I'm looking for a bigpicture treatment of algebraic Ktheory 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 Ktheory from Hochschild and cyclic homology. As far as I can tell, Ktheory 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 Ktheory is the right tool to use, or why someone would want to know that, e.g., K_{22}(Z) has an element of order 691. Explanations and pointers to readable literature would be greatly appreciated.

9$\begingroup$ I'd like to emphasize that I'm not necessarily looking for holistic, coherent answers. Fragmentary brain dumps are also welcome (and may receive upvotes). $\endgroup$– S. Carnahan ♦Oct 12, 2009 at 20:26

2$\begingroup$ Could someone provide some information about the application/manifestation of K theory in arithmetic other than Lichtenbaum conjecture and Veovodsky's work? (these are great problems and provide good motivation for a theory themselves, yet I am just curious. thx.) Or shall I open a new post for this question? $\endgroup$– Ying ZhangAug 13, 2010 at 4:27

$\begingroup$ Related: mathoverflow.net/questions/270490/… $\endgroup$– Tom CopelandApr 20, 2021 at 20:31
10 Answers
Algebraic Ktheory 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 localtoglobal 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 Ktheory.
From the perspective of homotopy theory, algebraic Ktheory 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_oospace structure, to be exact), and such an object has a naturally associated group completion. This is the Ktheory object of the category, and Ktheory is in some sense the universal functor that takes a category with a symmetric monoidal structure and turns it into an additive structure. The Ktheory of the category of finite sets captures stable homotopy groups of spheres. The Ktheory of the category of vector spaces (with appropriately topologized spaces of endomorphisms) captures complex or real topological Ktheory. The Ktheory 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 Ktheory is a universal thing that does this. The Qconstruction 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 dgrings 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 Kgroups 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".

2$\begingroup$ I wasn't talking specifically about Quillen's higher Ktheory functor, but about something homotopy equivalent, which produces a universal functor from symmetric monoidal categories to spectra that converts the symmetric monoidal structure into addition. $\endgroup$ Nov 18, 2009 at 13:30

9$\begingroup$ It is universal in a specific sense. See the thesis of Gonçalo Tabuada. $\endgroup$ Nov 25, 2009 at 3:26

2$\begingroup$ Nice to read! "The Ktheory of the category of finite sets captures stable homotopy groups of spheres." I often heard this but never seen a precise statement and a proof. Can anybody provide me a reference? $\endgroup$– roger123Mar 15, 2010 at 21:21

8$\begingroup$ That's the BarratPriddyQuillen theorem. See BarratPriddy, "On the homology of nonconnected monoids and their associated groups", and others reference Quillen, "The group completion of a simplicial monoid", Appendix Q in FriedlanderMazur's "Filtrations on the homology of algebraic varieties". Also see Segal, "Configurationspaces and iterated loopspaces" for another take. $\endgroup$ Mar 16, 2010 at 3:40

6$\begingroup$ Since there is now a paper on this, I would like to add that one can read about the universal characterization of higher Ktheory that Tyler is discussing here: arxiv.org/abs/1001.2282 $\endgroup$ Dec 25, 2011 at 12:47
I think a key point is that algebraic Ktheory is defined not only for rings, but also for schemes (and other kinds of "generalized spaces" in algebraic geometry). If you believe that generalized (EilenbergSteenrod) 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 Ktheory is in some sense the simplest and most widely studied such theory, although yes, computations are very hard.
Some other motivation:
Algebraic Ktheory allows you to talk about characteristic classes of vector bundles on schemes, with values in various cohomology theories, see for example Gillet: Ktheory and algebraic geometry.
Algebraic Ktheory 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 Lfunctions. 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 BirchSwinnertonDyer conjecture, and in many versions of these conjectures, algebraic Ktheory plays a crucial role. See for example the survey by Bruno Kahn in the Ktheory handbook, also availably at his webpage: http://people.math.jussieu.fr/~kahn/preprints/kcag.pdf
There are also many other useful things in the Ktheory handbook, such as the lectures by Gillet on Ktheory and intersection theory, also available here: http://www.math.uic.edu/~henri/preprints/KTheory_Chow_Groups6.pdf

1$\begingroup$ Yes, I agree with you. Algebraic K theory is for studying algebraic geometry. In particular, algebraic cycle. $\endgroup$ Nov 18, 2009 at 9:11
First, recall the slogan:
Small constructions are good for making calculations, but large constructions are good for proving theorems.
Ktheory is certainly a large construction.
In general, Ktheory 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 Ktheory or without, but Ktheory makes the extra structure in the chain complex visible (such as Poincare duality), which makes it possible to prove theorems. As a lowdimensional 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 (Ltheory), 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 Ktheoretic 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.
As a particular application of algebraic Ktheory, 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 Ktheory 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 Ktheory with supports and the Adam's operations. Besides that, all you need is Quillen's original paper to understand the intersection theory.
I suggest looking at the introduction to Waldhausen's original paper on algebraic Ktheory (Algebraic Ktheory of generalized free products, Part I, Ann. Math., 108 (1978) 135204).
Waldhausen started out as a 3manifold theorist, and he realized that certain phenomena in the topology of 3manifolds 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 Kgroups (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 (nonconnective) $K$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 $R$!
I found Mitchell's survey "On the LichtenbaumQuillen conjectures from a stable homotopytheoretic viewpoint" very motivating. I assumes only little background and is written for mutually introducing homotopy theorists with algebraic Ktheorists.
For me, one interesting thing about algebraic Ktheory is Ltheory (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 4manifold, 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 CWcomplex 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 Ktheoretically (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 Ktheory enters.
The main point seems to be that algebraic Ktheory 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.
Here's a reference that gives some of the history of algebraic ktheory. It might have something you're looking for. http://www.math.uiuc.edu/Ktheory/0343/khistory.pdf. Also Rosenberg's book "Algebraic Ktheory and Its Applications is good.
Grothendieck's original motivation for Ktheory was to give a natural setting for the intersection theory on algebraic varieties.

2$\begingroup$ This is Benjamin Antieau's answer, right? $\endgroup$ Apr 6, 2011 at 22:08

2$\begingroup$ I do not recall if I noticed his post at the time of posting mine; I recalled that I found other's answers being in other directions. Still one thing is that something is an application and another the original motivation. $\endgroup$ Apr 7, 2011 at 16:02

2$\begingroup$ @Zoran: the date and time of the post is right above the poster's name. Antieau's answer was four months before yours. $\endgroup$ Mar 7, 2013 at 16:46
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 earchive of Ktheory papers!
Still, yes, it's a very important and general way to learn about abstract rings.

1$\begingroup$ I used to spend a lot of time looking at the Ktheory archive (www.math.uiuc.edu/Ktheory/) but I have yet to find enlightenment. $\endgroup$– S. Carnahan ♦Oct 12, 2009 at 19:52

$\begingroup$ Well, what about the Ktheory periodicity? It's a computation, too. $\endgroup$ Oct 23, 2009 at 20:45

$\begingroup$ Though I agree the things they compute right now, in 2009, are absolutely not fun from my point of view. $\endgroup$ Oct 23, 2009 at 20:46

13$\begingroup$ Since my knowledge of Ktheory is very limited, I get a big confused by your comment, that Ktheory is for people who don't mind the fact that there's no general picture. It seems to me that Ktheory creates a link between many different fields and so, it somehow a tool to see the big picture. So how is it then related to the fact that there may be no general picture at all? $\endgroup$– BrianNov 9, 2011 at 14:42