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.
Best Answer
$\newcommand\Z{\mathbf{Z}}$ $\newcommand\Q{\mathbf{Q}}$
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.
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).
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.