Commit dd3a585b authored by Jannes Bantje's avatar Jannes Bantje

some restructuring, state another result

parent 185cac06
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......@@ -1161,22 +1161,40 @@ and we say, that $G$ satisfies the Baum--Connes conjecture, if this map is an is
\section{A Concrete Model for Coarse \K-Homology} % (fold)
\label{sec:concrete_model_coarse_k_homology}
\todo[inline]{Should contain the following: Rips complexes, Classifying space variants for BCC for $G$, BCC for Euclidean space}
\todo[inline]{Should contain the following: Rips complexes}
\begin{theorem}
% section a_concrete_model_for_coarse_k_homology (end)
\section{Assembly Maps for Classifying Spaces} % (fold)
\label{sec:assembly_maps_for_classifying_spaces}
\begin{theorem}{\cite[Thm.~7.4.4]{WilletYu}}
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
\[
\K_*(\B G) \cong \K^G_*(\E G) \To{\mu_{\E G}} \K_* \enbrace*{\RoeAlg(\E G)^G} \cong \K_*(\Cstar_\rho(G))
\K_*(\B G) \cong \K^G_*(\EG G) \To{\mu_{\EG G}} \K_* \enbrace*{\RoeAlg(\EG G)^G} \cong \K_*(\Cstar_\rho(G))
\]
(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)
For the classifying space of proper $G$-actions $\properE G$ we have:
\begin{theorem}{\cite[Thm.~7.4.8]{WilletYu}}
Let $G$ be a countable discrete group and $\properE G$ a classifying space for proper actions for $G$.
Then the Baum--Connes assembly map for $G$ identifies with the map
\[
\lim_{Y \subseteq \properE G} \K^G_* \enbrace*{\poperE G} \To{\mu_{\properE G}} \K_* \enbrace*{\RoeAlg(G)^G}
\]
which is the direct limit over all assembly maps $\mu_{Y,G}\colon \K^G_*(Y) \to \K_* \enbrace*{\RoeAlg(G)^G}$, where $Y \subseteq \properE G$ is a proper cocompact subset.
\end{theorem}
% section assembly_maps_for_classifying_spaces (end)
\section{Applications} % (fold)
\label{sec:applications_coarse}
\begin{theorem}{\cite[Thm.~7.5.1]{WilletYu}}
The coarse Baum--Connes conjecture holds for $\mathbb{R}^d$.
\end{theorem}
% section applications (end)
\section{Generalisations, that Johannes and Rudi consider} % (fold)
......
......@@ -290,7 +290,8 @@
\newcommand{\B}[1]{B{#1}} % classifying spaces
\newcommand{\E}[1]{E{#1}} % Universal cover
\newcommand{\EG}[1]{E{#1}} % Universal cover
\newcommand{\properE}[1]{\underline{E}{#1}} % classifying space for proper actions
\newcommand{\Inf}{I\mkern-.1mu n\mkern-.5mu f}
\newcommand{\Infhat}{I\mkern-.1mu \widehat{n}\mkern-.5mu f}
......
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