# Arithmetic Topology

One of my favorite viewpoints on math is that it is the study of metaphor.  By abstracting an existing metaphor, we invent new objects (groups were invented because Galois realized that ‘polynomials are like plane figures’); by inventing a new metaphor, we can import techniques from one field to another (the Italian school of algebraic geometry got a lot of mileage from ‘varieties are like manifolds’; the functor-of-points school, from ‘schemes are like functors on the category of rings’).  A poet might compare the evening to an etherized patient and leave it there, and in poetry this is fine, for in poetry we revel in mystery, allusion, in half-knowledge.  But in math, we can’t stand these things, and so we must grab our things and run to the nearest hospital, examining all the gurneys we can in the hope of better understanding the twilight.

One particular metaphor that’s captivated me recently goes by the unassuming name of ‘arithmetic topology,’ essentially due to Barry Mazur in an unpublished paper from 1963 (though as always, neverendingbooks has a more precise history of the idea).  The one-sentence idea, which should be all the convincing anyone needs to do math, is that the integers are like three-space and prime numbers are like knotted circles.  I know, I know.  And yes, it does go a lot further than that — more generally, there’s a fascinating story about Galois-group-looking things relating to the algebraic topology of honest-to-God topological things.  In the interest of always doing something that’s slightly too hard for you, I’m trying to understand this right now, and in doing so will hopefully learn a lot of algebraic geometry, algebraic number theory, and knot theory ‘by accident.’  I’ll write about it here as I do so, and I invite you to learn with me if you don’t know what I’m talking about / correct my errors and give me references if you do.  Below the fold, an introduction to what seems to be going on, and a rough outline of what I’ll probably be writing about.

BIRTHED WHOLE FROM THE MIND OF MAZUR

The ‘reason’ why an analogy between knots and primes should exist lies in a paper that, surprisingly, doesn’t mention knots at all: Mazur’s ‘Lectures on étale cohomology of number fields’.  This establishes a nondegenerate pairing

$H^*_{et}(X,\mathscr{F}) \times {\rm Ext}^{3-*}(\mathscr{F},\mathbb{G}_m) \to H^3_{et}(X,\mathscr{F}) \cong \mathbb{Q}/\mathbb{Z}$

where $X$ is the affine scheme of the ring of integers of a number field, and $\mathscr{F}$ is a constructible abelian sheaf on $X$.  (Most of that sentence was meaningless to me, a problem I hope to fix over the course of this series.)  This is a generalization of something called Artin-Verdier duality, which is in turn a generalization of a statement called Tate duality about Galois modules.

By federal law, every time you write about knot theory, you have to include at least one picture of your knot. Call your representative if you feel this is unfair!

But it also looks like Poincaré duality for a 3-manifold, at least if we’re able to interpret the Ext group as behaving like a homology group.  Based on that evidence, Mumford and Mazur started looking for more similarities between rings of integers of number fields and 3-manifolds.

For example, one obvious invariant of a 3-manifold is its fundamental group.  But schemes also have something like this: the étale fundamental group $\pi_1^{et}(X)$, which is a profinite group obtained as the inverse limit of the automorphism groups over $X$ of the finite étale covers of $X$.  When $X = {\rm Spec}(\mathbb{Z})$, this is trivial, so by, uh, the Poincaré Conjecture, $X$ should correspond to the three-sphere!  More complicated rings of integers will presumably correspond to more complicated 3-manifolds (and their étale fundamental groups will correspond to profinite completions of the topological fundamental groups of these manifolds).

What about the individual primes?  Well, if $\mathcal{O}$ is our ring of integers and $\mathfrak{p}$ is our prime, we get a residue field $\mathbb{F}_\mathfrak{p} = \mathcal{O}/\mathfrak{p}$ and an inclusion ${\rm Spec}(\mathbb{F}_\mathfrak{p}) \to {\rm Spec}(\mathcal{O})$.  The étale fundamental group of $\mathbb{F}_\mathfrak{p}$ is the profinite completion $\hat{\mathbb{Z}}$ of $\mathbb{Z}$: indeed, $\mathbb{F}_\mathfrak{p}$ is just some finite field $\mathbb{F}_p$, and the finite étale covers of $\mathbb{F}_p$ are just the fields $\mathbb{F}_{p^m}$, with Galois groups precisely  the finite quotients of $\mathbb{Z}$.  So the inclusion of the spectrum of this residue field into that of the ring of integers corresponds to the embedding of a circle into a 3-manifold — that is, to a knot!  (According to Morishita’s expository article about this, ${\rm Spec}(\mathbb{F}_\mathfrak{p})$ is actually étale-homotopically a $K(\hat{\mathbb{Z}}, 1)$, a fascinating claim I’ll have to investigate further.)

If we complete at the prime $\mathfrak{p}$, we get a scheme ${\rm Spec}(\mathcal{O}_\mathfrak{p})$, which apparently has ${\rm Spec}(\mathbb{F}_\mathfrak{p})$ as an étale deformation retract (that is, the inclusion is an étale homotopy equivalence.  Removing ${\rm Spec}(\mathbb{F}_\mathfrak{p})$ leaves the $\mathfrak{p}$-adic field ${\rm Spec}(k_\mathfrak{p})$.  These can be thought of respectively as a tubular neighborhood of our knot, and the torus given by removing the original knot.  Here we see some of the wrinkles arising in the metaphor.  For instance, since we’re now in positive characteristic, we have to deal with something called the tame étale fundamental group, and the tame étale fundamental group of ${\rm Spec}(k_\mathfrak{p})$ is something but not exactly like the fundamental group of the torus (up to profinite completion, I’m guessing) — its relations include an extra, characteristic-p dash of cayenne.

In order to compensate for that knot picture earlier, here’s a picture of a prime.

Since $\mathcal{O}$ is a Dedekind domain, any ideal $\mathfrak{a}$ factorizes uniquely as a product of primes.  Thus the closed subscheme cut out by $\mathfrak{a}$ corresponds to a link in the 3-manifold.  Apparently the linking number of two knots corresponds to the Legendre symbol of two primes.

The list goes on — number-theoretic invariants like ideal class groups correspond to topological invariants like homology groups, and so on.  Mazur’s unpublished paper linked above (which is in bad need of TeXing, sad to say) introduced an ‘Alexander polynomial’ for these primes in rings of integers.  I’m guessing most of the ideas flow from knot theory to number theory rather than vice versa, number theory being the ‘harder’ science.

If I can understand at least some of this over winter break, I’ll be quite happy.  I started looking at Mazur’s paper in order to understand the construction of the determinant of a coherent sheaf explained in its first section, but it’s clearly far too late for that now.  I’ll try to cover some of the following:

• The construction of the étale fundamental group and its tame version, and some of the étale-homotopical style of thinking Morishita uses so fluidly.  Lenstra has a set of notes that look like a good source for this.
• Etale cohomology of schemes and the duality described by Mazur.
• Something of the relevant theory of rings of integers of number fields underlying this.  I’m familiar with a bit of this but almost certainly not enough.
• Any relevant knot theory — the linking number, Alexander polynomial, and whatever else ends up getting translated into things about primes.  It’ll be equally interesting to see what concepts can’t cross the bridge.
• Where does this theory get applied?  Are there number-theoretical results that can only, or best, be proved by thinking with knots?  Has knot theory itself been affected?  Morishita’s article looks like a nice preliminary source list.
• Since I found the parts of it I’ve read to be both difficult and instructive, I’ll probably go through Mazur’s paper on the Alexander polynomial.
• The opening paragraphs of this unpublished Kapranov-Smirnov paper use the knot-prime dictionary to give a good argument for the existence of the field with one element $\mathbb{F}_{\mathrm{un}}$.  As far as I can tell, the argument is: ‘If ${\rm Spec}(\mathbb{Z})$ is the terminal object of the category of schemes, and it’s topologically three-dimensional, that’s pretty weird, right?’  I’d like to talk/learn a bit about this.

At this point, if I haven’t either learned motivic homotopy theory by accident or gone crazy and pulled a Jim McAfee, I will probably stop talking and we can all go on with our lives.  Until then, references, suggestions, and conversation are much appreciated.