Timeline for K-theory of topological groupoids
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 2, 2012 at 11:53 | comment | added | Dai Tamaki | @Alain Thank you for explaining the case of torsion free groups. Yes, what I wondered first is what is the condition for groupoids corresponding to torsion-freeness. Another question is what kind of topological conditions do we need to impose on the space of objects. I recently learned that some people (arxiv.org/abs/0909.1624) are studying etale topological groupoids whose space of objects is the Cantor set. I would like to know what happens to such cases. | |
| Sep 30, 2012 at 20:56 | comment | added | Alain Valette | I'd like to clarify one point. For groups, $\underline{EG}$ denotes the classifying space for $G$-proper actions. For $G$ a torsion-free group, we have $\underline{EG}=EG$, the universal cover of $BG$; indeed $EG$ classifies free $G$-action, and ``free=proper'' as a consequence of $G$ being torsion-free. There are easy examples where $K_*^G(\underline{EG})$ is not isomorphic to $K_*(BG)$ when $G$ has torsion. This is to say that, to answer your question for groupoids, you will need some assumption of this kind. @Paul: I confirm there are counterexamples to BC for groupoids. | |
| Sep 30, 2012 at 18:38 | comment | added | Dai Tamaki | @Fermando and @Paul: The Baum-Connes conjecture says that the assembly map from $K_{\ast}^{G}(\underline{EG})$ to $K_{\ast}(C_r(G))$ is an isomorphism. And $K_{\ast}^{G}(\underline{EG})$ is defined as an inverse limit of $KK$-theory. My question is when we can replace $K_{\ast}^{G}(\underline{EG})$ by a "purely topological" $K$-homology $K_{\ast}(BG)$. | |
| Sep 30, 2012 at 13:45 | comment | added | Paul Siegel | @Fernando Muro: The Baum-Connes conjecture asserts that $K_\ast(BG)$ is isomorphic to $K_\ast(C_r^\ast(G))$ where $C_r^\ast(G)$ is the reduced group C*-algebra for $G$ (actually, it asserts that a particular map is an isomorphism). This question seems to be purely topological: does the K-homology of $BG$ agree with the equivariant K-homology of $EG$? As a side note, I think there are counter-examples to Baum-Connes for topological groupoids. | |
| Sep 30, 2012 at 10:02 | comment | added | Fernando Muro | This is the Baum--Connes conjecture, isn't it? A big open problem. Many cases are known, many other are open, and there's no counterexample so far. | |
| Sep 30, 2012 at 3:47 | history | asked | Dai Tamaki | CC BY-SA 3.0 |