Geometric topology · algebraic K-theory · L-theory · assembly maps

Farrell–Jones Conjecture

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Do virtually cyclic subgroups contain enough information to reconstruct the algebraic K- and L-theory of every group through the Farrell–Jones assembly map?

H_{n}^{G} (E_{VCyc} G; K_{A}) \overset{≅}{\to} K_{n} (\int_{G} A)

Known results and sources

Farrell–Jones predicts that virtually cyclic subgroup data assembles exactly into the algebraic K- and L-theory of a group; the full coefficient statement remains open.

Research problem

Exact mathematical statement

Let $G$ be any group and let $\mathcal A$ be a small additive $G$ -category. The coefficient algebraic $K$ -theoretic Farrell–Jones conjecture asserts that, for every $n\in\mathbb Z$ , the assembly map

H_n^G(E_{\mathrm{VCyc}}G;\mathbf K_{\mathcal A}) \longrightarrow K_n\left(\int_G\mathcal A\right)

is an isomorphism. If $\mathcal A$ also carries an involution, the $L^{\langle-\infty\rangle}$ -theoretic Farrell–Jones conjecture asserts the analogous isomorphism for ultimate lower quadratic or symmetric $L$ -theory. The target here is the full coefficient statement for arbitrary groups; restrictions to regular rings, torsion-free groups, one orbit, or inverted primes are not the full conjecture. The full conjecture is not proved in this source.

Problem infographic

Problem at a glance

A scientific explainer places virtually cyclic subgroup geometry on the left, the equivariant homology assembly source in the center, and algebraic K- and L-theory of the group on the right, separating the K-theory and involutive L-theory statements and marking the general coefficient case open. — The conjecture says the assembly maps from the virtually cyclic family recover all algebraic K- and ultimate lower L-theory classes for arbitrary groups and coefficients. Many classes of groups are known, but the universal statement is open.

Current mathematical picture

Where work on Farrell–Jones Conjecture stands

Partially resolved

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: Anti-invariant chains or a complementary sign symmetry should kill the assembly identity carried by the fixed transfer. The source reaches finite algebraic generation and a global ordered coset-family model, but it has only finite support, not the continuous control and Or-spectrum factorization needed to annihilate the assembly obstruction. The arbitrary-coefficient generator theorem and exact Mackey…

Route status · Narrowed route

Main reductionFivefold normal-seed cover

Conditional on metric Kan–Thurston and inheritance, a fixed finite acyclic CAT(0) seed U mapping onto A5 yields a fivefold amplified cover whose acyclic kernel is normally generated by U.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeVerify the external metric Kan–Thurston, Farrell–Jones inheritance, Tor-unital localization, and finite-superperfect-seed inputs, and complete the arbitrary-coefficient/ringoid version of the normal-seed generator theorem.Task status · Work already reported in progress

Research-record correctionResearch-record correction

We corrected the cited passages. We clarified how the claims are connected. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Farrell–Jones Conjecture in numbers

1.4kretained lines of mathematical investigation1,380 in the current working snapshot

Argument development: 1,161 · 84%
Explored or eliminated routes: 74 · 5%
Computational analysis: 1 · 0%
Open obligations: 37 · 3%
Definitions and setup: 107 · 8%

9selected mapped statements1routes investigated3open questions2contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byF. Thomas Farrell, Lowell E. JonesProposed1993Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Write the global conjugacy-stable Mackey dg category with exact differential, intersection-sector, convolution-composition, coefficient, conjugation, and finite-support formulas.

Suggested move: Start from the finite complex P_U, calculate morphisms over U^lambda backslash Lambda_H slash U^mu, check triple-coset composition, and compare the resulting category with finite-propagation objects over the global ordered coset simplex.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Write the global conjugacy-stable Mackey dg category with exact differential, intersection-sector, convolution-composition, coefficient, conjugation, and finite-support formulas.Suggested move: Start from the finite complex P_U, calculate morphisms over U^lambda backslash Lambda_H slash U^mu, check triple-coset composition, and compare the resulting category with finite-propagation objects over the global ordered coset simplex.

Ready to work on

Prove NSCM-Or as a continuous family-controlled factorization of the full relative Or-spectrum endomorphism, then construct the duality-preserving NSCM-L refinement.Suggested move: Choose one controlled obstruction, germ, or localizing-motive model; realize the Morita counit and all higher coherences there; absorb nonuniform commutator-length growth; and run the U=1, G=1, virtually cyclic, and identity-cover degeneration tests before adding the quadratic or symmetric null-cobordism.

Ready to work on

Verify the external metric Kan–Thurston, Farrell–Jones inheritance, Tor-unital localization, and finite-superperfect-seed inputs, and complete the arbitrary-coefficient/ringoid version of the normal-seed generator theorem.Suggested move: Build a theorem-numbered source audit, settle the essential-image lemma in the chosen derived model, and carry the fixed integral seed complex through finite-object matrix reduction with orbit-pseudofunctor coherence.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusPartially resolved

The Full Farrell–Jones Conjecture is not known for all groups, and Lück's 2025 survey reports no known counterexample. It is proved for broad classes including hyperbolic groups, finite-dimensional CAT(0) groups, virtually solvable groups, the stated lattices, 3-manifold groups, S-arithmetic groups, mapping class groups, braid groups, and Coxeter groups. It remains open in general for classes including amenable groups, Out(F_n) for n ≥ 3, general Artin groups, Thompson groups, torsion-free one-relator groups, linear groups, residually finite groups, and automatic groups.

[2][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025
Authoritative summaryLück's survey and monograph consolidated the Full formulation, proved classes, inheritance properties, applications, and significant classes that remain open.[2][3]
2019
Peer reviewedBartels and Bestvina proved the Farrell–Jones conjecture for mapping class groups.[8]
2015
Peer reviewedWegner proved the K- and L-theoretic Farrell–Jones conjecture with coefficients in additive categories for virtually solvable groups.[7]
2012
Peer reviewedThe hyperbolic and finite-dimensional CAT(0) programs yielded topological rigidity via the K- and L-theoretic conjectures; Lück's current status theorem lists both as Full Farrell–Jones classes.[6][2]

Mathematical neighborhood

Related results and reusable starting points

Current focusFarrell–Jones conjecture

Stronger or generalized formFull Farrell–Jones Conjecture

The coefficient formulation is stronger than the original ring-coefficient statement. Lück's Full formulation adds K-, L-, higher-categorical, and finite-wreath-product stability and implies the recorded literature variants.

[4][2]

Dependency or reductionAssembly over virtually cyclic subgroups

The conjecture asserts bijectivity of assembly maps from equivariant homology over the family of virtually cyclic subgroups to algebraic K- or L-theory of the group ring or coefficient category.

[2]

Dependency or reductionUniversal finitely presented group reduction

Using subgroup and directed-colimit inheritance, the Full conjecture holds for all groups if and only if it holds for one universal finitely presented group. This is a one-group reduction, not a finite computation.

[2]

Solved special caseKnown Farrell–Jones group classes

Representative verified classes include hyperbolic groups, finite-dimensional CAT(0) groups, virtually solvable groups, and mapping class groups; the current survey lists further substantial classes.

[5][6]

Logical consequenceRigidity and group-ring consequences

The Full conjecture implies major consequences involving the Borel and Novikov conjectures, Whitehead-group vanishing for torsion-free groups, and conjectures of Bass, Kaplansky, and Serre.

[2][3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetFormal nonconnective algebraic K-theory and algebraic L-theory at the spectrum and coefficient-category level.
Formalization targetFormal equivariant homology theories and classifying spaces for families of subgroups.
Formalization targetFormal construction and functoriality of the Farrell–Jones assembly maps.
Formalization targetFormal controlled algebra and topology, flow spaces, transfers, and the geometric group-theory inputs for the known classes.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We clarified how the claims are connected. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements3 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction5 of 95
lemma3 of 93

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementVirtually cyclic subgroup data should recover all algebraic K- and L-theory of the group through assembly.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementConditional acyclic coverintermediate
retained route statementFivefold normal-seed coverintermediate
retained route statementNonuniform seed-width boundintermediate
retained route statementRing-level relative generatorintermediate
retained route statementGlobal ordered coset modelintermediate
retained route statementNSCM-Or factorization gateintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetVerify the external metric Kan–Thurston, Farrell–Jones inheritance, Tor-unital localization, and finite-superperfect-seed inputs, and complete the arbitrary-coefficient/ringoid version of the normal-seed generator theorem.in progress reported
Research targetWrite the global conjugacy-stable Mackey dg category with exact differential, intersection-sector, convolution-composition, coefficient, conjugation, and finite-support formulas.open
Research targetProve NSCM-Or as a continuous family-controlled factorization of the full relative Or-spectrum endomorphism, then construct the duality-preserving NSCM-L refinement.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.3 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
ComputationThe source proposes exact finite matrices for the seed contraction and small Mackey sectors as future test evidence, but contains no executed computational result needed for the current status.No code or attachment was executed during intake. Conditional and internal labels are preserved as source-reported research standing and do not constitute independent theorem verification. · reported unreproduced
Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: Anti-invariant chains or a complementary sign symmetry should kill the assembly identity carried by the fixed transfer. The source reaches finite algebraic generation and a global ordered coset-family model, but it has only finite support, not the continuous control and Or-spectrum factorization needed to annihilate the assembly obstruction. The arbitrary-coefficient generator theorem and exact Mackey composition are also unfinished, and L-theory needs a duality-preserving controlled null-cobordism.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

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Prepared starting pointWrite the global conjugacy-stable Mackey dg category with exact differential, intersection-sector, convolution-composition, coefficient, conjugation, and finite-support formulas.

Farrell–Jones Conjecture · ready to start

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Research contextPrepared context for any AI agent

Do virtually cyclic subgroups contain enough information to reconstruct the algebraic K- and L-theory of every group through the Farrell–Jones assembly map?

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Sources and references8 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Isomorphism conjectures in algebraic K-theoryoriginal source · F. Thomas Farrell, Lowell E. Jones · Journal of the American Mathematical Society · 1993 · DOI 10.1090/S0894-0347-1993-1179537-0 · MR 1179537 · accessed Aug 7, 2026
2
Survey on the Farrell-Jones Conjecturesurvey or monograph · Wolfgang Lück · arXiv · 2025 preprint · ARXIV 2507.11337 · accessed Aug 7, 2026
3
Isomorphism Conjectures in K- and L-Theorysurvey or monograph · Wolfgang Lück · Springer Cham · 2025 · DOI 10.1007/978-3-031-98976-6 · accessed Aug 7, 2026
4
Coefficients for the Farrell–Jones Conjecturepeer reviewed result · Arthur Bartels, Holger Reich · Advances in Mathematics · 2007 · ARXIV math/0510602 · DOI 10.1016/j.aim.2006.05.005 · accessed Aug 7, 2026
5
The K-theoretic Farrell–Jones conjecture for hyperbolic groupspeer reviewed result · Arthur Bartels, Wolfgang Lück, Holger Reich · Inventiones Mathematicae · 2008 · ARXIV math/0701434 · DOI 10.1007/s00222-007-0093-7 · accessed Aug 7, 2026
6
The Borel Conjecture for hyperbolic and CAT(0)-groupspeer reviewed result · Arthur Bartels, Wolfgang Lück · Annals of Mathematics · 2012 · DOI 10.4007/annals.2012.175.2.5 · MR 2993750 · accessed Aug 7, 2026
7
The Farrell–Jones conjecture for virtually solvable groupspeer reviewed result · Christian Wegner · Journal of Topology · 2015 · DOI 10.1112/jtopol/jtv026 · accessed Aug 7, 2026
8
The Farrell–Jones Conjecture for mapping class groupspeer reviewed result · Arthur Bartels, Mladen Bestvina · Inventiones Mathematicae · 2019 · ARXIV 1606.02844 · DOI 10.1007/s00222-018-0834-9 · accessed Aug 7, 2026

Important qualifications

Farrell–Jones has original, fibered, coefficient, K-theoretic, L-theoretic, higher-categorical, wreath-product-stable, and Full formulations; the record preserves these scopes rather than treating them as one unchanged sentence.
The known-class and open-class audit is based chiefly on Wolfgang Lück's July 2025 survey, checked on 2026-08-07; later class results would require an event-driven refresh.
No dedicated proof-assistant formalization, solver certificate, dataset, or independently reproduced computation for the conjecture was verified in the scoped search. Empty lists do not establish nonexistence.

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