Motivation for algebraic K-theory?

Question

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like Waldhausen's) and a lot of work devoted to calculation in special cases, e.g., extracting information about K-theory from Hochschild and cyclic homology. As far as I can tell, K-theory is extremely difficult to compute, it yields deep information about a category, and in some cases, this produces highly nontrivial results in arithmetic or manifold topology. I've been unable to piece these results into a coherent picture of why one would think K-theory is the right tool to use, or why someone would want to know that, e.g., K₂₂(Z) has an element of order 691. Explanations and pointers to readable literature would be greatly appreciated.

Answer 1

Algebraic K-theory originated in classical materials that connected class groups, unit groups and determinants, Brauer groups, and related things for rings of integers, fields, etc, and includes a lot of local-to-global principles. But that's the original motivation and not the way the work in the field is currently going - from your question it seems like you're asking about a motivation for "higher" algebraic K-theory.

From the perspective of homotopy theory, algebraic K-theory has a certain universality. A category with a symmetric monoidal structure has a classifying space, or nerve, that precisely inherits a "coherent" multiplication (an E_oo-space structure, to be exact), and such an object has a naturally associated group completion. This is the K-theory object of the category, and K-theory is in some sense the universal functor that takes a category with a symmetric monoidal structure and turns it into an additive structure. The K-theory of the category of finite sets captures stable homotopy groups of spheres. The K-theory of the category of vector spaces (with appropriately topologized spaces of endomorphisms) captures complex or real topological K-theory. The K-theory of certain categories associated to manifolds yields very sensitive information about differentiable structures.

One perspective on rings is that you should study them via their module categories, and algebraic K-theory is a universal thing that does this. The Q-construction and Waldhausen's S.-construction are souped up to include extra structure like universally turning a family of maps into equivalences, or universally splitting certain notions of exact sequence. But these are extra.

It's also applicable to dg-rings or structured ring spectra, and is one of the few ways we have to extract arithmetic data out of some of those.

And yes, it's very hard to compute, in some sense because it is universal. But it generalizes a lot of the phenomena that were useful in extracting arithmetic information from rings in the lower algebraic K-groups and so I think it's generally accepted as the "right" generalization.

This is all vague stuff but I hope I can at least make you feel that some of us study it not just because "it's there".

Answer 2

I think a key point is that algebraic K-theory is defined not only for rings, but also for schemes (and other kinds of "generalized spaces" in algebraic geometry). If you believe that generalized (Eilenberg-Steenrod) cohomology theories are useful/interesting in algebraic topology, then it is also reasonable to think that they might be interesting in algebraic geometry, and algebraic K-theory is in some sense the simplest and most widely studied such theory, although yes, computations are very hard.

Some other motivation:

Algebraic K-theory allows you to talk about characteristic classes of vector bundles on schemes, with values in various cohomology theories, see for example Gillet: K-theory and algebraic geometry.

Algebraic K-theory is intimately connected with motivic cohomology and algebraic cycles, see for example Friedlander's ICTP lectures available on his webpage, especially the 5th lecture on Beilinson's vision: http://www.math.northwestern.edu/~eric/lectures/ictp/

One of the major themes in arithmetic geometry is the study of special values of motivic L-functions. These values capture a lot of deep arithmetic invariants of number fields and varieties over number fields, and they seem to be mysteriously related to many other things, for example orders of stable homotopy groups of spheres. There are many results and conjectures about these values, most famously the Clay Millennium Birch-Swinnerton-Dyer conjecture, and in many versions of these conjectures, algebraic K-theory plays a crucial role. See for example the survey by Bruno Kahn in the K-theory handbook, also availably at his webpage: http://people.math.jussieu.fr/~kahn/preprints/kcag.pdf

There are also many other useful things in the K-theory handbook, such as the lectures by Gillet on K-theory and intersection theory, also available here: http://www.math.uic.edu/~henri/preprints/K-Theory_Chow_Groups-6.pdf

Answer 3

First, recall the slogan:
Small constructions are good for making calculations, but large constructions are good for proving theorems.
K-theory is certainly a large construction.
In general, K-theory seems to turn up in topology when the following slogan holds:
Chain compex good; homology bad.
You can often construct exactly the same invariant using K-theory or without, but K-theory makes the extra structure in the chain complex visible (such as Poincare duality), which makes it possible to prove theorems. As a low-dimensional topologist, the example I have in mind is Ranicki's symmetric signature. More basically, the key observation in Milnor's proof that the Alexander polynomial is palindromic is that if you construct it from the chain complex by using Reidemeister torsion, the Poincare duality becomes evident. Similarly, the Blanchfield pairing (linking pairing on the infinite cyclic cover of a knot complement) can be constructed as the symmetric signature (L-theory), which lets you see Poincare duality.
This seems completely typical. You can squish everything into the middle dimension and get information (Wall's finiteness obstruction; Alexander polynomial; Branchfield pairing...) or you can use a large K-theoretic construction to get the same information in a way which preserves the chain complex context from which it comes, revealing properties which come from the grading.

Answer 4

I suggest looking at the introduction to Waldhausen's original paper on algebraic K-theory (Algebraic K-theory of generalized free products, Part I, Ann. Math., 108 (1978) 135-204).

Waldhausen started out as a 3-manifold theorist, and he realized that certain phenomena in the topology of 3-manifolds would be explained if the Whitehead groups of classical knot groups were trivial. So he set out to prove this, and in order to do so he developed a plethora of methods for dealing with K-groups (including his definition involving the S. construction). The basic approach here is that the Whitehead group is the cokernel of the assembly map, which is a map $$H_*(BG; K(Z)) \to K_* (R[G]).$$ Here $G$ is some group (e.g. the fundamantal group of a knot complement), $R$ is some ring (e.g. $\mathbb{Z}$), and $R[G]$ is the group ring. The homology on the left is the homology theory represented by the (non-connective) $K$-theory spectrum.

The study of assembly maps for group rings is one area where $K$-theory computations look a bit more organized; one hopes to show that $K$-theory of group rings is actually a homology theory, i.e. that the assembly map is an isomorphism. Of course this homology theory itself is still quite complex, since it involves the $K$-theory of some ring $R$!

Answer 5

As a particular application of algebraic K-theory, let me mention the intersection product on regular schemes. Let X be a regular scheme over spec Z. Then, one can use the Quillen spectral sequence and Adam's operations on K-theory to produce an intersection product on the Chow groups tensored with Q. To my knowledge, this is the first definition of an intersection theory on a class of schemes larger than those smooth over a Dedekind domain. For details, see Soule's book Lectures on Arakelov Geometry. Chapter 1 of this book contains a very nice introduction to K-theory with supports and the Adam's operations. Besides that, all you need is Quillen's original paper to understand the intersection theory.

Answer 6

I found Mitchell's survey "On the Lichtenbaum-Quillen conjectures from a stable homotopy-theoretic viewpoint" very motivating. I assumes only little background and is written for mutually introducing homotopy theorists with algebraic K-theorists.

Answer 7

For me, one interesting thing about algebraic K-theory is L-theory (which I wish I understood better). This is in no way going to be coherent, but: Let's say you are interested in classifying manifolds. That's not going to be possible, because any finitely presented group is realizable as the fundamental group of some 4-manifold, and "most" finitely presented groups don't have solvable word problem. OK then, the next best thing is to try to classify manifolds within a fixed homotopy type. Surgery theory is a technique for doing this. Given a homotopy type, you construct a CW-complex X with that homotopy type, and your first question is whether X is homotopy equivalent to a manifold. Roughly speaking, this is determined by the normal bundle. So for X to be homotopy equivalent to a manifold, you want there to exist a suitably defined bundle map $(f,b)\colon M \to X$,which is normal bordant to a homotopy equivalence. This latter condition (for a bundle map to be normal bordant to a homotopy equivalence) is detected K-theoretically (I don't understand this well enough to attempt to try to explain it).
Note that homological techniques are sufficient for simply connected manifolds (Browder, Kervaire, Milnor, Novikov...), but are not useful for manifolds which are not simply connected. And this is where K-theory enters.
The main point seems to be that algebraic K-theory provides the machinery to work with quadratic forms over nonabelian group rings with involution ($\mathbb{Z}[\pi_1(M)]$ in our case).
See also Walhausen's survery, which I don't yet understand.

Answer 8

Here's a reference that gives some of the history of algebraic k-theory. It might have something you're looking for. http://www.math.uiuc.edu/K-theory/0343/khistory.pdf. Also Rosenberg's book "Algebraic K-theory and Its Applications is good.

Answer 9

Grothendieck's original motivation for K-theory was to give a natural setting for the intersection theory on algebraic varieties.

Answer 10

It's very much "thing in itself" (quote from my advisor). And indeed it's mostly of interest to people who (1) like to compute (2) don't mind the fact that there's "no general picture", which admittedly are a minority among mathematicians. In fact, there is (or was) a separate e-archive of K-theory papers!

Still, yes, it's a very important and general way to learn about abstract rings.