There sohould be a list of K-theory and K-homology groups for the the standard spaces, like circle, spheres, (non-commutative) tori, but despite I've googled for it, I have found nothing satisfying. Maybe someone can give a reference?



Lists of K-homology Groups
For many basic examples, the usual tools of (co)homology theory work just fine (and very similarly) in K-theory and K-homology.


Here's an example. How does one compute, say, the De Rham cohomology of $S^1$? There are lots of ways, but one way is to use the Mayer-Vietoris sequence - the same thing works in K-theory (and K-homology). Write $S^1 = U \cup V$ where $U$ is a small neighborhood of the upper half of the circle and $V$ is a small neighborhood of the lower half. The long exact sequence in K-theory looks like:


$$\to K^0(U \cap V) \to K^0(U) \oplus K^0(V) \to K^0(S^1) \to$$
$$ K^1(U \cap V) \to K^1(U) \oplus K^1(V) \to K^1(S^1) \to$$


We have $K^0(point) = \mathbb{Z}$ and $K^1(point) = 0$, and it's not hard to calculate that the map $K^0(U \cap V) \to K^0(U) \oplus K^0(V)$ is the map $\mathbb{Z}^2 \to \mathbb{Z}^2$ given by $(x,y) \mapsto x - y$. So we get $K^0(S^1) = K^1(S^1) = \mathbb{Z}$.


For noncommutative spaces, there is still a version of the Mayer-Vietoris sequence which helps with some computations: it uses a decomposition $A = I + J$ of a C\* algebra as the sum of two closed ideals. Combined with equivariant K-theory/K-homology (and Bott periodicity!) these sorts of computations are often fairly routine. The challenge, as usual, generally is in working with specific (co)cycles that have geometric content.


