Talk:K-theory

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In mathematics, the topic of K-theory spans the subjects of algebraic topology, abstract algebra and some areas of application like operator algebras and algebraic geometry. It leads to the construction of families of K-functors, which contain useful but often hard-to-compute information.

All well and good, but what exactly is K-Theory? Any article on a maths topic should begin with a defintion of some kind. I would add one myself, but am not up to the task. Tompw 15:25, 25 July 2005 (UTC)

K-theory is an extraordinary (this adjective was invented for K-theory I believe) cohomology theory. It constructs so-called K-groups from topological spaces (topological K-theory) and from C^* algebras (algebraic K-theory).--MarSch 12:20, 2 September 2005 (UTC)

Topological K-theory is indeed an extraordinary cohomology theory (ie the usual axioms without the dimension axiom). I don't think one can say that algebraic K-theory is an ECT. Charles Matthews 21:10, 2 September 2005 (UTC)

Given that 'extraordinary' was invented for K-theory, this seems something of a tautology. However... I still don't feel that the opening sentence is a definition. It's more of a classification. Tompw 16:17, 16 October 2005 (UTC)

OK, K-theory as of 2005 is a whole big area. It's a wretched name, but we don't have any power over that. From a top-down look, it is something like the homotopy theory of the general linear groups over any old ring. If you look at applications it is not obviously that at all; it is implicated in number theory and algebraic geometry in the biggest way. Not often I say this, but I'm not competent to give an expert discussion of how it fits together. The topological stuff is not so bad. The algebraic K-theory stuff is about trying to get good invariants in module theory (and then finding that even the K-theory of the integers Z looks very deep). Charles Matthews 17:08, 16 October 2005 (UTC)

Using Quillen's definitions, K-theory can be defined for any exact category in a way that captures the algebraic and topological theories (using categories of projective modules or vector bundles). This approach also allows the study of other interesting objects in this context, such as the K-theory of a scheme. I can write an overview of these ideas for the article to give it some sense of 'what K-theory is' if desired. MarcHarper 02:52, 30 September 2006 (UTC)

Having a survey would be very good. In line with the concentric style we favour, it should not 'write over' some gentler explanations at the start. We don't want people to have to cope with the axiomatics of any one perspective immediately. Charles Matthews 09:17, 3 October 2006 (UTC)

Some of the aximomatic work is already laid out in the Algebraic K-theory article, though it only discusses the application to the case of rings. Perhaps it more naturally belongs here? Marc Harper 20:09, 25 October 2006 (UTC)