Why was it reasonable to ask what the higher K-groups are? - MathOverflow

Why was it reasonable to ask what the higher K-groups are?

2011-06-12T18:50:53Z

To say I am a novice in $K$-theory is to overstate my experience with the field. I've been reading the various wiki articles so as to have some preparation before jumping in, and I couldn't answer the following question to myself:

I understand that $K$-theory had started with the Grothendieck-Riemann-Roch in mind, and that the only thing that was needed for that purpose from $K$-theory was just to define the Grothendieck Group ($K_0$). Once the idea of the Grothendieck group was established, this was generalized to topological spaces, as well as for other kinds of modules. Then comes the step I don't understand -- it seems that people were then trying to find the higher $K$-groups that would make $K$-theory into a cohomological theory. Milnor came up with Milnor $K$-theory, which I understand from wiki is different from later notions of higher $K$-theory. But why would one leap from the concept of the Grothendieck group to thinking that this construction is the $0^{th}$ step in a cohomological theory? What was the context/motivation for that?

Answer by André Henriques for Why was it reasonable to ask what the higher K-groups are?

2011-06-12T21:36:49Z

After having defined K₀, a natural things to do is to study its functoriality properties. You do that, and you notice some exact sequences... that you happen to be able to extend a bit by using the functors K₁, and then later K₂.

It is then natural (specially if you know what cohomology is) to try to find long exact sequences that extend the above sequences... A lot of people tried to do that.

Quillen's brilliant idea was to define algebraic K-theory as the homotopy groups of an appropriately constructed space. In that way, the long exact sequences came as natural consequences of known long exact sequences in topology.
Slogan: Homotopy theory is the mother of all long exact sequences.

Aside: I also recommend reading Thomason's work on algebraic K-theory of schemes: it's beautifully written!

Answer by Richard Borcherds for Why was it reasonable to ask what the higher K-groups are?

2011-06-12T22:00:16Z

The idea of considering higher K-groups comes from topology, and is due to Atiyah, Bott, and Hirzebruch. Atiyah and Hirzebruch defined topological K theory and observed that Bott periodicity says that $K(X)$ is more or less the same as $K(S^2X)$. This suggested to them defining a generalized cohomology theory of period 2 by using all the groups $K(S^nX)$ (this was the first example of a generalized cohomology theory). Once one realizes that topological $K^0$ can be extended to topological $K^n$, it does not take much imagination to suggest that algebraic $K^0$ also has an extension to algebraic $K^n$. (Of course, finding this extension was much harder than guessing it existed.)