Feeds:
Posts

## Topological Diophantine equations

The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let $R$ be a commutative ring, which in the most classical case might be a number field like $\mathbb{Q}$ or the ring of integers in a number field like $\mathbb{Z}$. Suppose we want to find solutions, over $R$, of a system of polynomial equations

$\displaystyle f_1 = \dots = f_m = 0, f_i \in R[x_1, \dots x_n]$.

Then it’s not hard to see that this problem is equivalent to the problem of finding $R$-algebra homomorphisms from $S = R[x_1, \dots x_n]/(f_1, \dots f_m)$ to $R$. This is equivalent to the problem of finding left inverses to the morphism

$\displaystyle R \to S$

of commutative rings making $S$ an $R$-algebra, or more geometrically equivalent to the problem of finding right inverses, or sections, of the corresponding map

$\displaystyle \text{Spec } S \to \text{Spec } R$

of affine schemes. Allowing $\text{Spec } S$ to be a more general scheme over $\text{Spec } R$ can also capture more general Diophantine problems.

The problem of finding sections of a morphism – call it the section problem – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map $f : E \to B$ between spaces; this amounts, very roughly speaking, to solving a “topological Diophantine equation.” The notation here is meant to evoke a particularly interesting special case, namely that of fiber bundles.

We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.

## 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.

## A transcript of my qualifying exam

I passed my qualifying exam last Friday. Here is a copy of the syllabus and a transcript.

Although I’m sure there are more, I’m only aware of two other students at Berkeley who’ve posted transcripts of their quals, namely Christopher Wong and Eric Peterson. It would be nice if more people did this.

## 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.)