For a topological groupoid $G$, there are two kinds of ``topological'' $K$-homology theories, $K_{\ast}^{G}(\underline{EG})$ the $K$-homology of $G$ with $G$-compact support, and $K_{\ast}(BG)$ the topological $K$-homology of the classifying space of $G$.

In the introduction of this paper, Baum, Connes, and Higson stated that $K_{\ast}^{G}(\underline{EG})$ is isomorphic to $K_*(BG)$ when $G$ is a torsion-free discrete group. I wonder

What happens to other cases? Especially when $G$ is a topological groupoid?

  • $\begingroup$ 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. $\endgroup$ – Fernando Muro Sep 30 '12 at 10:02
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    $\begingroup$ @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. $\endgroup$ – Paul Siegel Sep 30 '12 at 13:45
  • $\begingroup$ @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 $K_{\ast}(BG)$. $\endgroup$ – Dai Tamaki Sep 30 '12 at 18:38
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    $\begingroup$ 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. $\endgroup$ – Alain Valette Sep 30 '12 at 20:56
  • $\begingroup$ @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. $\endgroup$ – Dai Tamaki Oct 2 '12 at 11:53

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