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| Farrell–Jones conjecture | |
|---|---|
| Name | Farrell–Jones conjecture |
| Field | Algebraic K-theory; Algebraic L-theory; Geometric Topology |
| Proposed | 1993 |
| Proposers | F. Thomas Farrell; Lowell E. Jones |
| Status | Partially proven in many cases |
| Consequences | Borel conjecture; Novikov conjecture; Bass conjecture |
Farrell–Jones conjecture The Farrell–Jones conjecture asserts that for a discrete group G and a ring R the assembly maps in algebraic K-theory and L-theory from equivariant homology of classifying spaces to the K- and L-groups of group rings are isomorphisms. The conjecture, formulated by F. Thomas Farrell and Lowell E. Jones, links categorical and homotopical constructions with concrete algebraic invariants and has driven major advances relating Novikov conjecture, Borel conjecture, and manifold rigidity problems. Its study connects techniques from Geometric group theory, Controlled topology, Surgery theory, and higher algebraic structures appearing in the work of researchers at institutions such as Institute for Advanced Study, Max Planck Institute for Mathematics, and Clay Mathematics Institute.
The conjecture proposes that for a discrete group G and an additive category A with G-action the assembly map - in algebraic K-theory: from the equivariant homology of the classifying space for the family of virtually cyclic subgroups to K_*(A ⋊ G), - in algebraic L-theory: from the equivariant homology of the classifying space for the family of virtually cyclic subgroups to L_*(A ⋊ G), is an isomorphism. This uses models of classifying spaces such as E_{vc}G studied alongside constructions from Quillen K-theory, the Waldhausen S-construction, and Ranicki's algebraic surgery framework developed with input from Cappell and Wall. The statement is formulated using equivariant homology theories related to Segal and Elmendorf frameworks and is compatible with excision principles developed by Thomason and Neeman.
Motivation comes from problems in high-dimensional topology such as rigidity of aspherical manifolds and classification of pseudoisotopies arising from work of Novikov, Borel, Milnor, and Kervaire. Verification of the conjecture for a group G implies the Novikov conjecture for G and, in many cases, the Borel conjecture for closed aspherical manifolds with fundamental group G, tying to results by Farrell and Jones on nonpositively curved manifolds and to rigidity theorems of Mostow and Mostow–Prasad. It also relates to the Bass conjecture on Hattori–Stallings traces, the Kaplansky conjecture on group rings, and assembly phenomena studied in works by Rosenberg and Atiyah.
The conjecture has been established for large classes of groups using independent methods: word-hyperbolic groups via work of Bartels, Lück, and Reich; CAT(0) groups using flow-space and controlled algebra techniques by Wegner and collaborators; virtually solvable groups and lattices in Lie groups through efforts by Bartels, Farrell, Lück, and Reich; groups acting on trees via inheritance properties from Bass–Serre theory building on Serre; and linear groups over fields of characteristic zero drawing on work of Guentner, Higson, and Kasparov. Specific proofs combine methods from controlled Algebraic K-theory pioneered by Quinn and ‘‘transfer reducibility’’ developed by Bartels and Reich, and use permanence properties proved by Lück and Rüping.
Techniques include controlled topology and controlled algebra from Quinn and Pedersen–Weibel, homological and categorical methods from Dwyer–Weiss–Williams, geometric group theory tools from Gromov and Bestvina–Feighn, and equivariant homotopy theoretic machinery inspired by Elmendorf and Lewis. Key innovations are the use of finite decomposition complexity attributable to Guentner–Tessera–Yu, the development of flow spaces and dynamic methods by Bartels and Lück, and inheritance or permanence results using induction and spectral sequence arguments related to Atiyah–Hirzebruch type filtrations. Operator-algebraic input from Kasparov and coarse geometry approaches by Yu also influence proofs of the Novikov consequences.
If true for G, the conjecture implies explicit calculations and vanishing theorems for Nil-terms, UNil-terms, and lower K-theory groups for group rings R[G], refining classical results of Bass, Swan, and Cohn. It yields rigidity statements for manifold classification via algebraic L-theory surgery exact sequences pioneered by Wall and Ranicki, and gives injectivity results for assembly maps that settle instances of the Kaplansky conjecture and the integral Novikov conjecture as addressed by Rosenberg and Connes. Computational consequences have been used to determine K-theory of group C*-algebras in collaboration with work by Higson and Kasparov.
Related conjectures include the Baum–Connes conjecture for topological K-theory of reduced group C*-algebras and the Novikov conjecture on higher signatures; interactions with the Baum–Connes program have produced conditional results and counterexamples in exotic contexts studied by Higson–Lafforgue–Skandalis. Potential counterexamples to naive strengthenings of the Farrell–Jones statement arise from groups with pathological finiteness properties explored by Thompson-type constructions and by groups constructed using methods of Gromov and Olshanskii; however, no counterexample to the original formulation for discrete groups is known in the mainstream literature. The landscape continues to evolve through work at research centers including Mathematical Sciences Research Institute and collaborations across fields represented by prize-awarded authors such as Lück and Bartels.
Category:Conjectures in algebraic topology