## March 1, 2008

### Kim on Fundamental Groups in Number Theory

#### Posted by John Baez

My friend Minhyong recently wrote up a talk he gave at Leeds:

It starts with some pleasant observations of an elementary nature and works its way up to some ideas I find rather terrifying. Maybe we can ask him some questions and get him to explain what’s going on.

Apart from the rather intimidating concepts involved, I think part of my problem is that I’d need to look at some examples to get a feeling for what’s going on. And, I can’t tell if these examples will be comprehensible in full detail by mere mortals… or whether they’ll be big, unfathomable things.

For example, Minhyong talks about how the absolute Galois group of $\mathbb{Q}$,

$G = Gal(\overline{\mathbb{Q}}|\mathbb{Q}),$

acts on the ‘pro-finite étale fundamental group’

$\pi_1(X(\mathbb{C}))\widehat{} \cong \pi_1^{et}(\overline{X})$

where $X$ is an algebraic variety over $\mathbb{Q}$, $X(\mathbb{C})$ is the corresponding variety over $\mathbb{C}$, and $\overline{X}$ is the corresponding variety over the algebraic closure of $\mathbb{Q}$ — I think.

Now, I know the group $G$ is famously hard to understand in detail. But, to what extent can we compute this ‘pro-finite étale fundamental group’ in some simple but nontrivial examples, and to what extent can we understand the action of $G$?

For example, are there any examples where only the abelianized version of $G$ matters? That’s a group I know and love.

Posted at March 1, 2008 7:31 AM UTC

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## 43 Comments & 3 Trackbacks

### Re: Kim on Fundamental Groups in Number Theory

The description of the etale fundamental groups is rather explicit for smooth algebraic curves over $\bar{\mathbb{Q}}$ because of the isomorphism$\pi_1(X(\mathbb{C}),b)^\simeq \pi_1^{et}(\bar{X},b)$that you’ve referred to. For example, if X is a smooth compact curve of genus g minus one point, then it will be the free pro-finite group on g generators. It is notable that there is no known way to prove even that $\pi_1^{et}(\bar{X})$ is finitely-generated using just algebraic techniques. In any case, the Galois action is very hard to access in most cases.

The best way to understand the Galois action was described in the lecture, namely, via the isomorphism $\pi_1^{et}(\bar{X},b)\simeq \tilde{X}_b,$ where the right hand side refers to the fiber over $b$ of the system of finite covers that come together to form the pro-finite universal object $\tilde{X}$. So what you would try to do in practice is to guess the system over $\mathbb{C}$ geometrically, then try to find a model over $\mathbb{Q}$ for the system. This model, which can be inaccessible in general, is what allows us to compute the Galois action. If $X=\mathbb{A}^1\setminus \{0\}$ and the base-point is $b=1$, things are quite simple because then $X(\mathbb{C})=\mathbb{C}^*$, so you can guess that the pro-finite universal cover is the system $\tilde{X}:=\{X_n \rightarrow X\}_n$where $X_n\to X$ just refers to the $n$-th power map from $X$ to itself. To prove this fact, i.e., that this system has the right universal property, still requires a nice bit of elementary algebraic geometry. Anyways, an element of $\tilde{X}_b$ is a compatible collection of roots of unity. This is often denoted $\hat{\mathbb{Z}}(1)$, group-theoretically isomorphic to $\hat{\mathbb{Z}}$, the pro-finite completion of $\mathbb{Z}$. The action in this case exactly factors through $G^{ab}\simeq \mathbb{Z}^*$ in the natural way.

The annoying thing is that even here, when we view $\pi_1^{et}(\bar{X},b)$ as a sheaf over $Spec(\mathbb{Q})$, the classification of *torsors* still involves the whole group $G$, or at least a non-abelian quotient of it. I can explain this further as needed, but to get a sense of this, consider a usual manifold $B$ and the constant sheaf given by some group $L$. The principal $L$-bundles on $B$ are classified by $H^1(\pi_1, L)$, which for the given trivial action is just $Hom(\pi_1,L)$ modulo the conjugation action. That is to say, the action of $\pi_1$ on $L$ even factors through the *trivial* group. But the classification of bundles still reflects the structure of the fundamental group in a complicated way. Of course the structure of the Galois group $G$ is mysterious, and hence, the difficulty of classifying bundles. This captures pretty well the main issues I’m struggling with over $\Spec(\mathbb{Q})$.

In general, the geometric picture to keep in mind is a fiber bundle $X$ over some base space $B$, and the action of $\pi_1(B,y)$ on $\pi_1(X_y)$. So the ‘analogy,’ in this case is $Spec(\mathbb{Q}) \leftrightarrow B$ $X over Spec(\mathbb{Q}) \leftrightarrow the total space X$ $\bar{X} \leftrightarrow X_y$

Exercise: What is $y$?

Posted by: Minhyong Kim on March 1, 2008 7:08 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

In case some of my statements are misleading, I should point out that the classification of torsors for $\hat{\mathbb{Z}}(1)$ itself is actually not hard. Using Hilbert’s theorem 90, one finds the classifying set $(\mathbb{Q}^*)^{\wedge}$, the pro-finite completion of the multiplicative group of $\mathbb{Q}$. In virtually all other cases of fundamental groups of curves, the classifying space of torsors is difficult.

Posted by: Minhyong Kim on March 1, 2008 8:55 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Oops, another error. I shouldn’t have written the pro-finite completion of $\mathbb{Q}^*$.

It’s actually the inverse limit $\invlim_n(\mathbb{Q})^*/(\mathbb{Q}^*)^n$ I’ll explain later why I made this slip.

Posted by: Minhyong Kim on March 1, 2008 9:05 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

John hopes that the abelianization of $Gal(\overline{\mathbb{Q}} | \mathbb{Q})$ will make an appearance — a group he ‘knows and loves’. I have googled, and I have looked at Minhyong’s notes, but I haven’t found the answer to my question. So go on, what is the abelianization of $Gal(\overline{\mathbb{Q}} | \mathbb{Q})$?

Posted by: Tom Leinster on March 2, 2008 1:58 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Tom writes:

So go on, what is the abelianization of $Gal(\overline{\mathbb{Q}}|\mathbb{Q})$?

First of all, it’s $Gal(\mathbb{Q}^ab|\mathbb{Q})$, where $\mathbb{Q}^ab$ is the ‘maximal abelian extension of $\mathbb{Q}$’ — the biggest algebraic extension of $\mathbb{Q}$ whose Galois group over $\mathbb{Q}$ is abelian. Concretely, we get $\mathbb{Q}^{ab}$ by taking $\mathbb{Q}$ and throwing in all roots of unity.

Second of all, $Gal(\mathbb{Q}^ab|\mathbb{Q})$ has a nice concrete description. Everything I know and love is in This Week’s Finds — or will be eventually. So, to see how we get this concrete description, try this quote by Edward Frenkel in week221:

…recall the classical Kronecker-Weber theorem which describes the maximal abelian extension $\mathbb{Q}^ab$ of the field $\mathbb{Q}$ of rational numbers (i.e., the maximal extension of $\mathbb{Q}$ whose Galois group is abelian). This theorem states that $\mathbb{Q}^ab$ is obtained by adjoining to $\mathbb{Q}$ all roots of unity; in other words, $\mathbb{Q}^ab$ is the union of all cyclotomic fields $\mathbb{Q}(1^{1/N})$ obtained by adjoining to $\mathbb{Q}$ a primitive $N$th root of unity

$1^{1/N}$

The Galois group $Gal(\mathbb{Q}(1^{1/N})|\mathbb{Q})$ of automorphisms of $\mathbb{Q}(1^{1/N})$ preserving $\mathbb{Q}$ is isomorphic to the group $(\mathbb{Z}/N)^*$ of units of the ring $\mathbb{Z}/N$. Indeed, each element $m$ in $(\mathbb{Z}/N)^*$, viewed as an integer relatively prime to $N$, gives rise to an automorphism of $\mathbb{Q}(1^{1/N})$ which sends

$1^{1/N}$

to

$1^{m/N}$

Therefore we obtain that the Galois group $Gal(\mathbb{Q}^ab|\mathbb{Q})$, or, equivalently, the maximal abelian quotient of $Gal(\overline{\mathbb{Q}}|\mathbb{Q})$, where $\overline{\mathbb{Q}}$ is an algebraic closure of $\mathbb{Q}$, is isomorphic to the projective limit of the groups $(\mathbb{Z}/N)^*$ with respect to the system of surjections

$(\mathbb{Z}/N)^* \to (\mathbb{Z}/M)^*$

for $M$ dividing $N$. This projective limit is nothing but the direct product of the multiplicative groups of the rings of $p$-adic integers, $\mathbb{Z}_p^*$, where $p$ runs over the set of all primes. Thus, we obtain that

$Gal(\mathbb{Q}^ab|\mathbb{Q}) = \prod_p \mathbb{Z}_p^*$

If you stare at this and don’t let your eyes glaze over, you’ll see that at the end he’s computing the inverse limit of the multiplicative groups $(\mathbb{Z}/N)^*$ and getting

$\prod_p \mathbb{Z}_p^*$

So, the group you’re asking about has many nice descriptions, which is one reason I like it so much.

Another reason is that a Galois group is like a first homotopy group ($\pi_1$), so an abelianized Galois group is like an abelianized first homotopy group, namely a first homology group ($H_1$). Homotopy groups are hard, but homology groups are easier. Here we see that very nicely. $Gal(\overline{\mathbb{Q}}|\mathbb{Q})$ is big, scary and mysterious — at least one mathematician has gone insane contemplating it. Its abelianization is big, impressive — yet friendly and eminently comprehensible.

This is the idea behind class field theory: don’t mess with general Galois groups; stay in the safe waters of abelian Galois groups, where you’re secretly doing homology theory instead of homotopy theory.

Minhyong is leaving these safe waters, so what’s he’s doing is deeper and — to me, at least — more frightening.

Posted by: John Baez on March 2, 2008 5:02 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

In relation to fundamental groups, you could say that abelian class field theory *is* the study of the Galois action on $\pi_1$ of $\mathbb{C}^*$. Given a curve $X$ over $\mathbb{Q}$, one of the many ways of thinking about the action is via the exact sequence $0\rightarrow \pi_1(\bar{X},b) \rightarrow \pi_1(X,b)\rightarrow G \rightarrow 0$ analogous to the exact sequence of a fibration. There is then an *outer* action of $G$ on $\pi_1(\bar{X},b)$ given by lifting and conjugating. But because I’ve been choosing the point $b$ to be rational, there is a splitting $G\rightarrow \pi_1(X,b)$ that allows us to turn this into an actual action.

Now, for $A^1\setminus\{0\}$, $\pi_1$ is abelian and the action factors through $G^{ab}\simeq \hat{\mathbb{Z}}^*$. When $X$ is an elliptic curve $\pi_1\simeq \hat{\mathbb{Z}}^2$ so that we have a homomorphism $G \rightarrow GL_2(\hat{\mathbb{Z}}),$ which has probably been studied more than any other subject in number theory over the last few decades. Notice here that a non-abelian quotient of $G$ comes up. Already this is hard enough and the Langlands’ program deals mostly with this kind of non-commutativity. For a hyperbolic curve, however, we have a more severely non-abelian representation: $G\rightarrow Aut(\pi_1(\bar{X},b))$ into the autormophism group of a non-abelian group. So, to a certain extent, we are moving beyond non-abelian class field theory in the sense of Langlands. It’s interesting that Weil had already distinguished between these two kinds of non-commutativity in his 1938 paper on vector bundles.

By the way, to me it is rather clear that Schreier theory, gerbes, etc. eventually have to make an appearance to organize the theory properly. Exactly how is yet unclear. But one portion where it should be relevant is in the dualtiy theorems’ for Galois cohomology that came up in the discussion on three-manifolds. It was mentioned by someone there that these duality theorems are also a version of class field theory. At some point, I will try to explain how such duality theorems are used in Diophantine geometry. If your institution subscribes to Springer, you can look into this paper of Kato. It will be rather hard going if you’re not a hard-core number theory. But the introduction, the first few sections, and the last section might be amusing nevertheless. Of course there, the whole discussion is about elliptic curves. For hyperbolic curves, one will eventually need a duality theorem for non-abelian cohomology $H^1(G,\pi)$ (or at least some subspace of it corresponding to some geometric conditions). I’m hoping I can eventually gain some insight from the people here on how such a theorem might be formulated, probably involving some higher’ construction.

There is a non-arithmetic version of this question which should have a good answer but is already difficult (for me): Suppose $\Sigma$ is three-manifold or a Riemann surface and $L$ is a locally constant sheaf of non-abelian groups on $\Sigma$. Can one then formulate a Poincare duality theorem for $H^1(\Sigma,L)$ that generalizes the case where $L$ is a sheaf of vector spaces?

Posted by: Minhyong Kim on March 2, 2008 9:39 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Since John already alluded to the Kronecker-Weber theorem I can’t resist adding one comment: The fact that the action $\rho:G \rightarrow Aut (\pi_1(\mathbb{C}^*,1))$ factors through $G^{ab}$ is simple algebra, namely, the fact that $Aut(\mathbb{Z}/n)$ is the abelian group $(\mathbb{Z}/n)^*$. The fact that the map is *surjective* is a bit of Galois theory: each element of $(\mathbb{Z}/n)^*$ can actually be realized as a field automorphism of $\mathbb{Q}(e^{2\pi i/n})$, i.e., the corresponding power operation on the roots of unity extends to a field automorphism. But the *injectivity* of $\rho$ is the difficult Kronecker-Weber theorem.

When one looks at the representation on $\pi_1$ of $X=\mathbb{P}^1\setminus\{0,1,\infty\}$, that is, we remove just one more point, it turns out that the homomorphism $\rho^{na}: G \rightarrow Aut( \pi_1(\bar{X},b))$ is actually injective on *all* of $G$. This is a theorem of Belyi, which, perhaps naturally, is quite easy in comparison to KW.

Posted by: Minhyong Kim on March 2, 2008 10:04 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

I found Matsumotos and Wojtkowiaks notes usefull too. BTW, I wonder if Galois himself viewed Galois groups in a similar geometric way?

Conc. Katos article:
The link expired. Do you mean this article? Some parts of it seem to require knowkedge of crystalline cohomology. The standart reference is Berthelot-Ogus intro book, here and here some short surveys.

Posted by: Thomas Riepe on March 2, 2008 12:36 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Great: thanks John.

You said that abelian Galois groups correspond to homology theory, and general Galois groups to homotopy theory. Some of us also go around saying that strict $n$-category theory corresponds to homology theory, and weak $n$-category theory to homotopy theory. So the obvious conclusion is that (somehow) abelian Galois groups correspond to strict $n$-category theory and general Galois groups to weak $n$-category theory. Do you have a direct sense of what that correspondence is?

Posted by: Tom Leinster on March 2, 2008 5:26 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Tom wrote:

Some of us also go around saying that strict $n$-category theory corresponds to homology theory, and weak $n$-category theory to homotopy theory.

It doesn’t make quite as good of a bumper sticker slogan, but to be accurate, I think you guys should go around saying “strict stable $\infty$-groupoids correspond to homology theory, while weak $\infty$-groupoids correspond to homotopy theory”.

So, when we go from homotopy theory to homology theory, we’re taking two simplifying steps! We’re making everything strict… but we’re also making everything ‘stable’, which in the strict case just means ‘abelian’.

These steps are edges of the Cosmic Cube of $\infty$-Categories on page 10 here.

It can be good to take just one of these steps, e.g. making your $\infty$-groupoid stable but not strict. Then you can go around saying “weak stable $\infty$-groupoids correspond to stable homotopy theory”.

Stable homotopy theory is a kind of halfway house between the wild, scary world that is full-fledged homotopy theory and the cozy corner called homology theory. As you probably know, many homotopy theorists (like Peter May) are secretly stable homotopy theorists at heart.

Weak $\infty$-categories are even more wild and scary than weak $\infty$-groupoids — we’re not really getting into that dangerous territory here!

Anyway, all this stuff becomes quite a bit simpler when our $\infty$-groupoids are mere 1-groups. Then the ‘strict versus weak’ distinction becomes less important, and the ‘abelian versus nonabelian’ distinction is the main thing left. And that’s about all I was talking about in my remarks above.

So the obvious conclusion is that (somehow) abelian Galois groups correspond to strict $n$-category theory and general Galois groups to weak $n$-category theory. Do you have a direct sense of what that correspondence is?

Again, while braver people may want to tackle your question at full strength, I prefer to focus on just what we’re losing when we go from the nonabelian group $\pi_1$ to the abelian group $H_1$… and how this plays itself out in number theory.

I find it easier to think about this using topology… and then at the last minute say that I secretly meant not ordinary topology, but ‘étale topology’. (Then I hope nobody asks me exactly what I meant by that!)

So, I’d say that ordinary Galois groups of algebraic number fields are a special case of ‘étale fundamental groups’: subgroups of these classify things like ‘covering spaces’ that are really field extensions. Abelianized Galois groups of algebraic number fields are a special case of ‘étale first homology groups’: subgroups of these classify things like ‘abelian covering spaces’ that are really field extensions with abelian Galois groups.

I don’t know if this helps. I could say more, but maybe I’ve already said too much — it might be too technical for you, and it might also be somewhat incorrect!

Maybe we should talk about ‘abelian covering spaces’.

Posted by: John Baez on March 3, 2008 1:55 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

strict stable $\infty$-groupoids correspond to homology theory, while weak $\infty$-groupoids correspond to homotopy theory

Suppose somebody were interested in generalized cohomology, nonabelian cohomology and generalized differential cohomology; would he be well advised to concentrate on models that involve strict stable $\infty$-groupoids?

Cosmic Cube of $\infty$-Categories on page 10 here.

Is there a hidden fourth dimension to these cubes where $\mathbb{Z}$-categories and spectra appear?

The cube on p. 12 has “chain complexes” in the top right rear box. Is that properly speaking just “non-negatively graded chain complexes”?

In the language of these cubes, what would correspond to the replacement of a space $B G$ by its “rationalization” $\prod_i K(\mathbb{Q},f(i))$ (mentioned here) which can, I think, be read as replacing an $n$-groupoid $B G$ (thought of as the one-object $n$-groupoid obtained from an $n$-group $G$) by an abelian $m$-group $\prod_i B^{f(i)} \mathbb{Q}$, for $m \gt\gt n$, possibly $m = \infty$?

Posted by: Urs Schreiber on March 3, 2008 10:04 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Urs wrote:

Suppose somebody were interested in generalized cohomology, nonabelian cohomology and generalized differential cohomology; would he be well advised to concentrate on models that involve strict stable $\infty$-groupoids?

No!

A generalized cohomology theory is the same as a spectrum, which is the same as a $\mathbb{Z}$-groupoid. (By ‘the same as’, I mean there’s an equivalence of some sort.)

A generalized cohomology theory with vanishing cohomology groups below the $0$th is the same as a connective spectrum, which is the same as a stable $\infty$-groupoid.

A product of possibly shifted ‘ordinary’ cohomology theories is the same as a product of Eilenberg–Mac Lane spectra, which is the same as a strict stable $\mathbb{Z}$-groupoid, which is the same as a $\mathbb{Z}$-graded chain complex of abelian groups.

As a special case, a $\mathbb{N}$-graded chain complex of abelian groups is the same as a simplicial abelian group, or a strict $\infty$-category internal to $AbGp$.

Nonabelian cohomology is very much about $\infty$-groupoids that are neither stable nor strict!

Is there a hidden fourth dimension to these cubes where $\mathbb{Z}$-categories and spectra appear?

Sorta, though I think of that as an intensification of the passage from $n$-groupoids to $\infty$-groupoids. I suppose we could add two dimensions and speak of $[n,m]$-groupoids, which have morphisms going from $n$-morphisms on up to $m$-morphisms, where $m$ can actually be $\infty$ and $n$ can actually be $-\infty$.

The cube on p. 12 has “chain complexes” in the top right rear box. Is that properly speaking just “non-negatively graded chain complexes”?

Yes.

(That’s what I often mean when I say ‘chain complexes’, actually. When we get to $\mathbb{Z}$-graded things, the difference between ‘chain complexes’ and ‘cochain complexes’ dissolves to some extent, so we might just call those ‘complexes’. But in practice, it’s good to always ask what people mean when they talk about chain complexes or cochain complexes!)

Posted by: John Baez on March 3, 2008 8:21 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Nonabelian cohomology is very much about $\infty$-groupoids that are neither stable nor strict!

Oh, right, certainly when I mentioned nonabelian and stable in the same sentence I wasn’t properly paying attention. After all, stable means: “as abelian as possible”.

But strict? I was asking in followup to your discussion with Tom, where you mentioned that

$\left. \array{ strict \\ homology } \right\} \to \left\{ \array{ weak \\ homotopy } \right.$

So I thought I’d ask if in this sense the study of cohomology (abelian or not!?) is that of strict $\infty$-things.

Posted by: Urs Schreiber on March 3, 2008 8:35 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Nonabelian cohomology is very much about $\infty$-groupoids that are neither stable nor strict!’

Is there some place I can read about this statement?

Posted by: Minhyong Kim on March 4, 2008 11:49 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Nonabelian cohomology is very much about $\infty$-groupoids that are neither stable nor strict!

Is there some place I can read about this statement?

Maybe two remarks:

the “stable” here is a red herring which I think I brought up by not concentrating. Since “stable” means “as abelian as possible” it is clear that nonabelian cohomology is not about stable things.

Concerning strictness, notice John’s reply below, where he says that non-strict $\infty$-groupoids are needed for generalized cohomology.

Now, I must say that it seems to me that there is at least some overlap between generalized cohomology theories and nonabelian cohomology. For instance K-theory is a generalized cohomology theory, and the classifying space for $K^0$ may be taken to be $B U \times \mathbb{Z}$. But that, in turn, can be regarded as a space for nonabelian cohomology, coming from the strict 1-groupoid $\mathbf{B} U \times \mathbb{Z}$ (first factor the one-object groupoid corresponding to the group $U$).

Also, for what it’s worth, it seems the person who first thought about nonabelian cohomology (or among the first), apparently John Roberts #, always thought (and still does in his more recent work #) in terms of strict $\infty$-categories.

In fact, he was half-way the inventor of strict $\infty$-categories, being the one who invented “complicial sets” and conjecturing that these are the nerves of strict $\infty$-groupoids.

This was later proven by Ross Street, who recalls that story on p. 10, 11 of An Australian conspectus of higher categories.

There it says:

[Roberts] had worked on (strict) $n$-categories because he thought they were what he needed as coefficient structures in non-abelian cohomology.

Posted by: Urs Schreiber on March 5, 2008 9:57 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

I wrote:

This was later proven by Ross Street, who recalls that story on p. 10, 11 of An Australian conspectus of higher categories.

In fact, Ross Street, motivated by remarks by Jack Duskin (p. 16) then went on to fully develop the notion of “nonabelian cohomology” (p. 18). And he does so entirely within the context of strict $\infty$-categories ($\omega$-categories) – well, or maybe simplicial $\omega$-categories….

A more detailed discussion of this is in

I have absorbed and greatly profited from the first couple of pages of that. I wish I had penetrated more into the material further towards the end. I know I would greatly profit again, but it is tough going.

In section 9 Street discusses the Gray tensor product on strict $\infty$-categories which we are talking about here.

Posted by: Urs Schreiber on March 5, 2008 11:02 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Minhyong wrote:

John wrote:

Nonabelian cohomology is very much about $\infty$-groupoids that are neither stable nor strict!

Is there some place I can read about this statement?

Here’s what I wish I’d read when I was trying to understand nonabelian cohomology:

There’s a lot of stuff here, but starting on page 6, in the section ‘Grothendieck’s dream’, you’ll see a quick account of how fibrations of $n$-groupoids

$1 \to F \to E \to B \to 1$

should be classified by ‘nonabelian cohomology’

$hom(B, AUT(F)),$

and then you’ll see a more detailed working out of this in the case of 1-groupoids that are actually just gorups! Then we are classifying short exact sequences of groups by ‘Schreier theory’, which is the simplest and oldest form of nonabelian cohomology: the 2nd cohomology of the group $B$ with coefficients in the nonabelian group $F$!

For more, jump to around page 22, where I give a fairly general definition of nonabelian cohomology for $n$-groupoids that are neither stable nor strict.

Urs writes:

In fact, Ross Street, motivated by remarks by Jack Duskin (p. 16) then went on to fully develop the notion of “nonabelian cohomology” (p. 18). And he does so entirely within the context of strict $\infty$-categories ($\omega$-categories) – well, or maybe simplicial $\omega$-categories.

Now we’re talking about further extensions of the concept of nonabelian cohomology: Ross Street ultimately wants to say that the ‘cohomology’ of a weak $\infty$-category $B$ with coefficients in a weak $\infty$-category $A$ is itself a weak $\infty$-category, simply $hom(B,A)$! But, he and John Roberts and Jack Duskin did a lot of work on this idea using strict $\infty$-categories, simply because these only these were well-understood.

In my initial attempt to explain what’s going on, when I said ‘cohomology’ I meant cohomology as topologists use the term: ordinary cohomology or generalized cohomology, not these ultra-general notions of ‘nonabelian cohomology’.

Why? Because when you make the notion of cohomology that general, it boils down to nothing but the concept of ‘hom’! That’s a great viewpoint, but it’s not yet what your average guy means by cohomology.

Posted by: John Baez on March 5, 2008 6:04 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

I had trouble agreeing with your suggestion that co/homology was about strict stable $\infty$-groupoids, not just because you mentioned ‘nonabelian’ cohomology, but also because you mentioned ‘generalized’ cohomology. Both of these stretch the old concept of cohomology.

Leaving out some of the fine print that cluttered my previous overly technical reply, we may say that ‘ordinary’ co/homology is about strict stable $\infty$-groupoids, while ‘generalized’ co/homology is about weak stable $\infty$-groupoids. So, you could say ‘generalized’ means ‘not necessarily strict’.

Topologists would say it another way, which amounts to the same thing: ‘generalized’ co/homology is about spectra, rather than mere chain complexes. (A spectrum is like a weak version of a chain complex.)

It would be wonderful if I could use my knowledge of these subjects to somehow cancel out some of my miserable ignorance of the actual topic of this blog entry: the application of ideas from homotopy theory to number theory.

Posted by: John Baez on March 4, 2008 6:33 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

A comment from the peanut gallery: whenever I see the symbols “$Gal(\overline{\mathbb{Q}}|\mathbb{Q})$” my eyes glaze over and I think whimsically about the Grothendieck-Teichmuller group. It was explained to me by a number theory guru (who has also given a course on it). One day John will explain it in simple terms to all of us on This Week’s Finds, as he indicated in TWF201 :-)

As far as I understand it (which may be way out), it says the following : the absolute Galois group $Gal(\overline{\mathbb{Q}}|\mathbb{Q})$ is (conjecturally) isomorphic to the group of deformations of the braided monoidal category $\hat{Braid}$.

Recall the braided monoidal category $Braid$. It has objects the natural numbers $n \in \mathbb{N}$, and the hom-sets are empty unless $n=m$, when they are just the braid group:

(1)$\Hom_{Braid}(n,n) = Braid_n.$

The braided monoidal category $\hat{Braid}$ is the same, except we take the profinite completions of the hom-sets:

(2)$\Hom_{\hat{Braid}}(n,n) = \hat{Braid_n}.$

A deformation of this braided monoidal caetgory means forming a new braiding and a new associator by postcomposing the old ones with pure braids, such that the new ones still satisfy the pentagon and hexagon conditions. These form a group, which is marvelously conjecturally isomorphic to $Gal(\overline{\mathbb{Q}}|\mathbb{Q})$.

Number theory and braided monoidal categories!

Posted by: Bruce Bartlett on March 2, 2008 12:16 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

I’ve never tried too hard to understand the GT theory. But in case some people haven’t seen it before, I will remark that the connection with G (=$Gal(\bar{\mathbb{Q}}/\mathbb{Q})$) comes exactly from the embedding $\rho^{na}:G \hookrightarrow Out (\pi^{et}_1(\bar{X}))$ alluded to above, where $X$ is the projective line minus three points. (Actually, I spoke of the automorphisms rather than the outer automorphisms, but it’s still an injection when you pass to the quotient. This allows us to suppress reference to a basepoint.) One definition of the profinite GT group is as a subgroup of $Out (\pi^{et}_1(\bar{X}))$ satisfying certain natural constraints that come from viewing $X$ as the configuration space of four points on $\mathbb{P}^1$, and then examining relations to the configuration space of five points. Roughly speaking, you require the automorphisms to act on the $\pi_1$ of both configuration spaces, and be compatible with natural maps between them. Sorry I’m not being more precise at the moment. But I will say that these constraints define a subgroup $GT\subset Out (\pi^{et}_1(\bar{X})).$It’s not too hard to see that $Im(\rho^{na})$ is contained in GT. Some people then conjecture that this inclusion is actually an equality.

Posted by: Minhyong Kim on March 2, 2008 7:46 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Thanks for reminding me, Bruce!

Grothendieck–Teichmüller theory is one of those amazing conjectural connections between beautiful fundamental structure that are exactly what I like to talk about on This Week’s Finds…. so I must come back to it someday. I’ve been meaning to, off and on. I just never feel I understand it well enough.

I guess you know, Bruce, that the ‘projective line minus three points’, which Minhyong is talking about, is an algebraic version of the ‘pair of pants’ that string theorists and TQFT dudes are always talking about:

So, somehow Grothendieck–Teichmüller theory is secretly all about a relation between $Gal(\overline{\mathbb{Q}}|\mathbb{Q})$ and string theory! There should be some stunning implications for mathematical physics, but I’ve never succeeded in guessing what they are.

Posted by: John Baez on March 3, 2008 2:20 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Sheesh kebabs, everything is connected.

Posted by: Bruce Bartlett on March 3, 2008 9:49 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

These days, I don’t mind putting this sentiment in a somewhat more pretentious form: All serious inquires seem to lead to the same structures.

MK

Posted by: Minhyong Kim on March 3, 2008 10:42 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

It reminds me of the little sign people put on the windows of certain expensive art galleries: Serious inquiries only. If we had too much traffic on this blog (we don’t), we could hang a sign like that on our door.

Posted by: John Baez on March 3, 2008 6:42 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Minhyong wrote:

In relation to fundamental groups, you could say that abelian class field theory *is* the study of the Galois action on $\pi_1$ of $\mathbb{C}^*$.

Cool! So, just like $\mathbb{Z}$ is secretly all about ordinary covers of the circle, the abelianization of the holy mysterious group

$G = Gal(\overline{\mathbb{Q}}|\mathbb{Q})$

is all about these funny ‘étale covers’ of this funny circle-like thing.

Let me check to see if I can repeat what you said. (I want to learn to act like an expert, so I can infiltrate gatherings of arithmetic geometers and learn secret stuff.)

Just as the fundamental group of the circle is $\mathbb{Z}$, the étale fundamental group of the affine line minus a point is

$G^ab = \hat{\mathbb{Z}}^*$

Right? Hmm… maybe it’s not kosher to talk about “the étale cohomology of $A^1 \setminus \{ 0 \}$”. You seem to set $X = A^1 \setminus \{ 0 \}$ and then talk about the étale cohomology of $\overline{X}$. I guess we need to explicitly state that we’re working over $\overline{\mathbb{Q}}$.

Posted by: John Baez on March 3, 2008 8:53 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Close, but not quite!

Let me say something about your last point. When you say “affine line” in algebraic geometry, you technically have to say what field (or ring, or scheme, or…) you want your affine line to be over. This is kind of like saying “trivial fibration” in usual geometry—you have to say what the base is. Also the etale fundamental group depends on what the base is. This is just like usual geometry where the fundamental group of the total space of a fibration depends on what the base is. And just like in usual geometry, the fundamental group is an extension of the fundamental group of the base by the fundamental group of the fiber.

So, it’s perfectly fine to talk about the fundamental group of the $A^1-\{0\}$ over $Q$. It’s just the semi-direct product of the fundamental group of $A^1-\{0\}$ over $\bar{Q}$ and $\mathrm{Gal}(\bar{Q}/Q)$. (As Minhyong mentioned above, it’s a split extension because the fibration has a section, i.e. the affine line minus one point still has a rational point.) Which brings us to the action…

The etale fundamental group of a variety over $\bar{Q}$ is just the pro-finite completion of the usual fundamental group of the corresponding complex-analytic space. In the case of the affine line minus one point, we get $\hat{Z}$, not $\hat{Z}^*$. This is good. $\hat{Z}$ is much more like $Z$ than $\hat{Z}^*$ is.

The $\hat{Z}^*$ comes in with the Galois action. $\hat{Z}^*$ acts on $\hat{Z}$ by multiplication, and there is a perfectly natural map (called the “cyclotomic character”) $\mathrm{Gal}(\bar{Q}/Q)\to \hat{Z}^*$. Composing these two then gives the action of the absolute Galois group on the etale fundamental group of the affine line minus one point.

It might be worth looking at this on the finite quotient $Z/nZ$. Then $\mathrm{Gal}(Q(\zeta_n)/Q)=(Z/nZ)^*$, and again the action on $Z/nZ$ is given by multiplication. The original thing is just a way of collecting all this data as $n$ varies into one convenient package where you don’t have to keep track of which finite quotient you’re talking about.

By the way, there is one other fundamentally abelian example—elliptic curves with “complex multiplication”. Ignoring algebraic structure, these are just tori of the form $C/Z+\tau Z$, where $\tau$ is a imaginary quadratic irrational. Maybe if there’s enough public demand, Minhyong will be inspired to explain the Galois action there…

Posted by: James on March 4, 2008 12:11 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

What you write is correct. If we put $X=A^1\setminus \{0\}$, then it is the $\pi_1^{et}$ of $\bar{X}$ that becomes $\hat{\mathbb{Z}}$. The distinction between the fundamental groups over $\mathbb{Q}$ and $\bar{\mathbb{Q}}$ has to do with the fact that $X$ has etale covers $X_F$ coming from just constant field extensions $F$ of $\mathbb{Q}$. That is, if we define $X_F$ to be the pull-back of $X$ via the map $Spec(F) \rightarrow Spec(\mathbb{Q}),$ then $X_F \rightarrow X$ is an etale map. Going to the limit over all such $F$, we get $\bar{X}$. And then ‘composing’ with a system of coverings for $\bar{X}$ will finally yield the universal cover of $X$ itself. This is why we get the exact sequence $0 \rightarrow \pi_1(\bar{X}) \rightarrow \pi_1(X) \rightarrow G \rightarrow 0$ I mentioned before. So $\pi_1(X)$ actually subsumes the Galois group $G$ as a quotient, and hence, is horribly complicated. $\pi_1(\bar{X})$, on the other hand, can be computed using ‘standard geometry.’ To see that the system of $n$-th power maps gives a universal cover, you just need to use the so-called ‘Riemann-Hurwitz formula.’ In that article, the formula is given for a Riemann surface, but it’s identical for an algebraic curve over $\bar{\mathbb{Q}}$. Now if you have a covering $Y\rightarrow \bar{X},$ it extends to a covering of the compactifications $Y^' \rightarrow \mathbb{P}^1$ that is ramified only over $0$ and $\infty$. It is a relatively simple exercise using the formula to see that $Y^'$ must be of genus zero and that the inverse image of $\{0,\infty\}$ is a two-point set. From this, one quickly deduces that it is isomorphic to an $n$-th power map.

When I said that abelian class field theory is the study of the Galois action on $\pi_1^{et}(\bar{X})$, I meant provisionally just that this action is the same as the action on the roots of unity. But there are more precise things one can say, for which I will try eventually to muster up the energy!

But for now, I will repeat that non-abelian class field theory in the sense of Langlands attempts to understand Galois actions on $H_i^{et}(\bar{V}, \hat{\mathbb{Z}})$ for general varieties $V$ in situations where the image of the Galois group inside $Aut(H_i^{et}(\bar{V}, \hat{\mathbb{Z}}))$ is non-abelian. (Note that for $X$, $\pi_1(\bar{X})=H_1(\bar{X})$.) Wiles’ great theorem deals with the case where $V$ is an elliptic curve over $\mathbb{Q}$ . However, replacing abelian homology by non-abelian fundamental groups as the object of the action is genuinely an additional layer of non-commutativity that is expected to yield more information about the arithmetic geometry of the entities involved.

Posted by: Minhyong Kim on March 4, 2008 12:24 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Thanks, James and Minhyong, for gently correcting some of my silly mistakes. I’ve always been vaguely puzzled about the interaction between $\hat{\mathbb{Z}}$ and $\hat{\mathbb{Z}}^*$ in number theory, and I got them tangled up in my last comment. I’m too tired to think clearly about this now, but I see you’ve offered me the royal road to forever quelling this puzzlement: we’ve got both $\hat{\mathbb{Z}}$ and $\hat{\mathbb{Z}}^*$ sitting nicely in the same situation here, the latter acting on the former.

I’m sorry for not diving directly into the more excitingly nonabelian examples, Minhyong — I’ve just got to get some basics straightened out first. But, I’m not immune to the lure of nonabelian Galois representations. For the last month or so, James Dolan have spent our biweekly afternoon chats discussing elliptic curves and the Modularity Theorem. I’m finally getting a clear picture of what modular curves look like! We’re drawing nice pictures of lots of them, relating them to dessins d’ enfants, and so on. Eventually this should connect with some of the things you’re trying to explain here… but not, alas, tonight.

Posted by: John Baez on March 4, 2008 6:53 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Well, I’m quite enjoying this exchange. I’m just tending to write quickly and compactly so I’ll be able to put down some halfway decent responses before I run out of steam.

Here are some general observations regarding levels of non-commutativity: We will continue to denote by $G$ the Galois group of the rationals. If $V$ is a usual representation on some abelian group or vector space, it is rather important to examine the *image* of $G$ inside $Aut(V).$ When this image is abelian, matters simplify considerably. This happens when $V$ is $\mu_n,$ the group of $n$-th roots of unity. As was pointed out now several times, the image then is $(\mathbb{Z}/n)^*.$ The image is also abelian in a limit of this situation over $n$. As James was pointing out, one other nice abelian image occurs for an elliptic curve $E$ with complex multiplication, say $y^2=x^3-1.$

Recall that complex multiplication on an elliptic curve occurs when over the complex numbers, it is of the form $\mathbb{C}/\lt 1,\tau \gt$ for $\tau$ an imaginary quadratic number, say $\tau=(-1+(-3)^{1/2})/2.$ This causes it to have more endomorphisms over $\mathbb{C}$ than just $\mathbb{Z}$, in fact, a rank-two subring of $K=\mathbb{Q}(\tau).$ But if the curve is defined over $\mathbb{Q}$, then the endomorphism ring is exactly $O_K$, the ring of algebraic integers in $K$. (As a general fact, this is not so easy to prove and is part of the more sophisticated theory of complex multiplication.) Furthermore, all these endomorphisms are defined over $K$. Thus, on $\pi_1(\bar{E},0),$ there is an action of $G$, but also an action of $O_K$. The statement about the field of definition of the endomorphisms implies that if we restrict the Galois action to $G_K,$ the index-two subgroup of $G$ fixing $K$, then the $G_K$-action commutes with the action of $O_K$. Finally, and this is the key point, $\pi_1(\bar{E})$, which is rank two over $\hat{\mathbb{Z}}$, is actually of rank *one* over $\hat{O}_K,$ the profinite completion of $O_K$. So, for any $g\in G_K$, we get that for any $v\in \pi_1(\bar{E})$, $g v=\chi(g)v$ for some $\chi(g) \in \hat{O}^*_K$. That is, the image of the $G_K$ lies in $\hat{O}^*_K\subset Aut (\pi_1(\bar{E})).$ So most of $G$ has abelian image. The full image is a semi-direct product of an abelian group by an order two group corresponding to the action of $Aut(K/\mathbb{Q})$ (complex conjugation). This is the main reason people managed to do so much with CM elliptic curves before more general cases. For elliptic curves without CM, which is the generic situation, the image of $G$ is almost all of $Aut (\pi_1(\bar{E}))$, creating much complexity.

I said earlier that even with abelian image, the classification of *torsors* still involves non-abelian Galois groups. I thought I would mention the simplest case of this phenomenon. For $\mu_n$, we can construct torsors by taking the set of $n$-th roots of some other number, say $\{2^{1/n}\}.$ For $n\gt 1$, this is definitely a non-trivial torsor, and the Galois action on it factors through the Galois group of $\mathbb{Q}(2^{1/n}, e^{2\pi i/n}),$ which is a solvable group. Incidentally, this is an instance of the homotopy classes of paths’ torsor on $A^1\setminus \{0\}$ I mentioned near the beginning.

Posted by: Minhyong Kim on March 4, 2008 11:38 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Since much of what I write will be tiresome to the general public, here is another small puzzle at the level of definitions:

Possibly you’ve seen, e.g., here, the definition of the etale fundamental group of a scheme X with base point b as $Aut(F_b)$ where $F_b:Cov^{et}(X) \rightarrow Finite Sets$ is the fiber functor that sends a finite etale cover $Y\rightarrow X$ to its fiber $Y_b$ over $b$. As John mentioned it is true that the Galois group $G=Gal(\bar{\mathbb{Q}}/\mathbb{Q})$ can be recovered as the fundamental group of $Spec(\mathbb{Q})$, unifying homotopy theory and Galois theory. The question is: where is the base-point in this assertion?

Posted by: Minhyong Kim on March 5, 2008 12:15 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Even if missing the (rational) point of the discussion in this thread, I’d like to ask about different definitions of such homotopy groups. E.g., if I remember correctly, Nori, Voevodsky and Kurke defined such things. Are there survey articles comparing them?

Posted by: Thomas Riepe on March 5, 2008 1:21 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

You probably know that Voevodsky’s theory was originally supposed to be modeled on *stable* homotopy theory, hence the triangulated categories and so forth. The triangulated category he ends up with has the expected properties of the derived category of the category of mixed motives. That is, one has a list of properties that the *abelian* category of mixed motives should have, from which one infers a list for the derived category of that abelian category. Voevodsky constructs such a triangulated category. Although it’s a wonderful theory, many anabelian types’ seem to consider the whole construction too close to abelian.

Subsequently, there was an unstable theory developed by Morel, Levine, and so on. I haven’t looked into it at all, but according to some of the experts, the relation to Grothendieck’s etale pi_1 is rather unclear.

Posted by: Minhyong Kim on March 7, 2008 9:41 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

where is the base-point in this assertion?

the “base-point” is the algebraic closure “q-bar”. that is, the fundamental group_oid_ of spec(q) is (at least morally) the groupoid of algebraic closures of the field q.

Posted by: james dolan on March 7, 2008 11:49 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

That’s right! This is of course one of the interesting elementary points of the theory. A *point* of a scheme $X$ (that is, one that will be used as a basepoint for the fundamental group) is a map $Spec(F) \rightarrow X$ where $F$ is separably closed field. So, for $\mathbb{Q}$, a choice $\mathbb{Q} \hookrightarrow \bar{\mathbb{Q}}$ of an algebraic closure will correspond to a map $Spec(\bar{\mathbb{Q}}) \rightarrow Spec(\mathbb{Q}) )$ that can then be considered a base-point. In many expositions on arithmetic $\pi_1$’s, you’ll see the point emphasized that the fundamental group is just a Galois group of some sort. However, this is a natural interpretation only when you are consider such large’ base-points. That is, if $X$ is a variety, the fundamental group becomes a Galois group naturally if one take the base-point to be something like $Spec(F)\rightarrow Spec(K) \rightarrow X$ where $F$ is the field of functions on $X$ and $K$ is an algebraic closure. This point of view obscures the case where we take a small’ base point that starts from, say, a rational point $x$ of a variety $X$ over $\mathbb{Q}$. If one focuses on those, than one can study the *variation* (as a function of $x$) of $\pi_1(X,x)$ or $\pi_1(X;b,x)$ profitably, which is actually the main topic of the lecture that started this discussion. It is in such arithmetic contexts one sees a fruitful realization of the observation that $\pi_1(X,x)$ and $\pi_1(X,y)$ are isomorphic but nor canonically isomorphic, which seems rather pedantic in introductory algebraic topology.

Posted by: Minhyong Kim on March 7, 2008 11:58 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

A neat introduction to etale homotopy groups was in Artin and Mazur’s SLNM 100. The links with hypercoverings is made clear in fairly simple terms and is clearly relevant to lots of other topics discussed in this blog. The original Pursuit of Stacks by Grothendieck was in that vein, so the weak $n$-groupoid as model for homotopy type paradigm was there in the 1970s and in relation exactly to etale homotopy theory.

Years ago I had a student who investigated the extent to which the profinite group cohomology theory then around could be modernised’ via a liberal addition of crossed modules, crossed $n$-fold extensions (a la Huebschmann) etc. He also looked at tensor product constructions for profinite groups and a whole lot more. Most of the extensions of the discrete theory went through without pain but some needed hard grind. Some of this was published and I have typed up an extended version of his work, adding a whole lot more background, with the idea of publishing it. It is nearly 400 pages long so be warned but it may be relavant to some of the deliberations here. I found that non-Abelian cohomology is difficult to push through to the profinite case because many of the approaches do use Aut(G) and that is not too clear in the profinite group case. I can give more precision on this if someone want.

Posted by: Tim Porter on March 7, 2008 8:36 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

R. Vakil just placed a draft of a preprint “Universal covering spaces and fundamental groups in algebraic geometry as schemes” on his website.

Posted by: Thomas on January 19, 2009 6:39 PM | Permalink | Reply to this
Read the post Afternoon Fishing
Weblog: The n-Category Café
Excerpt: A question on a picture of Spec(Q).
Tracked: April 23, 2009 12:34 PM
Read the post The Elusive Proteus
Weblog: The n-Category Café
Excerpt: Continuing the thought that mathematics and physics share in having difficult to articulate general principles.
Tracked: June 5, 2009 2:00 PM
Read the post Galois Theory in Two Variables
Weblog: The n-Category Café
Excerpt: I have just returned from attending three days of the Final Workshop of the Newton Institute Non-Abelian Fundamental Groups in Arithmetic Geometry Programme. I was kindly invited by Café visitor Minhyong Kim, whose lecture we discussed a while a...
Tracked: December 18, 2009 10:17 AM

### Re: Kim on Fundamental Groups in Number Theory

I’d like to know what you folks think of the Galois theory chapter in Fuchs and Tabachnikov’s book Mathematical omnibus - Thirty lectures on classic mathematics (the chapter is available on the Google Books link).

There they give a very nice elementary geometric proof for why the equation

(1)$x^5 - x + a$

is not solvable in radicals. My impression is that their approach is somehow in the spirit of what you guys are saying above.

They argue geometrically as follows. When $a=0$, the equation has five roots, $0, \pm 1, \pm i$. As $a$ varies, these roots move around in the complex plane. There are four “dangerous” values: $\pm \frac{4}{5 \sqrt[4]{5}}$ and $\pm \frac{4i}{5 \sqrt[4]{5}}$, where some of the roots coalesce.

What happens as $a$ traverses a closed loop, beginning at the origin, and looping around one of these dangerous points? By doing an elementary calculation, one finds that the roots don’t return to their starting values, but rather undergo a permutation.

Then one proves that the permutation (“holonomy”) is quite complicated: more complicated than can be obtained from any formula for the roots in terms of radicals. This completes the proof! You can’t solve it in radicals because radicals don’t have complicated enough holonomy.

I think that’s really cool. It’s geometric. It manifestly brings in the notion of a braid (those roots trace out a braid when $a$ goes in a closed loop). So it is manifestly close to Grothendieck-Teichmuller stuff.

What do the experts think? Is this indeed in the spirit of all the “actions of Galois groups on covers…” above? The Galois group is featuring as a functor from the homotopy groupoid $\Gamma = \mathbb{C} - 4 nasty points$ to Set:

(2)$Z : \Gamma \rightarrow \Set$

On objects, it sends

(3)$a \mapsto \{ roots of polynomial x^5 - x + a = 0\}$

On morphisms, it sends

(4)$path from a to b \mapsto parallel transport.$

Seems to work nicely!

Posted by: Bruce Bartlett on April 19, 2010 11:31 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

the Galois theory chapter in Fuchs and Tabachnikov’s book

This is a gem! The graphical display of the movement of the roots under the action of the fundamental group is great.

Posted by: David Roberts on April 21, 2010 7:56 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Here is an article by Zoladek describing in detail how Arnold used that in courses for high school pupils. I’m pretty sure that such visual ideas were used by Abel, Galois and others.

Posted by: Thomas on April 21, 2010 8:56 AM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

It looks like a nice exposition, even though I gave it only a glance. Note that they prove insolvability for the *general* such polynomial, not for a specific $a$. That is, they are interested in the multi-valued functions that come up as solutions when $a$ is just regarded as the coordinate function on the plane. Using the language of usual Galois theory, the gist of such arguments should run like this:

Let $G\subset S_5$ be the image of the fundamental group in the automorphism group of the fiber over some fixed point, say 1. Then

(1) $G$ acts transitively on the fiber;

(2) $G$ contains a transposition.

This automatically implies that $G=S_5$, because (1) implies there is an element of order 5 (by Sylow’s theorem), that is, a five cycle, and then, elementary group theory shows that a five cycle and a transposition generate all of $S_5$. Meanwhile, I presume they are implicitly using (or showing) the fact that coverings obtained by radicals give solvable groups.

I thought I’d mention that even for a *specific* number $a$, the usual technique in elementary algebra actually uses such monodromy arguments. Except the monodromy is going around loops in $\Spec(\Z)$. That is, even when the initial covering just lies over $Spec(Q)$, we can ‘stretch it out’ to a covering of $Spec(Z)$ with just a few dangerous values, and use arguments similar to that in the book you refer to. A standard textbook example is

$x^5-x+1,$

which has exactly three real roots. This implies that the field automorphism given by complex conjugation induces a transposition. That is, we are using a ‘loop around infinity’ to get at the permutation action. Since a loop around infinity is a bit exotic, another example is

$x^5-x-7.$

You can check that this equation again has exactly three roots in $F_7$ and two more lying in a quadratic extension of $F_7$. So there will be a pair of roots transposed by the automorphism group of $\overline{F_7}$. Because 7 is not among the ‘dangerous points’ (this is in fact just saying that the roots remain distinct mod 7) one sees that a lift of this transposition to the Galois group of $Q$ must also transpose two roots. So in this case, monodromy around the little knot at 7 is enough to show that the Galois group is $S_5$.

There are interesting examples where you actually use the monodromy around the dangerous points (the ramified primes, in the language of number theory), but the only ones I know of are rather involved. For example, take the curve

$E:$ $y^2=x^3-16x+16.$

You probably know that the points form a group. Now look at all the points $(x_i,y_i)$ that have order 11 with respect to this group law. There are 120 such points. Then let

$Q(E[11])$

be the field generated by all the $x_i$ and $y_i$. The automorphism group of this field is

$GL_2(F_{11}).$

As far as I know, the proof definitely uses a careful analysis of the loop around the dangerous prime (the ‘inertia action’), which is 37 in this case.

For those of you who like $F_1$, it’s tempting to speculate about differential (or infinitesimal) versions of monodromy over spaces like $Spec(Z)$. For example, as I understand it, the early transcendence results for certain analytic functions used the infinitude of their monodromy. Would it be possible to give a proof of transcendence for numbers using infinitude of monodromy in some sense? This is the kind of down-to-earth consequence of $F_1$-theory it would be nice to see explored.

Posted by: Minhyong Kim on April 21, 2010 12:22 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Argh. Just after posting, I realized that the argument for

$x^5-x+1$

is incorrect. It seems to have only one real root.

But

$x^5-x+1/4$

should work.

Posted by: Minhyong Kim on April 21, 2010 12:35 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

In case any analysts would like to review their elementary algebra, perhaps I should also set exercises: For

$x^5-x-7$

and

$x^5-x+1/4,$

which knots should we use to produce the 5-cycle, now that we have the transposition? (I hope I did it correctly!)

Posted by: Minhyong Kim on April 21, 2010 1:24 PM | Permalink | Reply to this

### Re: Kim on Fundamental Groups in Number Theory

Minhyong has written a very nice account of fundamental groups in arithmetic geometry here.

Posted by: David Corfield on May 10, 2010 11:56 AM | Permalink | Reply to this

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