Commit 185cac06 authored by Jannes Bantje's avatar Jannes Bantje

add important reformulation of BC for groups

parent 3da49850
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......@@ -1008,7 +1008,7 @@ The basic result about the equivariant \K-homology of the the balanced product i
The following theorem requires even longer computations:
\begin{theorem}{\cite[Thm.~6.5.15]{WilletYu}}
\begin{theorem}{\cite[Thm.~6.5.15]{WilletYu}}\label{thm:iso_equiv_X_to_X/G}
Sag $G$ acts freely on $X$.
Then there are $*$-homomorphisms $\Phi \colon \LocAlg(H_X)^G \to \LocAlg(H_{X/G})$ and $\Psi \colon \LocAlg(H_{X/G}) \to \LocAlg(H_X)^G$ which give mutually inverse $*$-isomorphisms on the quotients
\(
......@@ -1163,6 +1163,15 @@ and we say, that $G$ satisfies the Baum--Connes conjecture, if this map is an is
\todo[inline]{Should contain the following: Rips complexes, Classifying space variants for BCC for $G$, BCC for Euclidean space}
\begin{theorem}
Assume $G$ is finitely generated and torsion free and admits a classifying space $\B G$ which is a finite CW complex.
Then the Baum--Connes assembly map for $G$ (acting on itself) identifies with the composition
\[
\K_*(\B G) \cong \K^G_*(\E G) \To{\mu_{\E G}} \K_* \enbrace*{\RoeAlg(\E G)^G} \cong \K_*(\Cstar_\rho(G))
\]
(where we used the isomorphisms from \cref{thm:iso_equiv_X_to_X/G,thm:equiv_roe_alg_group_cstar})
\end{theorem}
% section a_concrete_model_for_coarse_k_homology (end)
\section{Applications} % (fold)
......
......@@ -290,6 +290,7 @@
\newcommand{\B}[1]{B{#1}} % classifying spaces
\newcommand{\E}[1]{E{#1}} % Universal cover
\newcommand{\Inf}{I\mkern-.1mu n\mkern-.5mu f}
\newcommand{\Infhat}{I\mkern-.1mu \widehat{n}\mkern-.5mu f}
......
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