Farrell–Jones conjecture in algebraic K- and L-theory

Let $G$ be a discrete group and let $R$ be an associative ring with unit. Write $RG$ for the group ring.

• $\mathcal{VCyc}$ is the family of virtually cyclic subgroups of $G$, those that are finite or contain $\mathbb Z$ as a subgroup of finite index.
• $E_{\mathcal{VCyc}}G$ is a classifying space for this family. It is a $G$-CW complex such that the fixed set $(E_{\mathcal{VCyc}}G)^H$ is contractible for every $H\in\mathcal{VCyc}$ and empty for every other subgroup $H$. It exists and is unique up to $G$-homotopy equivalence.
• $H^G_*(-;\mathbf K_R)$ is the $G$-equivariant homology theory of Davis and Lück built from non-connective algebraic $K$-theory [3]. On orbits it takes the values $H^G_n(G/H;\mathbf K_R)=K_n(RH)$ for all $n\in\mathbb Z$.
• If $R$ carries an involution, $H^G_*(-;\mathbf L^{\langle-\infty\rangle}_R)$ is the analogous theory built from algebraic $L$-theory with decoration $\langle-\infty\rangle$. On orbits it takes the values $L^{\langle-\infty\rangle}_n(RH)$.

The projection $E_{\mathcal{VCyc}}G\to G/G=\mathrm{pt}$ induces the assembly maps

$$\mathrm{asmb}^K_n\colon H^G_n(E_{\mathcal{VCyc}}G;\mathbf K_R)\longrightarrow K_n(RG), \qquad \mathrm{asmb}^L_n\colon H^G_n(E_{\mathcal{VCyc}}G;\mathbf L^{\langle-\infty\rangle}_R)\longrightarrow L^{\langle-\infty\rangle}_n(RG).$$

Conjecture.

• $K$-theoretic Farrell–Jones conjecture. For every group $G$, every ring $R$ and every $n\in\mathbb Z$, the map $\mathrm{asmb}^K_n$ is an isomorphism.
• $L$-theoretic Farrell–Jones conjecture. For every group $G$, every ring with involution $R$ and every $n\in\mathbb Z$, the map $\mathrm{asmb}^L_n$ is an isomorphism.

The problem is to prove both statements for all groups, or to give a group, a ring and a degree for which one of them fails. The family $\mathcal{VCyc}$ cannot be shrunk to the finite subgroups in general: already for $G=\mathbb Z$ and a non-regular ring $R$, the $K$-theory of $R[\mathbb Z]$ contains Nil-terms that the finite subgroups do not see. [1, 2, 4]

If $G$ is torsion-free and $R$ is regular, the $K$-theoretic conjecture reduces to the statement that the classical assembly map $H_n(BG;\mathbf K(R))\to K_n(RG)$ is an isomorphism. If $G$ is torsion-free, the $L$-theoretic conjecture reduces in the same way to $H_n(BG;\mathbf L^{\langle-\infty\rangle}(R))\to L^{\langle-\infty\rangle}_n(RG)$. [2]

Background

Farrell and Jones formulated the conjecture in 1993 [1]. It grew out of their work on topological rigidity of nonpositively curved manifolds, which used controlled topology and the geodesic flow. The conjecture predicts that the $K$- and $L$-theory of a group ring are assembled from the virtually cyclic subgroups together with homological data. It is the algebraic counterpart of the Baum–Connes conjecture for group $C^*$-algebras [2].

The conjecture implies many other problems. For a torsion-free group $G$, the $K$-theoretic version for $R=\mathbb Z$ gives $\mathrm{Wh}(G)=0$, $\widetilde K_0(\mathbb ZG)=0$ and $K_{-i}(\mathbb ZG)=0$ for $i\ge1$. Together with the $L$-theoretic version it implies the Borel conjecture for closed aspherical manifolds of dimension at least $5$: any homotopy equivalence between two such manifolds is homotopic to a homeomorphism. The $L$-theoretic version implies the Novikov conjecture. Further consequences include the Kaplansky idempotent conjecture and the Bass conjecture for wide classes of rings [2, 4]. In practice one proves a stronger version with coefficients in additive categories [5], because that version passes to subgroups, finite products and directed colimits.