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.