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I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about Algebraic $K$-theory mention that we don't know $K(\mathbb{Z})$ and that somehow $K(\mathbb{Z})$ is worth computing because it contains lots of arithmetic information. I'd like to better understand what kinds of arithmetic information it contains. I've been unable to answer number theorists who've asked me this before. A related question is about what information is contained in $K(S)$ where $S$ is the sphere spectrum.

I am aware of Vandiver's Conjecture and that it is equivalent to the statement that $K_n(\mathbb{Z})=0$ whenever $4 | n$. I also know there's some connection between $K$-theory and Motivic Homotopy Theory, but I don't understand this very well (and I don't know if $K(\mathbb{Z})$ helps). It seems difficult to search for this topic on google. Hence my question:

Can you give me some examples of places where computations in $K(\mathbb{Z})$ or $K(S)$ would solve open problems in arithmetic or would recover known theorems with difficult proofs?

I'm hoping someone who has experience motivating this field to number theorists will come on and give his/her usual spiel. Here are some potential answers I might give a number theorist if I understood them better...The wikipedia page for Algebraic K-theory mentions non-commutative Iwasawa Theory, L-functions (and maybe even Birch-Swinnerton-Dyer?), and Bass's conjecture. I don't know anything about this, not even whether knowing $K(\mathbb{Z})$ would help. Quillen-Lichtenbaum seems related to $K(\mathbb{Z})$, but it seems it would tell us things about $K(\mathbb{Z})$ not the other way around. Milnor's Conjecture (or should we call it Voevodsky's Theorem?) is definitely an important application of $K$-theory, but it's the $K$-theory of field of characteristic $p$, not $K(\mathbb{Z})$.

There was a previous MO question about the big picture behind Algebraic K Theory but I couldn't see in those answers many applications to number theory. There's a survey written by Weibel on the history of the field, and that includes some problems it's solved (e.g. the congruence subgroup problem) but other than Quillen-Lichtenbaum I can't see anything which relies on $K(\mathbb{Z})$ as opposed to $K(R)$ for other rings. If $K(\mathbb{Z})$ could help compute $K(R)$ for general $R$ then that would be something I've love to hear about.

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Teena Gerhardt and her collaborators have done a number of computations for rings like Z[x]/(x^2) in which the answers are given (essentially) in terms of K(Z). So this would be in the direction of your last sentence. –  Dan Ramras May 5 '13 at 19:58
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You might enjoy section 1.1 of Clark Barwick's Göttingen talks, which serves exactly the purpose of motivating homotopy theory for number theorists. There's not so much specifically on K(Z) though... dl.dropbox.com/u/1741495/papers/barwick.pdf –  Peter Arndt May 5 '13 at 21:24
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@David: I have the feeling that $K_*(\mathbb{S})$ helps much more in geometry than in arithmetic, more precisely for the study of homeomorphism groups of high-dimensional manifolds and h-cobordisms. See folk.uio.no/rognes/papers/plmf.pdf or math.nagoya-u.ac.jp/~larsh/papers/023/whitehead.pdf for example. –  Lennart Meier May 6 '13 at 0:14
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Something that Persiflage wrote only a few days ago is related to your question : galoisrepresentations.wordpress.com/2013/05/04/… –  Chandan Singh Dalawat May 6 '13 at 3:20
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By the way, we DO know the K theory of Z, except in degree 8,12,16,20,24.... (where it is conjectured to be 0), see math.uiuc.edu/K-theory/0691/KZsurvey.pdf –  user15817 May 6 '13 at 12:44
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2 Answers

up vote 45 down vote accepted

$\newcommand\Z{\mathbf{Z}}$ $\newcommand\Q{\mathbf{Q}}$

I'm a number theorist who already thinks of the algebraic $K$-theory of $\Z$ as part of number theory anyway, but let me make some general remarks.

A narrow answer: Since (following work of Voevodsky, Rost, and many others) the $K$-groups of $\Z$ may be identified with Galois cohomology groups (with controlled ramification) of certain Tate twists $\Z_p(n)$, the answer is literally "the information contained in the $K$-groups is the same as the information contained in the appropriate Galois cohomology groups." To make this more specific, one can look at the rank and the torsion part of these groups.

  1. The ranks (of the odd $K$-groups) are related to $H^1(\Q,\Q_p(n))$ (the Galois groups will be modified by local conditions which I will suppress), which is related to the group of extensions of (the Galois modules) $\Q_p$ by $\Q_p(n)$. A formula of Tate computes the Euler characteristic of $\Q_p(n)$, but the cohomological dimension of $\Q$ is $2$, so there is also an $H^2$ term. The computation of the rational $K$-groups by Borel, together with the construction of surjective Chern classes by Soulé allows one to compute these groups explicitly for positive integers $n$. There is no other proof of this result, as far as I know (of course it is trivial in the case when $p$ is regular).

  2. The (interesting) torsion classes in $K$-groups are directly related to the class groups of cyclotomic extensions. For example, let $\chi: \mathrm{Gal}(\overline{\Q}/\Q) \rightarrow \mathbf{F}^{\times}_p$ be the mod-$p$ cyclotomic character. Then one can ask whether there exist extensions of Galois modules:

$$0 \rightarrow \mathbf{F}_p(\chi^{2n}) \rightarrow V \rightarrow \mathbf{F}_p \rightarrow 0$$

which are unramified everywhere. Such classes (warning: possible sign error/indexing disaster alert) are the same as giving $p$-torsion classes in $K_{4n}(\Z)$. The non-existence of such classes for all $n$ and $p$ is Vandiver's conjecture. Now we see that: The finiteness of $K$-groups implies that, for any fixed $n$, there are only finitely many $p$ such that an extension exists. An, for example, an explicit computation of $K_8(\Z)$ will determine explicitly all such primes (namely, the primes dividing the order of $K_8(\Z)$). As a number theorist, I think that Vandiver's conjecture is a little silly --- its natural generalization is false and there's no compelling reason for it to be true. The "true" statement which is always correct is that $K_{2n}(\mathcal{O}_F)$ is finite.

Regulators. Also important is that $K_*(\Z)$ admits natural maps to real vector spaces whose image is (in many cases) a lattice whose volume can be given in terms of zeta functions (Borel). So $K$-theory is directly related to problems concerning zeta values, which are surely of interest to number theorists. The natural generalization of this conjecture is one of the fundamental problems of number theory (and includes as special cases the Birch--Swinnerton-Dyer conjecture, etc.). There are also $p$-adic versions of these constructions which also immediately lead to open problems, even for $K_1$ (specifically, Leopoldt's conjecture and its generalizations.)

A broader answer: A lot of number theorists are interested in the Langlands programme, and in particular with automorphic representations for $\mathrm{GL}(n)/\Q$. There is a special subclass of such representations (regular, algebraic, and cuspidal) which on the one hand give rise to regular $n$-dimensional geometric Galois representations (which should be irreducible and motivic), and on the other hand correspond to rational cohomology classes in the symmetric space for $\mathrm{GL}(n)/\Q$, which (as it is essentially a $K(\pi,1)$) is the same as the rational cohomology of congruence subgroups of $\mathrm{GL}_n(\Z)$. Recent experience suggests that in order to prove reciprocity conjectures it will also be necessary to understand the integral cohomology of these groups. Now the cohomology classes corresponding to these cuspidal forms are unstable classes, but one can imagine a square with four corners as follows:

stable cohomology over $\mathbf{R}$: the trivial representation.

unstable cohomology over $\mathbf{R}$: regular algebraic automorphic forms for $\mathrm{GL}(n)/\Q$.

stable cohomology over $\mathbf{Z}$: algebraic $K$-theory.

unstable cohomology over $\mathbf{Z}$: ?"torsion automorphic forms"?, or at the very least, something interesting and important but not well understood.

From this optic, algebraic $K$-theory of (say) rings of integers of number fields is very naturally part of the Langlands programme, broadly construed.

Final Remark: algebraic K-theory is a (beautiful) language invented by Quillen to explain certain phenomena; I think it is a little dangerous to think of it as being an application of "homotopy theory". Progress in the problems above required harmonic analysis and representation theory (understanding automorphic forms), Galois cohomology, as well as homotopy theory and many other ingredients. Progress in open questions (such as Leopoldt's conjecture) will also presumably require completely new methods.

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Beautiful answer! –  Mariano Suárez-Alvarez May 5 '13 at 23:45
    
As a narrow-minded person I find your narrow answer best! But to be even more concrete: Are there explicit arithmetic application of the Voevodsky-Rost result (or the integral computation of $K_*i(\mathbb{Z})$ for $i$ not divisible by $4$) which do not contain K-theory in their statement? –  Lennart Meier May 6 '13 at 0:06
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@Rebecca: Thanks, this is a great answer! I'm somewhat amazed that the Langlands program is involved with algebraic K-theory, but you explain this point very well. And that connection to Galois cohomology groups is great. Also, welcome to MathOverflow! –  David White May 6 '13 at 4:11
    
@Frictionless Jellyfish: I have to say, I find your broader answer very mysterious. Could you expand on your four analogies? –  Daniel Litt Jul 2 '13 at 3:57
    
@DanielLitt: Dear Daniel, The stable cohomology (with $\mathbb C$ coefficients, say, although FJ takes $\mathbb R$-coefficients --- but it doesn't really matter, as long as it's a field of char. zero) for $\mathrm{SL}$ comes (from an automorphic point of view) from the trivial subrepresentation of $L^2(\mathrm{SL}_n(\mathbb Z) \backslash \mathrm{SL}_n(\mathbb R))$. (``Comes from'' refers to a generalization of Eichler--Shimura theory, in which $(\mathfrak g, \mathfrak k)$-cohomology of automorphic representations is identified with group cohomology.) The unstable cohomology comes from ... –  Emerton Jul 28 '13 at 5:03
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Let $p$ is an odd prime and $C$ the $p$-Sylow of the class group of $\mathbb{Q}(\zeta_p)$. If $C^\sigma$ denotes the group fixed by complex conjugation then Vandiver's conjecture is that $C^\sigma = 0$. Both Kurihara and Soulé have made some partial progress towards this conjecture, and their methods rely on knowledge of the torsion piece of the groups $H_i(\mathbb{Z})$. A good introduction is Soulé's 14-page paper on the matter entitled "Perfect forms and the Vandiver conjecture". Kurihara's paper "Some remarks on conjectures about cyclotomic fields and $K$-groups of $\mathbb{Z}$" is another very readable source, which points out many arithmetic applications.

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Thanks for your answer. It seems Vandiver's Conjecture is the standard motivation for trying to understand $K_*(\mathbb{Z})$. I was mostly interested in other problems in arithmetic which could be solved by computations in $K_*(\mathbb{Z})$, so I'm going to hold off on accepting your answer because I'm hoping for more. Still, it sounds like Kurihara's paper might contain some further applications, so I'll look into that paper soon. –  David White May 5 '13 at 22:53
    
That's ok, I was in a hurry typing this and admittedly did not read the question as thoroughly as I should have. –  Jason Polak May 5 '13 at 22:57
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