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## The cohomology of the n-torus

The goal of this post is to compute the cohomology of the $n$-torus $X = (S^1)^n \cong \mathbb{R}^n/\mathbb{Z}^n$ in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is $\mathbb{Z}$ by default. At the end we will interpret this computation in terms of cohomology operations.

## 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.)

## Groupoid cardinality

Suitably nice groupoids have a numerical invariant attached to them called groupoid cardinality. Groupoid cardinality is closely related to Euler characteristic and can be thought of as providing a notion of integration on groupoids.

There are various situations in mathematics where computing the size of a set is difficult but where that set has a natural groupoid structure and computing its groupoid cardinality turns out to be easier and give a nicer answer. In such situations the groupoid cardinality is also known as “mass,” e.g. in the Smith-Minkowski-Siegel mass formula for lattices. There are related situations in mathematics where one needs to describe a reasonable probability distribution on some class of objects and groupoid cardinality turns out to give the correct such distribution, e.g. the Cohen-Lenstra heuristics for class groups. We will not discuss these situations, but they should be strong evidence that groupoid cardinality is a natural invariant to consider.

My current top candidate for a mathematical concept that should be and is not (as far as I can tell) consistently taught at the advanced undergraduate / beginning graduate level is the notion of a groupoid. Today’s post is a very brief introduction to groupoids together with some suggestions for further reading.

## Split epimorphisms and split monomorphisms

• What is the “easiest way” a morphism can be a monomorphism (resp. epimorphism)?
• What are the absolute monomorphisms (resp. epimorphisms) – that is, the ones which are preserved by every functor?
• A morphism which is both a monomorphism and an epimorphism is not necessarily an isomorphism. Can we replace either “monomorphism” or “epimorphism” by some other notion to repair this?
• If we wanted to generalize surjective functions, why didn’t we define an epimorphism to be a map which is surjective on generalized points?

The answer to all of these questions is the notion of a split monomorphism (resp. split epimorphism), which is the subject of today’s post.

## Centers, 2-categories, and the Eckmann-Hilton argument

The center $Z(G)$ of a group is an interesting construction: it associates to every group $G$ an abelian group $Z(G)$ in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor $\text{Grp} \to \text{Ab}$ (unlike the abelianization $G/[G, G]$). We might wonder, then, exactly what kind of construction the center is.

Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category $C$ we may associate to an object $c \in C$ its automorphism group $\text{Aut}(c)$ or endomorphism monoid $\text{End}(c)$), and this is a canonical construction which again doesn’t extend in an obvious way to a functor $C \to \text{Grp}$ or $C \to \text{Mon}$. (It merely reflects some special part of the bifunctor $\text{Hom}(-, -)$.)

It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group $G$, but of the identity functor $G \to G$, where $G$ is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.

## Euler characteristic as homotopy cardinality

Let $X$ be a finite CW complex with $c_0$ vertices, $c_1$ edges, and in general $c_i$ different $i$-cells. The Euler characteristic

$\displaystyle \chi(X) = \sum_{i \ge 0} (-1)^i c_i$

is a fundamental invariant of $X$, and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula $V - E + F = 2 = \chi(S^2)$ for a convex polyhedron. But where exactly does this expression come from? The modern story involves the homology groups $H_i(X, \mathbb{Q})$, but actually one can work on a more intuitive level characterized by the following slogan:

The Euler characteristic is a homotopy-invariant generalization of cardinality.

More precisely, the above expression for Euler characteristic can be deduced from three simple axioms:

1. Cardinality: $\chi(\text{pt}) = 1$.
2. Homotopy invariance: If $X \sim Y$, then $\chi(X) = \chi(Y)$.
3. Inclusion-exclusion: Suppose $X$ is the union of two subcomplexes $A, B$ whose intersection $A \cap B$ is a subcomplex of both $A$ and $B$. Then $\chi(X) = \chi(A) + \chi(B) - \chi(A \cap B)$.

Of course, this isn’t enough to conclude that there actually exists an invariant with these properties. Nevertheless, it’s enough to motivate the search for a proof that such an invariant exists.

## SU(2) and the quaternions

The simplest compact Lie group is the circle $S^1 \cong \text{SO}(2)$. Part of the reason it is so simple to understand is that Euler’s formula gives an extremely nice parameterization $e^{ix} = \cos x + i \sin x$ of its elements, showing that it can be understood either in terms of the group of elements of norm $1$ in $\mathbb{C}$ (that is, the unitary group $\text{U}(1)$) or the imaginary subspace of $\mathbb{C}$.

The compact Lie group we are currently interested in is the $3$-sphere $S^3 \cong \text{SU}(2)$. It turns out that there is a picture completely analogous to the picture above, but with $\mathbb{C}$ replaced by the quaternions $\mathbb{H}$: that is, $\text{SU}(2)$ is isomorphic to the group of elements of norm $1$ in $\mathbb{H}$ (that is, the symplectic group $\text{Sp}(1)$), and there is an exponential map from the imaginary subspace of $\mathbb{H}$ to this group. Composing with the double cover $\text{SU}(2) \to \text{SO}(3)$ lets us handle elements of $\text{SO}(3)$ almost as easily as we handle elements of $\text{SO}(2)$.

## SO(3) and SU(2)

In order to study the hydrogen atom, we’ll need to know something about the representation theory of the special orthogonal group $\text{SO}(3)$. This post consists of a few preliminaries along the way to doing this. I’ll be somewhat vague about a few things that 1) I don’t have much experience with, and 2) that would detract from the main narrative anyway.

Lemma 1: Fix a field $F$. The group of fractional linear transformations $PGL_2(F)$ acts triple transitively on $\mathbb{P}^1(F)$ and the stabilizer of any triplet of distinct points is trivial.
Lemma 2: The group of fractional linear transformations on $\mathbb{P}^1(\mathbb{C})$ preserving the upper half plane $\mathbb{H} = \{ z \in \mathbb{C} | \text{Im}(z) > 0 \}$ is $PSL_2(\mathbb{R})$.