Lists of K-homology Groups - MathOverflow

Lists of K-homology Groups

2011-03-23T14:56:51Z

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 by Paul Siegel for Lists of K-homology Groups

2011-03-23T18:53:31Z

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 by Alain Valette for Lists of K-homology Groups

2011-06-24T09:05:44Z

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.