# What is the purpose of K-Theory?

I have recognized that there is a theory called K-Theory in mathematics is used also for applications in mathematical physics. There is existing algebraic K-Theory and topological K-Theory. Are these theories very similar?

For algebraic K-Theory by Milnor I have seen that the K-Groups are given by

$K_n = T^n / a \otimes (1-a)$ (Wikipedia).

Here, $T^n$ is the $n$-fold Tensor product. For n=2 one obtains abelian matrices. I don't understand this theory in depht. What is the reason that K- theory was introduced? (Is the theoretical physics application topological or algebraic?) And is there material (lecture Video or good pdf script) where the algebraic K-theory is explained?

I would greatly appreciate an answer.

• What is your background and what is the motivation? – quid Nov 22 '15 at 20:56
• "Barely was the ink dry on Hirzebruch's note when new problems arose about the Riemann-Roch relation ... " J. Dieudonne, A History of Algebraic and Differential Topology 1900-1960, p. 598. – Heinrich Nov 22 '15 at 21:24
• Concerning your last question, for "K_0" a good source is Manin's paper "Lectures on the K-functor in algebraic geometry". Also is Atiyah's book (more focused on the topological analogue). For higher $K_n$, Springer LNM 341 has a lot of information (the books by Srinivas and by Weibel are also excellent). – Heinrich Nov 22 '15 at 21:51
• Algebraic K-Theory for rings is explained well by Rosenberg. For topological K theory the book of Wegge-Olsen is a good introduction. – user42761 Feb 26 '16 at 20:12
• If you want an application, it is used to classify the charges of Dp-branes in String Theory. – user204299 Feb 26 '16 at 21:13

Here's a very quick introduction to the background ideas. On the topological side, start off with a nice space $X$. (I'm being deliberately vague about what 'nice' entails, but I at least want something paracompact and often want compact. The example to keep in mind here is a finite-dimensional smooth manifold without boundary.) Given vector bundles $\xi, \xi'\to X$, we can form $\xi \oplus \xi'$ (as well as the tensor product $\xi \otimes \xi'$). We'd like to make this operation into a group. We don't exactly have an identity element, but we can consider the trivial bundle $\theta^n \to X$ and define $\xi, \xi'$ stably equivalent if $\xi \oplus \theta^n$ and $\xi' \oplus \theta^m$ are equivalent for some $n$. In particular, $\xi$ is called stably trivial if we can take $\xi'$ trivial above. Now every bundle $\xi$ has an inverse $\xi^\perp$, and we have a nice group $\tilde K_0(X)$. With a bit of work, we can turn that into a generalized cohomology theory.

On the algebraic side, we can take a ring $A$ (say, commutative) and look at modules over it. Projective modules are, by definition, direct summands of free modules. We can therefore construct a group or ring of projective modules modulo free modules as above, taking free modules instead of trivial bundles. The connection is the general idea that projective modules over a (nice) ring are like vector bundles over a (nice) space. More precisely, Swan's theorem states that for a sufficiently nice space $X$, the map $\xi \to \Gamma(\xi)$ is an equivalence between finite-dimensional (real or complex) vector bundles and finitely-generated projective modules over $A = C^\infty(X)$ (real- or complex-valued). It's not quite that simple, though; for one thing, there isn't an obvious way of defining higher $K_*(A)$, and the ways that do exist are complicated.

In short, algebraic $K$-theory starts with the observation that the dimension of vector spaces over a field is a very useful thing! The start is the study of the $K_0$ group of a ring, which is «the best thing for $A$-modules that feels like the dimension of vector spaces».

The next player in $K$-theory is the $K_1$ of a ring $A$, which again measures how far we are from the nice situation of linear algebra: there one can take, by applying row and column operations, matrices into very simple forms. This cannot be done for general rings, and $K_1$ tells you how badly it fails.

Higher $K$-theory is considerably more difficult to motivate and to explain, but can be felt in the same way.

You should browse Jonathan Rosenberg's beautiful book on $K$-theory.

algebraic K-Theory and topological K-Theory. Are these theories very similar?

Algebraic K-theory is far more difficult and harder to define.

For algebraic K-Theory by Milnor I have seen that the K-Groups are given by

Milnor does what is now called "Milnor K-theory of fields" in his book, with an appendix explaining the relation (due to Tate) of $K_2$ to the quadratic reciprocity law with Hilbert symbols. This was before the general definition of algebraic K-theory by Quillen. When people write algebraic K-theory today they mean the Quillen version (or other later developments) of higher algebraic K-theory, and specify Milnor K-theory when they mean that.

What is the reason that K- theory was introduced? (Is the theoretical physics application topological or algebraic?)

Physics uses topological K-theory of manifolds, whose motivation is to organize vector bundles over a space into an algebraic invariant, that turns out to be useful. Some applications to the quantum Hall effect using noncommutative geometry (Bellissard) involve K-theory of operator algebras.

Algebraic K-theory started from $K_i$ defined by hand for small $i$, with relations to classical constructions in algebra and number theory, followed by Quillen's homotopy-theoretic definition for all $i$ (and later approaches due to Waldhausen and others). The connections to algebra and number theory often persist for larger values of $i$ but in ways that are subtle and conjectural, such as special values of zeta- and L-functions.

And is there material (lecture Video or good pdf script) where the algebraic K-theory is explained?

Nothing very accessible for algebraic K-theory. Blackadar's book for K-theory of operator algebras, and Atiyah's book for topological K-theory as it stood in the 1960's, are readable without a lot of algebraic prerequisites.

• By the way, I am pretty sure there have been several variants of this question on MathOverflow. – zyx Feb 26 '16 at 20:08
• For example, the first page of the k-theory tag has elaborations of the themes of prior motivation, number theoretic applications, and the role of Milnor K theory. mathoverflow.net/questions/tagged/kt.k-theory-homology – zyx Feb 26 '16 at 20:12
• The relation of K2 to quadratic reciprocity is an appendix by Tate in Milnor's paper, is in another appendix to another book by Milnor on symmetric bilinear forms, but is in the last chapter rather than an appendix in Milnor's book on K-theory. The description in the answer mixes up these things somewhat. – zyx Feb 27 '16 at 18:10