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K-theoretic Farrell–Jones Conjecture with Coefficients for Additive Categories

Establish that for every discrete group G and every additive category A with G-action, the assembly map μ: H^G_*(E_{VCyc} G; K_A) → K_*(A ⋊ G) is an isomorphism.

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Background

The version with coefficients generalizes the Farrell–Jones conjecture to the setting of additive categories equipped with G-actions, replacing the group ring by the crossed product category. This stronger form is essential for the authors’ methods, as it enables the use of equivariant additive functors and the Going-Down principle.

While the conjecture with coefficients is known for many groups (including those cited in the paper), its validity for all discrete groups remains open and is a key assumption throughout the authors’ results.

References

All these groups satisfy a more general $K$-theoretic Farrell--Jones conjecture with coefficients Conjecture 3.2, which says that for any additive category $\mathcal A$ with $G$-action, the assembly map \begin{equation}\label{eq:FJcoeff} \mu\colon HG_*(\mathcal E_{\mathrm{VCyc}G,\mathbb K_{\mathcal{A})\to K_*(\mathcal A\rtimes G) \end{equation} is an isomorphism.

eq:FJcoeff:

μ ⁣:H∗G(EVCycG,KA)→K∗(A⋊G)\mu\colon H^G_*(\mathcal E_{\mathrm{VCyc}}G,\mathbb K_{\mathcal{A}})\to K_*(\mathcal A\rtimes G)

— Bernoulli shifts on additive categories and algebraic $K$-theory of wreath products (2401.14806 - Kranz et al., 26 Jan 2024) in Section 1 (Introduction), Equation (2), label eq:FJcoeff