# Lists of K-homology Groups

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?

• Perhaps you could specify which flavour K-theory you are talking about- topological, algebraic, analytic...? Mar 23 '11 at 15:23
• What one calls "standard spaces" depends very much on one's perspective. Mar 23 '11 at 17:10
• The reduced $K$-theory of a sphere is the same as the $K$-theory of a point, except that all the degrees get shifted by the dimension of the sphere. The (unreduced) $K$-theory of a sphere is the direct sum of its reduced K-theory and of the $K$-theory of a point. Mar 23 '11 at 17:12

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.
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.