\u00c1lvaro S\u00e1nchez (Universitat Aut\u00f2noma de Barcelona) delivered the talk:<\/p>\n
Natural embedding of H0 (G) into K0 (Cr \u2217 (G)) for rank 3 Deaconu-Renault groupoids, and HK conjecture I Abstract: Take an \u00e9tale groupoid G such that G (0) is compact, metrizable and totally disconnected. By definition of Cr \u2217 (G), we can always consider the canonical inclusion \u03b9 : C(G (0)) \u2192 Cr \u2217(G), which induces an homomorphism in K-theory K0 (\u03b9) : K0(C(G (0))) \u2192 K0 (Cr \u2217 (G)). Then the differential map \u03b41 : Cc (G, Z) \u2192 C(G (0) , Z) defining the homology groups verifies (K0 (\u03b9) \u25e6 \u03b41 )(u) = 0 (since \u03b41 (u) = \u03c7s(U ) \u2212 \u03c7r(U ) ). From here we deduce that im\u03b41 \u2286 ker(K0 (\u03b9)). Since every \u00e9tale groupoid has a countable basis consisting in open bisections, it follows that there exists a canonical homomorphism \u03a6 : H0 (G) \u2192 K0 (Cr \u2217(G)) such that \u03a6([f ]) := K0 (\u03b9)(f ). The question about when this map is injective is open, even for some simple groupoids. We prove this result for the Deaconu-Renault groupoids of \u00c1lvaro S\u00e1nchez (Universitat Aut\u00f2noma de Barcelona) delivered the talk: Natural embedding of H0 (G) into K0 (Cr \u2217 (G)) for rank 3 Deaconu-Renault groupoids, and HK conjecture I Abstract: Take an \u00e9tale groupoid G such that G (0) is compact, metrizable and totally disconnected. By definition of Cr \u2217 (G), we can always consider the … <\/p>\n
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\nNow, since there are no non-unit elements in G (0), K0 (C(G (0) )) = C(G (0) , Z). For any U compact open bisection of G, u = \u03c7U is a partial isometry of Cr \u2217 (G) such that uu\u2217 = \u03c7s(U ) ,and u\u2217 u = \u03c7r(U ) , which means \u03c7s(U ) and \u03c7r(U ) belong to the same equivalence class inK0(Cr\u2217 G)).<\/p>\n
\nrank 3, i. e. the groupoids arising from N3 acting over a Cantor set by surjective local homeomorphisms. We use this to prove the HK-conjecture for this family of groupoids.<\/p>\n","protected":false},"excerpt":{"rendered":"