Blog posts

Topology, particularly homotopy theory, is hard. The scenes where these kind of mathematics happen are immensely complicated; the category of topological spaces; the category of spectra. The problem is that there is simply too much information to try to capture by using simple tools that we can actually understand properly. Trying to classify topological spaces or spectra is a feat that many deem impossible, it is simply too difficult. So, how can we try to fix this? We take inspiration from other fields, where similar situations occur and then try to translate into our own situation. Take for example abelian groups. We have no classification of all abelian groups, but there are certain constructions that help us study them. We have a classification of finitely generated abelian groups, where we can decompose any abelian group $A$ into understandable pieces: a free part $\mathbb{Z}^r$ and a torsion part $\mathbb{Z}/p_1^{k_1}\oplus \ldots \oplus \mathbb{Z}/p_t^{k_t}$. For all abelian groups things get more complicated, but there are nice groups that have classifications, like divisible groups, which are direct sums of copies of $\mathbb{Q}$ and Prüfer groups $\mathbb{Z}(p^\infty)$. The general approach seems to be to split the complicated groups into smaller pieces, or to study them via some easier groups. If we just consider $\mathbb{Z}$ for a moment, we can for a prime $p$ study ...

In the next couple years I will need to understand the ins and outs of different cohomology theories and the spectra that represents them. Some of the most important of these (for my research) can be described using $MU$ — the complex cobordism spectrum. We briefly met this spectrum — or at least its cohomology theory — when we discussed formal group laws. There we explained briefly a theorem of Quillen, stating that the universal formal group law over the Lazard ring corresponds to complex cobordism cohomology. We did not cover what complex cobordism actually is, so that is the plan for this post. ...

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. ...

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”. We have met several of the tools before, like the long exact sequence from a fibration and the Freudenthal suspension theorem, but we will also meet some new ones, like the $J$-homomorphism and the $h$-cobordism group. These two are methods for calculating the stable homotopy groups of spheres, or at least some of their subgroups. For the low dimensional cases, these subgroups will luckily be the entire groups. Due to the length of the post I have split it into two: one covering the unstable homotopy groups, mostly focusing on the Hopf fibration, and one covering the stable groups, mostly focusing on the image of the $J$-homomorphism. Before we start we recall the definition of the homotopy groups of spheres. ...

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. But the question is, what is it the homotopy category of? As we remarked in the post on tensor triangulated categories, it is the homotopy category of the category of spectra, and it is here that the different approaches lie. What exactly is the category of spectra, and which spectra are we even talking about? Is it the sequential spectra? or maybe the orthogonal spectra? or perhaps the symmetric ones? maybe $S$-modules or excisive functors? All these names of course means nothing to us yet, as we haven’t properly looked at any of them. We did however meet the $\Omega$-spectrum in an earlier post, but which of the above types does it belong to? ...

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. In fact, we actually used it, or at least almost when we said: “Hence the suspension functor should shift the degrees of the homotopy groups up by one” in this post. Today we make this precise, and look at a cool thing that happens as a consequence of this. ...

Recently we have covered a lot of heavy topology and abstract mathematics, so today I thought we would cover something else — something maybe a bit easier to grasp. We will introduce the concept of formal group laws, and a bit on why they are interesting. Introduction and definition To not just spew out the definition straight away, we look at a situation where formal group laws arise very naturally. Let $G$ be a one-dimensional commutative Lie group (Think here of the real numbers $\mathbb{R}$ or the circle group $S^1$). This group has a continuous product, $m:G\times G\longrightarrow G$, which locally can be described by a real-valued function in two variables. This function has a Taylor expansion around the origin, which is a power series in two variables. Denote this power series by $F$. This power series satisfies some axioms because of the group structure we have on $G$, these axioms are identity, commutativity and associativity. In more mathematical terms this means that the following statements hold ...

Even though this blog is not centered around a specific topic, we have during the last year looked more frequently at certain topics than others, such as (co)homology theory, homotopy theory and category theory. We will continue this trend today as we will try to find a solid reason for a particular object to exist. These objects were briefly mentioned in the earlier post on tensor triangulated categories, namely spectra. These objects are hugely important to the field of algebraic topology, one reason being that they are intimately linked to cohomology. This intimate connection is the study of todays blog post1. ...

How can we use travel-journaling to study some highly abstract and highly complicated mathematical machinery? Can we get anything out of such an analogy? In this post we do just that, study the homotopy hypothesis and infinity groupoids through the lens of a travel ledger.

I have recently handed in and defended my master thesis in mathematics, so I though I would go through its abstract and try to explain what it’s all about. We look at formality of DG-algebras, Massey products, A_infinity-algebras and how we can use these to some interesting results.