Lists of K-homology Groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:50:14Z http://mathoverflow.net/feeds/question/59309 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59309/lists-of-k-homology-groups Lists of K-homology Groups Kolya Ivankov 2011-03-23T14:56:51Z 2011-06-24T09:05:44Z <p>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?</p> http://mathoverflow.net/questions/59309/lists-of-k-homology-groups/59343#59343 Answer by Paul Siegel for Lists of K-homology Groups Paul Siegel 2011-03-23T18:53:31Z 2011-03-23T18:53:31Z <p>For many basic examples, the usual tools of (co)homology theory work just fine (and very similarly) in K-theory and K-homology.</p> <p>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:</p> <p>$$\to K^0(U \cap V) \to K^0(U) \oplus K^0(V) \to K^0(S^1) \to$$ $$K^1(U \cap V) \to K^1(U) \oplus K^1(V) \to K^1(S^1) \to$$</p> <p>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 $K^0(S^1) = K^1(S^1) = \mathbb{Z}$.</p> <p>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.</p> http://mathoverflow.net/questions/59309/lists-of-k-homology-groups/68709#68709 Answer by Alain Valette for Lists of K-homology Groups Alain Valette 2011-06-24T09:05:44Z 2011-06-24T09:05:44Z <p>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.</p>