequences that are not covered in my old post on them, as well as Elias’ post, and that is creating spectral sequences from exact couples....

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....

Recently I gave a talk about the homotopy groups of spheres, and as usual, I try to collect my thoughts on this blog before (or after) presenting. The homotopy groups of spheres have featured several times on this blog, and we have made some effort into calculating them for some small dimensions. In the talk I wanted to showcase some methods used to calculate these groups, as well as doing some of the “calculations”....

A little while ago we discussed the definition of a tensor triangulated category, and in that post we mentioned an example that we didn’t explicitly define, namely the stable homotopy category. The goal for todays post is to fix this. There are many ways of defining it, and some are actually better than others. As the name suggests, the stable homotopy category is a homotopy category, which we have discussed before in the fibration series....

For the last few posts we have covered some theory surrounding cohomology theories, and today we want to do something else, namely again look at some homotopy theory. It’s been a long time since we have covered homotopy groups, but today we return once again. In particular I want to cover a theorem and its consequences — the Freudenthal suspension theorem. This is one of the central theorems in the homotopy theory of topological spaces, and is one of the more important theorems we left out from the fibration series....

The last few posts have all been of relatively long length and have all taken some time to construct and write. I initially also wanted to produce shorter posts just discussing an example or a calculation etc, and today I tried to do just that, but failed. The post became somewhat longer than intended, but it is really informal and intuitive, so its fine in my opinion. In a previous post we discussed both weak homotopy equivalences and regular homotopy equivalences, and we have also encountered the Whitehead theorem, which says that any weak homotopy equivalence between CW-complexes is in fact a regular homotopy equivalence....

Some time ago I saw this problem of hanging a picture on the wall using a string and two nails in such a way that if you remove one of the nails from the wall, the picture falls down. This is a bad way to hang pictures you immediately say, and I would agree. I saw some solution to the problem, and didn’t think about it for many years, until this week when I figured out that we need homotopy, in particular the fundamental group, to do it!...

This is part 5 of a series leading up to and exploring model categories. For the other parts see the series overview. As promised in the previous part, we are going to calculate $\pi_4(S^3)$. I think we will have to use all of the machinery (plus some new) that we have been through during this series to do the calculation. What more could we possibly need you ask? Last time we developed the machinery to calculate the cohomology of the total space of a fibration, but we want to compute homotopy....

This is part 3 of a series leading up to and exploring model categories. For the other parts see the series overview. For an introduction to the material, the definitions, motivation and some examples, please read part 1 and part 2 about fibrations and fiber bundles. This and the the following parts of this series will be about their usefulness, especially in computing homology and homotopy groups. This will be done through two different techniques, namely the long exact sequence of homotopy groups, and the spectral sequence associated to a fibration....