### The Algebraic K-Theory of the Integers

#### Posted by John Baez

The category of groups $\mathbb{Z}^n$ and isomorphisms between these is symmetric monoidal under $\oplus$. You can build a space out of simplexes where the 0-simplexes are objects of this category, the 1-simplexes are morphisms, the 2-simplexes are commutative triangles, the 3-simplexes are commutative tetrahedra, and so on forever. This space has an operation, coming from $\oplus$, that obeys the commutative monoid axioms up to homotopy. If you ‘group complete’ this space by throwing in formal inverses, you get a space that’s an abelian group up to homotopy. It’s called the **algebraic $K$-theory spectrum** of the integers.

The algebraic $K$-theory spectrum of the integers has homotopy groups $\pi_0 = \mathbb{Z}$, $\pi_1 = \mathbb{Z}/2$, $\pi_3 = \mathbb{Z}/48$, and so on. These groups are called the **algebraic $K$-theory groups** of the integers, $K_n(\mathbb{Z})$.

The algebraic $K$-groups of the integers, and other rings, are a big deal. A lot of fairly recent progress on understanding them is due to Voevodsky, Rost, Rognes and Weibel, whose work culminated in a proof of the Bloch–Kato Conjecture connecting $K$-theory to Galois cohomology. But lots of other mathematicians deserve credit too.

Three quarters of the algebraic $K$-groups of the integers are known. That is, they’re all known except for the groups $K_{4n}(\mathbb{Z})$, which will be known if someone can prove the Kummer–Vandiver Conjecture. This is a conjecture about prime numbers.

The Kummer–Vandiver Conjecture been verified for all primes less than 163,000,000. Nonetheless, some unimpressed Wikipedia editor says

there is no particularly strong evidence either for or against the conjecture.

Brutal! But it makes sense, because there’s a probabilistic argument saying that *if* the conjecture is *false*, you’d expect just one counterexample in the first $10^{100}$ primes.

For an advanced tour of this subject, try

- Charles Weibel, Algebraic
*K*-theory of rings of integers in local and global fields.

For more detail, starting at the very beginning, try *The K-book*, also by Weibel.

What I’d like to understand is the relation between stable homotopy groups of spheres and algebraic K-theory groups of the integers. For example, the third stable homotopy group of spheres is $\mathbb{Z}/24$, and this is connected to $K_3(\mathbb{Z}) = \mathbb{Z}/48$. Topologists understand this connection: there’s a homomorphism from the $n$th stable homotopy group of spheres, called $\pi_n^s$, to $K_n(\mathbb{Z})$. But I’d like to understand it.

If I had to cook up such this homomorphism, here’s what I’d guess. The stable homotopy groups of spheres are the homotopy groups of a space called the ‘sphere spectrum’. So it’s enough to get a map from the sphere spectrum to the algebraic $K$-theory spectrum of the integers. And the sphere spectrum is built from a symmetric monoidal category just like the algebraic $K$-theory spectrum is! It’s built from the category of finite sets $0,1,2,\dots$ and bijections between these, made symmetric monoidal using disjoint union. So, I’d use the obvious functor sending any finite set $n$ to the group $\mathbb{Z}^n$, and any bijection of finite sets to the corresponding isomorphism of groups. This gives a map of symmetric monoidal categories, and thus a map of spectra, and thus a bunch of homomorphisms $\pi_n^s \to K_n(\mathbb{Z})$.

Is this right? Are these the right homomorphisms?

If I knew a lot more about this stuff, I’d also understand this claim: the reason $K_3(\mathbb{Z})$ is $\mathbb{Z}/48$ is that 24 is the largest natural number $n$ such that every number relatively prime to $n$ squares to 1 mod $n$. Let’s check that:

$1^2 = 1$

$5^2 = 25 = 24 + 1$

$7^2 = 49 = 24 \times 2 + 1$

$11^2 = 121 = 24 \times 5 + 1$

$13^2 = 169 = 24 \times 7 + 1$

$17^2 = 289 = 24 \times 12 + 1$

$19^2 = 361 = 24 \times 15 + 1$

$23^2 = 529 = 24 \times 22 + 1$

Yup!

Notice anything interesting about these numbers less than and relatively prime to 24?

## Re: The Algebraic K-Theory of the Integers

No doubt plenty to add to nLab: algebraic K-theory.

Regarding the relation between stable homotopy groups of spheres and algebraic K-theory groups of the integers, we have at nLab: Field with one element