Do we have a "topological assembly map" in the Baum-Connes conjecture?

Question

In the equivariant Atiyah-Singer index theorem, when $G$ is a compact group acting on a manifold $M$ and $R(G)$ is the representation ring of $G$. We have the analytic index $$ \text{a-ind}: K^*_G(TM)\rightarrow R(G) $$ given by considering the kernel and cokernel of a elliptic operator as representations of $G$.

We also have the topological index $$ \text{a-ind}: K^*_G(TM)\rightarrow R(G) $$ given by Bott periodicity or the Chern character and Todd class, etc.

Now if the group $G$ is not compact, we cannot apply the above method in the naive way. However, in the content of noncommutative geometry we have the assembly map $$ \mu: K^G_*(M)\rightarrow K _ * (C^* _ r (G) ) $$ which is a generalization of the analytic index map and the Baum-Connes conjecture claims that if we take $M$ to be the unversal proper $G$ space $\mathcal{E}G$, then the assemble map $$ \mu: K^G_*(\mathcal{E}G)\rightarrow K _ * (C^* _ r (G) ) $$ is an isomorphism. For more details we can refer to Alain Valatte's book Introduction to the Baum-Connes Conjecture.

My question is not about the Baum-Connes conjecture. but the following: Since the assembly map is an analytic index, does it make sense to define a topological index in this setting?

I've heard that the word "assembly map" originally came from surgery theory in topology but I'm not sure whether or not that gives us the desired definition.

Answer 1

There is an interpretation of the Baum-Connes assembly map as the index of a $C^*_r(G)$-linear operator in case $G$ is discrete and torsion-free. The group $K_*(BG)$ can be represented by triples $(M, E, f)$, where $M$ is a closed spin$^c$-manifold, $E$ is a complex vector bundle and $g \colon M \to BG$ is a continuous map with respect to an equivalence relation given in Section 5 here.

Let $\mathcal{V} = \widetilde{M} \times_{\lambda} C^*_r(\pi_1(M))$ be the bundle of right Hilbert $C^*_r(\pi_1(M))$-modules over $M$, where $\lambda \colon \pi_1(M) \to Aut(C^*_r(\pi_1(M)))$ (module automorphisms) is the left multiplication with elements of $\pi_1(M)$. Let $D$ be the Dirac operator on $M$. We can modify $D$ to act on smooth sections of the bundle $S \otimes E \otimes \mathcal{V}$ over $M$, where $S$ is the spinor bundle. Mishchenko and Fomenko have developed a theory of how to associate and index to these $A$-linear operators for some $C^*$-algebra $A$. This can be found here. The map $[M,E,f] \mapsto [ind(D^{E \otimes \mathcal{V}}_+)]$ is the assembly map.

They also prove an index theorem for these operators, which involves a topological index on the right hand side. For this you first need a Chern character, which can be obtained for nice enough $C^*$-algebras from the Künneth theorem for $K$-theory, i.e. we have isomorphisms $$ K_0(C(M,A)) \otimes \mathbb{R} = K_0(C(M) \otimes A) \otimes \mathbb{R} \cong \left(K_0(C(M)) \otimes K_0(A) \oplus K_1(C(M)) \otimes K_1(A) \right) \otimes \mathbb{R} $$ which yields a map $K_0(C(M,A)) \to K_0(C(M)) \otimes K_0(A) \otimes \mathbb{R}$ by projection to one summand and finally a Chern character $$ ch \colon K_0(C(M,A)) \to H^{\rm even}(M; K_0(A) \otimes \mathbb{R}) $$ by applying the usual Chern character to $K_0(C(M))$. The group $K_0(C(M,A))$ is the natural home for (formal differences of) isomorphism classes of Hilbert $A$-module bundles over $M$. In particular $E \otimes \mathcal{V}$ represents an element in $K_0(C(M,C^*_r(\pi_1(M)))$. A corollary of the Mishchenko-Fomenko index theorem is now that $$ {\rm ind}(D^{E \otimes \mathcal{V}}_+) = \int_M \hat{A}(M)\cdot ch(E \otimes \mathcal{V}) $$ and there you have your topological index (if I have done everything correct). I glossed over some things here, e.g. the grading on $S$. I also recommend the notes by Higson about the Baum Connes conjecture. (I recommend reading everything by Higson. He does a marvelous job at explaining things!)