Chern characters for proper equivariant homology theories and applications to \(K\)- and \(L\)-theory (Q2769421)

An equivariant homology theory \({\mathcal H}^?_*\) assigns to each discrete group \(G\) a \(G\)-homology theory \({\mathcal H}^G_*\) graded over \(*\in\mathbb{Z}\) which is defined for proper \(G\)-CW-complexes. The various homology theories \({\mathcal H}_*^G\) for different groups \(G\) are linked by induction structures. An example is equivariant topological \(K\)-homology. We assign to an equivariant proper homology theory \({\mathcal H}^?_*\) an equivariant Chern character, provided that certain conditions are satisfied. It computes \({\mathcal H}_*^G(X)\) rationally in terms of the Bredon homology of \(X\) with coefficients in the system \(G/H \mapsto {\mathcal H}^G_q (G/H)\). These conditions are for instance satisfied if there exist Mackey structures on the collection of the \({\mathcal H}^G_q(*)\), where \(G\) runs through the finite groups and \(q\in\mathbb{Z}\) is fixed. This is the case in many interesting situations such as equivariant topological \(K\)-homology, equivariant bordisms, etc.NEWLINENEWLINENEWLINEThe construction of the equivariant Chern character applies also to the sources of the assembly maps in the Farrell-Jones Conjecture with respect to the family \({\mathcal F}\) of finite subgroups and in the Baum-Connes Conjecture. Thus we get an explicit calculation of \(\mathbb{Q} \otimes_\mathbb{Z} K_n(RG)\) and \(\mathbb{Q}\otimes_\mathbb{Z} L_n(RG)\) for a commutative ring \(R\) with \(\mathbb{Q}\subset R\), provided the Farrell-Jones Conjecture with respect to \({\mathcal F}\) is true, and of \(\mathbb{Q}\otimes_\mathbb{Z} K_n^{\text{top}}(C^*_r(G,F))\) for \(F=\mathbb{R}\), \(\mathbb{C}\), provided the Baum-Connes Conjecture is true, in terms of the group homology of the centralizers of the finite cyclic subgroups of \(G\) and the values \(K_q(R)\), \(L_q(R)\) and \(K_n^{\text{top}} (F)\).