What is the purpose of K-Theory?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.