Farrell–Jones conjecture

Can information about small subgroups, joined with information about how they sit inside a larger group, recover the larger group’s algebraic invariants exactly? A group specifies how transformations combine. For a two-element group, the nonidentity transformation applied twice does nothing; a group ring allows sums of these transformations with numerical coefficients as well as composition. Its K- and L-theories record stable algebraic information, including modules and quadratic forms. Farrell–Jones asks whether this information can always be assembled from virtually cyclic subgroups: subgroups containing a cyclic subgroup of finite index. Every finite group counts as virtually cyclic. The target includes coefficients with a group action, not just one numerical coefficient ring.[1] [2]

A group describes how reversible moves combine. Can information from simpler collections of moves, together with how they sit inside the group, recover its algebraic information exactly? A group ring allows sums of moves with coefficients: if a move $s$ undoes itself, $s^2=1$ and $(1+s)^2=2+2s$. Over such a ring, modules generalize vector spaces. K-theory includes records of modules and their invertible transformations after stabilization; L-theory includes records of quadratic forms, with standard pieces allowed in their comparisons.

The simpler collections form the virtually cyclic subgroup family: finite groups, or groups containing a cyclic subgroup of finite index. A cyclic group is generated by repeating one move; finite index means that finitely many cosets, or shifted copies, cover the subgroup. The two-element group above already belongs to this family. For a general group $G$, let $E_{\mathrm{VCyc}}G$ be its classifying space for that family. The coefficient systems may be more general than one ring and carry a group action. For every additive $G$-category $\mathcal A$ and $n\in\mathbb Z$, require the assembly map $H_n^G(E_{\mathrm{VCyc}}G;\mathbf K_{\mathcal A})\to K_n(\mathcal A[G])$ to be an isomorphism. If $\mathcal A$ also has a compatible involution, require the analogous map to $L_n^{\langle-\infty\rangle}(\mathcal A[G])$. The source includes how subgroup data are joined; the target is not merely the direct sum of subgroup invariants.[1] [2]

The statement is established for substantial families, including hyperbolic groups and virtually solvable groups. The two-element example is already in the virtually cyclic family, so it does not test reconstruction for an arbitrary larger group. Published 2026 extension results cover specified suspensions and automorphism groups with geometric and peripheral assumptions. The missing claim ranges over every group, every allowed coefficient category and every integer degree.[1] [2]

The selected survey’s Conjectures 8.7 and 8.8 are the exact additive-category coefficient targets. Known classes, including the surveyed full-version results, satisfy these targets, but arbitrary groups remain outside a general proof. The published 2026 Theorems A–C retain hypotheses on the relatively hyperbolic groups, their peripheral subgroups or the automorphism setting. A proof for the finite-wreath-product package would imply the page’s statements, but that package is stronger and is not silently part of the requested scope.[1] [2]

Algebraic invariants can measure whether two spaces that look the same after continuous deformation are actually equivalent as manifolds. For example, a region connecting two high-dimensional manifolds can have the same homotopy information as each end without being a simple product cylinder; a Whitehead-group obstruction distinguishes these possibilities under the usual cobordism hypotheses. Farrell–Jones supplies algebraic input for such obstruction calculations and for rigidity results. The motivation is to connect algebraic data to classification of spaces, not merely to make an assembly map easier to compute.[1] [2]

For a connected closed manifold of dimension at least five, the Whitehead group of its fundamental group classifies h-cobordisms under the s-cobordism theorem: trivial obstruction corresponds to a product cobordism. K-theory also detects whether certain finitely dominated spaces have finite cell models, while L-theory enters surgery obstructions. The survey connects suitable torsion-free K- and L-assembly statements to Borel rigidity in dimensions at least five. These are distinct geometric uses of the algebraic invariants. They explain why organizing subgroup data can matter to space classification, without claiming that the conjecture alone classifies every manifold or removes the other hypotheses.[1] [2]