Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one?


Using general spectra stuff, one gets a Kuenneth spectral sequence for K-homology. The question is then, phrased this way, why this spectral sequence (in the case of K-homology) degenerates suitably to give rise to a short exact sequence?



Kuenneth short exact sequence for K-homology
Let me answer my own question.


Markus Land referred me to a remark on top of page 62 in his PhD thesis ( <http://hss.ulb.uni-bonn.de/2016/4432/4432.htm> ), where he argues why we have a short exact UCT sequence relating K- and L-theory.


The same arguments also apply to the Kuenneth formula for K-homology: since the graded ring $\pi\_\ast(KU)$ has global dimension 1, the Kuenneth spectral sequence collapses on the $E\_2$-page and gives rise to the sought short exact sequence.


The reason why $\pi\_\ast(KU)$ has global dimension 1 is because $\pi\_\ast(KU)$ is a Laurent polynomial ring over $\mathbb{Z}$ in one variable, and the category of graded modules over this is equivalent to the category $\mathtt{Ab} \times \mathtt{Ab}$.


