diff --git a/contents/coarse.tex b/contents/coarse.tex index 588860c3f52115e9c6cf9cf357220446f54de553..f177314af2290a2bba0fa7b7683db22644596fbb 100644 --- a/contents/coarse.tex +++ b/contents/coarse.tex @@ -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) diff --git a/math.tex b/math.tex index 68b7579e896239596570007e80c00681ab27c1f9..439cfe07fdcc9ce5582aef0b50bff62688fc7d37 100644 --- a/math.tex +++ b/math.tex @@ -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}