Lists of K-homology Groups

Question

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?

Answer 1

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.

Answer 2

The Chern character in $K$-homology gives that, for a finite CW-complex $X$, up to torsion (i.e. after tensoring with $\mathbb{Q}$), $K_j(X)$ is isomorphic to $\bigoplus_{i=0}^\infty H_{j+2i}(X,\mathbb{Q})$ (standard homology groups with rational coefficients). If $\dim X=2$ (and if I remember correctly there is also something in dimension 3), then this isomorphism actually holds over $\mathbb{Z}$: see Michel Matthey, Mapping the homology of a group to the $K$-theory of its $C^*$-algebra. Illinois J. Math. 46 (2002), no. 3, 953–977.