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## The Picard groups

Let $R$ be a commutative ring. From $R$ we can construct the category $R\text{-Mod}$ of $R$-modules, which becomes a symmetric monoidal category when equipped with the tensor product of $R$-modules. Now, whenever we have a monoidal operation (for example, the multiplication on a ring), it’s interesting to look at the invertible things with respect to that operation (for example, the group of units of a ring). This suggests the following definition.

Definition: The Picard group $\text{Pic}(R)$ of $R$ is the group of isomorphism classes of $R$-modules which are invertible with respect to the tensor product.

By invertible we mean the following: for $L \in \text{Pic}(R)$ there exists some $L^{-1}$ such that the tensor product $L \otimes_R L^{-1}$ is isomorphic to the identity for the tensor product, namely $R$.

In this post we’ll meander through some facts about this Picard group as well as several variants, all of which capture various notions of line bundle on various kinds of spaces (where the above definition captures the notion of a line bundle on the affine scheme $\text{Spec } R$).

## Five proofs that the Euler characteristic of a closed orientable surface is even

Let $\Sigma_g$ be a closed orientable surface of genus $g$. (Below we will occasionally write $\Sigma$, omitting the genus.) Then its Euler characteristic $\chi(\Sigma_g) = 2 - 2g$ is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be $2 - 2g$, roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications.

## Hypersurfaces, 4-manifolds, and characteristic classes

In this post we’ll compute the (topological) cohomology of smooth projective (complex) hypersurfaces in $\mathbb{CP}^n$. When $n = 3$ the resulting complex surfaces give nice examples of 4-manifolds, and we’ll make use of various facts about 4-manifold topology to try to say more in this case; in particular we’ll be able to compute, in a fairly indirect way, the ring structure on cohomology. This answers a question raised by Akhil Mathew in this blog post.

Our route towards this result will turn out to pass through all of the most common types of characteristic classes: we’ll invoke, in order, Euler classes, Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes.

## 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.