Almost the same question was already asked on MO Motivation for algebraic K-theory? However, to my taste, the answers there consider the subject from a more modern point of view.

When I open a book on algebraic K-theory (I am not an expert) I see various complicated very ingenious constructions which become equivalent for mysterious (to me) reasons. What I do not understand, what problem exactly Quillen tried to solve?

Are there any properties that any higher algebraic K-theory is expected to satisfy a priori like long exact sequences (assuming that $K^0$ is known)? I realize that in any formulation it has to be functorial, but this does not tell much unless more information is known.

In order to give an example of an answer which would be kind of convincing for me, let me try to summarize my understanding of topological $K$-theory. Since I am not an expert, this understanding is incomplete and down to earth, but this is what I am looking for in the algebraic case. In what follows, the first two paragraphs are about the motivation of topological $K$-theory, while the other two are about concrete applications of it.

1) Grothendieck defined algebraic $K^0$ ring for schemes in order to formulate his generalization of Riemann-Roch-Hirzebruch theorem. The construction used algebraic vector bundles. Topological $K^0$ was defined by analogy using topological vector bundles.

2) Higher topological $K$-groups were introduced in order to have the Mayer-Vietoris long exact sequence for pairs of spaces.

3) There is an important construction of elements of $K^0$-groups: any elliptic (pseudo-) differential operator defines an element in $K$-theory of the tangent bundle. This construction is necessary for the formulation of the Atiyah-Singer theorem.

4) After some identifications based on the Bott periodicity, higher rational $K$-ring becomes isomorphic to the rational cohomology ring via the Chern character. Thus $K^*/torsion$ is a lattice in $H^*(\cdot,\mathbb{Q})$ which is different from the lattice $H^*(\cdot,\mathbb{Z})/torsion$. This canonical and non-obvious integral structure was used to prove some non-embeddability theorems.

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    $K_0$, $K_1$ and $K_2$ pre-existed Quillen's construction (and those of other people) of higher $K$-theory, and they were known to fit together in nice exact sequences. Experience shows that such things can be extended into long exact sequences of functors and that this is useful in various standard ways —for example, in proving things by induction. The book by Rosenberg studies in nice detail the first functors and constructs them «classically»; it is very informational. (I'd guess that $K_2$ was discovered extending the exact sequence for $K_0$ and $K_1$...) – Mariano Suárez-Álvarez Mar 31 '14 at 8:07
  • @Mariano: Concerning your parenthetic remark at the end, it's probably hard at this point to separate out the ingredients leading to Milnor's $K_2$, but the increasing visibility of Steinberg's work on central extensions in the treatment of the Congruence Subgroup Property certainly played a major role for Milnor. Bass also had much to do with these developments (and with the search for higher algebraic K-groups). – Jim Humphreys Mar 31 '14 at 12:51
  • Thanks for all comments. @JimHamphreys: if one could spell out what does that mean that"Bass also had much to do with ... the search for higher algebraic K-groups", that might probably answer my question. Object with what properties Bass was searching for? – orbits Mar 31 '14 at 13:02
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    @semyonalesker According to Grayson, Bass had this to say about the motivation for the definition of $K_{1}$: "The idea was that a bundle on a suspension is trivial on each cone, so the gluing on the `equator' is defined by an auto-morphism of a trivial bundle, $E$, up to homotopy. Thus, topologically, $K_{-1}$ is $\operatorname{Aut}(E) / \operatorname{Aut}(E)^{\circ}$, where $\operatorname{Aut}(E)^{\circ}$ is the identity component of the topological group $\operatorname{Aut}(E)$. Since unipotents are connected to the identity (by a straight line, in fact) they belong to... – Tom Harris Mar 31 '14 at 13:15
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    If you can read French, Loday had written something about it page 2-3: – Philippe Gaucher Mar 31 '14 at 13:55

I am neither a K-theorist nor a historian, so I don't know all the things the led to Quillen to his definition(s) of higher K-theory, but I thought I'd mention one striking application that can be found in his original paper. The Chow group of a variety $CH^p(X)$ is the group of codimension $p$-cycles modulo rational equivalence. For lots of reasons, it was desirable to express this in terms of sheaf cohomology. For $p=1$, $CH^1(X)=Pic(X)= H^1(X, O_X^*)$ was known for a long time. This can be recast into K-theoretic terms, by observing, using Bass' definition, that $O_X^*$ can be identified the sheaf associated to $U\mapsto K_1(O(U))$. I believe Bloch extended this to $CH^2$ using Milnor's $K_2$. And finally Quillen proved that $CH^p(X)=H^p(X,K_p(O_X))$ for any regular variety, and any $p$ using his definition.

This is really just an elaboration of Mariano's comment. As he said, Quillen did not jump straight from the definition of $K_{0}(R)$ to define $K_{n}$ for all $n$. As you have noted, algebraic $K_{0}$ motivated the definition of topological $K^{0}$, which was then extended. Since that theory proved to be so fruitful, it was hoped that an algebraic analogue could be found, at least in the simplest case of an affine scheme, which then reduces to the $K$-theory of the coordinate ring $R$. One reason I'd guess that this might be a reasonable hope is that if $R$ is Dedekind then $K_{0}(R)$ already contains interesting arithmetic information (the class group).

Bass defined $K_{1}(R)$ in analogy with the way one constructs the elements of $K^{-1}(X)$ by glueing trivial bundles on cones via a bundle automorphism to obtain a bundle on the suspension. This turned out to be the correct thing to do, as evidenced by the existence of an exact sequence $$ \bigoplus_{\mathfrak{m}}K_{1}(R/\mathfrak{m}) \rightarrow K_{1}(R) \rightarrow K_{1}(F) \rightarrow \bigoplus_{\mathfrak{m}}K_{0}(R/\mathfrak{m}) \rightarrow K_{0}(R) \rightarrow K_{0}(F) \rightarrow 0,$$ where $R$ is Dedekind, $F$ is its field of fractions and $\mathfrak{M}$ runs over all the maximal ideals of $R$. As expected, this also contained useful information -- due to Milnor, Bass and Serre's solution of the congruence subgroup problem, for a number ring $\mathcal{O}_{k}$, the group $K_{1}(\mathcal{O}_{k})$ is the group of units of $\mathcal{O}_{k}$.

The development of $K_{2}$ I am less clear on, but I believe Milnor's book on algebraic $K$-theory gives details of the proof that Milnor's $K_{2}$ does extend the exact sequence above in the desired way, along with many others given by Bass.

I should note also that the lower $K$-theory groups were of interest to topologists at this time as well, since the Whitehead group $\operatorname{Wh}(\pi)$, which in some sense classifies $h$-cobordisms, is a quotient of $K_{1}(\mathbb{Z}[\pi])$. Work of (I think) Hatcher and Wagoner showed that a similar quotient of $K_{2}(\mathbb{Z}[\pi])$ encoded similarly interesting topological information.

So in answer to your original question, yes there were long exact sequences that higher algebraic $K$-theory was expected to satisfy, namely the natural extension of the exact sequence above (for a Dedekind domain). Quillen's definition of the higher algebraic $K$-groups of an exact category allowed him to prove just that, as a consequence of the Resolution, Devissage and Localization theorems (A question for someone else: can this long exact sequence be constructed using the original definition of higher algebraic $K$-theory for rings?).

I hope this is in some way useful, although I am sure there is a lot more to the story. The ideas above, and many more, are covered in a lot more detail in Dan Grayson's nice survey Quillen's work in algebraic $K$-theory.

  • Note that "A question for someone else" needs a clearer formulation. (Certainly there is a lot of history here in any case.) – Jim Humphreys Mar 31 '14 at 12:41
  • Oh yes, I wrote that badly. The question should be: can the long exact sequence above be extended in the expected way to all $n$ using the definition of $K_{n}(R)$ via Quillen's plus construction? – Tom Harris Mar 31 '14 at 13:11

As an appendix to Tom Harris's nice answer, it might be worth mentioning that the idea of defining algebraic K-groups as the homotopy groups of something was certainly in the air before Quillen, e.g. in the work of Swan, Gersten, and Karoubi-Villamayor.

In particular, Karoubi-Villamayor (pre-Quillen) defined the higher K-theory of a ring $R$ as the homotopy groups of the simplicial complex that has $GL(R[t_0,\ldots,t_n]/(\sum t_i=1)$ in the $n$th place (with face maps defined by setting various $t_i=0$). This turned out to work well only when the ring $R$ is regular, in which case it coincides (non-obviously) with Quillen's definition.

Thanks to the link mentioned above by Philippe Gaucher, I learned something about Quillen's motivation of the $+$-construction (though still there are many gaps in my understanding of further developments). Below is a very short summary of what I could understand. The original reference is "On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field", Daniel Quillen, Ann. Math., Vol. 96, No. 3 (Nov., 1972), pp. 552-586.

The most surprising thing I learned is that Quillen was solving a problem completely unrelated to algebraic $K$-theory: he computed the group cohomology $H^*(GL_n(\mathbb{F}_q),\mathbb{F}_l)$ of the linear group over finite field with coefficients in another finite field $\mathbb{F}_l$ where $l$ is a prime number not dividing $q$. As a first step based on the Brauder's theory of characters of finite groups, Quillen partly reduced this problem to the topological question of describing the homotopy fiber of the map $BU\overset{\Psi^q-Id}{\to} BU$ where $BU$ is the classifying space of the infinite unitary group, and $\Psi^q$ is the $q$-th Adams operation. He proved that this homotopy fiber has the same homology as the classifying space $BGL(\mathbb{F}_q)$, but the $\pi_1$ group of the former is equal to the abelinization of the $\pi_1$ of the latter. Then Quillen realized that one can attach 2- and 3-cells to $BGL(\mathbb{F}_q)$ in a tricky way to get the space with the desired properties.

Trying to understand and generalize his construction, Quillen observed that for any unital ring $R$ if one attaches 2- and 3-cells in the same way to $BGL(R)$, one obtains space with the same homology, but with the abelinized fundamental group; the space is denoted $BGL(R)^+$.

It immediately follows from the construction that $\pi_1(BGL(R)^+)=K_1(R)$. Quillen was able to prove that $\pi_2(BGL(R)^+)=K_2(R)$; the $K_2$ group was already known at the time. It looks likely that these two computations combined with the general aesthetic beauty and elegance of the $+$-construction were suggestive to study higher homotopy groups of the $+$-construction. However at this point my understanding of the intuition stops, since apparently instead of following this way, Quillen introduced a very different (but eventually equivalent) mysterious Q-construction which was used to define higher $K$-groups of schemes and to prove a Mayer-Vietoris type sequence for them.

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