The Baum-Connes conjecture asserts that for a group G the analytic assembly map, which is nowadays usually defined using KK-theory, \[\mu_*^{K\!K}\colon RK_*^G(\underline{EG}) \to K_*^{top}(C^*_r G)\] is an isomorphism. This map can be factored through the algebraic K-theory of the group ring SG, where S denotes the Schatten-class operators on an \(\infty\)-dimensional, separable Hilbert space. The resulting map \[\mu_*^{alg}\colon RK_*^G(\underline{EG}) \to K_*^{alg}(\mathcal{S}G)\] is often called the algebraic assembly map and it has the feature that it can be transformed by certain Chern(-Connes-Karoubi) characters to a map \[\mu_*^{H\!P}\colon H^G_{[*]}(\underline{EG};\mathbb{C}) \to H\!P_*(\mathbb{C}G)\] from the equivariant (Bredon) homology of \(\underline{EG}\) to periodic cyclic homology of \(\mathbb{C}G\). This blog post is about the resulting map \[\mu_*^{ch}\colon RK_*^G(\underline{EG}) \to H\!P_*(\mathbb{C}G)\] obtained by composing some of the maps occurring in the above discussion.
Details of the above constructions can be found in my paper ‘Burghelea conjecture and asymptotic dimension of groups’ jointly written with Marcinkowski (arXiv:1610.10076) and in the paper ‘Operator ideals and assembly maps in K-theory’ by Cortiñas-Tartaglia (arXiv:1202.4999). The idea of using the Schatten-class operators as coefficients to define an algebraic version of the analytic assembly map is due to Yu (arXiv:1106.3796).
In a recent preprint (arXiv:2012.12359) Rouse-Wang-Wang constructed a Chern assembly map \[ch^{top}_\mu\colon K_{top}^*(G) \to H\!P_*(\mathbb{C}G)\,.\] Here the left-hand side is the domain of the assembly map \[\mu_*^{BC}\colon K_{top}^*(G) \to K^{top}_*(C^*_r G)\] as originally defined by Baum-Connes (employing heavily shriek maps in equivariant K-theory). In arXiv:1402.3456 Rouse-Wang defined a comparison map \[\lambda_G\colon K_{top}^*(G) \to RK_*^G(\underline{EG})\] and showed that it relates the assembly maps \(\mu_*^{BC}\) and \(\mu_*^{K\!K}\). When reading the introduction of their first mentioned preprint, the following question immediately came to my mind:
- Does the comparison map \[\lambda_G\colon K_{top}^*(G) \to RK_*^G(\underline{EG})\] relate the map \[\mu_*^{ch}\colon RK_*^G(\underline{EG}) \to H\!P_*(\mathbb{C}G)\] to the one defined by Rouse-Wang-Wang: \[ch^{top}_\mu\colon K_{top}^*(G) \to H\!P_*(\mathbb{C}G)\,?\] I strongly assume that this is indeed the case, so the question basically is ‘How hard (or easy) is it to prove this?’
Interestingly, in the paper by Rouse-Wang where they define the comparison map \(\lambda_G\), they do not prove that it is an isomorphism and instead leave this as an open problem. Again, I assume that it is actually an isomorphism for any group G, and so we end with the question:
- How hard (or easy) is it to prove that the comparison map \(\lambda_G\) is an isomorphism?