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The homotopy groups are only groups

Often in mathematics we define constructions outputting objects which a priori have a certain amount of structure but which end up having more structure than is immediately obvious. For example:

• Given a Lie group $G$, its tangent space $T_e(G)$ at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.
• Given a space $X$, its cohomology $H^{\bullet}(X, \mathbb{Z})$ is a priori a graded abelian group, but it ends up having the structure of a graded ring.
• Given a space $X$, its cohomology $H^{\bullet}(X, \mathbb{F}_p)$ over $\mathbb{F}_p$ is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod-$p$ Steenrod algebra.

The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?

In this post we will answer this question for the homotopy groups $\pi_n(X)$ of a space: that is, we will show that, in a suitable sense, each individual homotopy group $\pi_n(X)$ is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)

I’ve added a new page of reading recommendations, mostly for undergraduates, to the top. The emphasis is intended to be on well-written and accessible books. Comments and suggestions welcome.

The p-group fixed point theorem

The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let $G$ be a finite $p$-group acting on a finite set $X$. Let $X^G$ denote the subset of $X$ consisting of those elements fixed by $G$. Then $|X^G| \equiv |X| \bmod p$; in particular, if $p \nmid |X|$ then $G$ has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

Cantor’s theorem, the prisoner’s dilemma, and the halting problem

Cantor’s theorem is somewhat infamous as a mathematical result that many non-mathematicians have a hard time believing. Trying to disprove Cantor’s theorem is a popular hobby among students and cranks; even Eliezer Yudkowsky1993 fell into this trap once. I think part of the reason is that the standard proof is not very transparent, and consequently is hard to absorb on a gut level.

The goal of this post is to present a rephrasing of the statement and proof of Cantor’s theorem so that it is no longer about sets, but about a particular kind of game related to the prisoner’s dilemma. Rather than showing that there are no surjections $X \to 2^X$, we will show that a particular kind of player in this game can’t exist. This rephrasing may make the proof more transparent and easier to absorb, although it will take some background material about the prisoner’s dilemma to motivate. As a bonus, we will almost by accident run into a proof of the undecidability of the halting problem.

Operations, pro-objects, and Grothendieck’s Galois theory

Previously we looked at several examples of $n$-ary operations on concrete categories $(C, U)$. In every example except two, $U$ was a representable functor and $C$ had finite coproducts, which made determining the $n$-ary operations straightforward using the Yoneda lemma. The two examples where $U$ was not representable were commutative Banach algebras and commutative C*-algebras, and it is possible to construct many others. Without representability we can’t apply the Yoneda lemma, so it’s unclear how to determine the operations in these cases.

However, for both commutative Banach algebras and commutative C*-algebras, and in many other cases, there is a sense in which a sequence of objects approximates what the representing object of $U$ “ought” to be, except that it does not quite exist in the category $C$ itself. These objects will turn out to define a pro-object in $C$, and when $U$ is pro-representable in the sense that it’s described by a pro-object, we’ll attempt to describe $n$-ary operations $U^n \to U$ in terms of the pro-representing object.

The machinery developed here is relevant to understanding Grothendieck’s version of Galois theory, which among other things leads to the notion of étale fundamental group; we will briefly discuss this.

Previously we described $n$-ary operations on (the underlying sets of the objects of) a concrete category $(C, U)$, which we defined as the natural transformations $U^n \to U$.

Puzzle: What are the $n$-ary operations on finite groups?

Note that $U$ is not representable here. The next post will answer this question, but for those who don’t already know the answer it should make a nice puzzle.

Operations and Lawvere theories

Groups are in particular sets equipped with two operations: a binary operation (the group operation) $(x_1, x_2) \mapsto x_1 x_2$ and a unary operation (inverse) $x_1 \mapsto x_1^{-1}$. Using these two operations, we can build up many other operations, such as the ternary operation $(x_1, x_2, x_3) \mapsto x_1^2 x_2^{-1} x_3 x_1$, and the axioms governing groups become rules for deciding when two expressions describe the same operation (see, for example, this previous post).

When we think of groups as objects of the category $\text{Grp}$, where do these operations go? They’re certainly not morphisms in the corresponding categories: instead, the morphisms are supposed to preserve these operations. But can we recover the operations themselves?

It turns out that the answer is yes. The rest of this post will describe a general categorical definition of $n$-ary operation and meander through some interesting examples. After discussing the general notion of a Lawvere theory, we will then prove a reconstruction theorem and then make a few additional comments.