ProofAtlas

Operator algebras · noncommutative geometry · topological K-theory

Baum–Connes Conjecture Without Coefficients

Collaboration beta

Does the proper geometry of a group recover every K-theory class of its reduced group C*-algebra, and only those classes?

μG:K*top(G)⟶K*(Cr*(G))is an isomorphism
Known results and sources
A dark mathematical landscape places a geometric group-action network opposite layered operator-algebra and K-theory forms, separated by an unresolved central aperture in the assembly connection.
The assembly map asks whether proper group geometry and reduced group C*-algebra K-theory contain exactly the same information; the coefficient-free conjecture remains open in general.

Research problem

Exact mathematical statement

For every second-countable locally compact group GG, the coefficient-free Baum–Connes assembly map

μG:K*top(G)⟶K*(Cr*(G))\mu_G:K_*^{\mathrm{top}}(G)\longrightarrow K_*(C_r^*(G))

is conjectured to be an isomorphism. Equivalently, every class on the reduced group C*C^*-algebra side should come from proper equivariant geometry, and two geometric classes should map to the same analytic class only when they were already equal. The conjecture is known for many important group classes but remains open in general. This statement is explicitly without coefficients; the stronger conjecture with arbitrary coefficients is a different statement and is false in full generality. The current research program develops a countable-discrete technical core, while the extension to all second-countable locally compact groups is a separate open stage.

Problem infographic

Problem at a glance

Scientific explainer for Baum–Connes without coefficients. Proper equivariant geometry maps through the unresolved assembly map to the reduced group C*-algebra. The general operator panel uses Haar-space L²(G,dμ), compactly supported continuous convolution kernels, projections, and unitaries; finite-support matrices appear only in the separately labelled discrete model. Onto and one-to-one remain questions, and the general locally compact statement remains open.
The coefficient-free assembly map is known to be an isomorphism for many major group classes, but whether it is both onto and one-to-one for every second-countable locally compact group remains open.

Current mathematical picture

Where work on Baum–Connes Conjecture Without Coefficients stands

Open conjecture

The v10 packet strengthens the finite quotient-table program without claiming Baum–Connes. It replaces a bounded rigidity search with a complete minimal-transversal certificate, reduces the common-square first stage to eleven marked-symmetry orbits with only two nonamenable descendant families, and retains the stronger four-point formal-subset and dimension-free lifting layers. The common-square descendants and the analytic transfer interfaces remain open.

Strongest supported footholdTwo-sided quotient-table lifting established

Exact unitary and projection relations are lifted to the full group algebra of the table group, together with a support-dependent almost-unitary bound.

Evidence posture · Reported result
Leading routeCommon-square descendant closure

Reproduce the governing finite fixtures and close the two marked common-square descendant trees before moving to the size-two, ERR, and ETR families or returning to the analytic transfer steps.

Route status · Active route
Useful failureSuperseded four-point closure census

This v8 prerequisite is inactive: the governing formal-subset argument bypasses simultaneous closure and realizability for the structural theorem. Retain the census only as optional independent checking material.

Route status · Eliminated route
Main reductionFinite-support route established

Target classes are reduced to finite-support controlled convolution matrices, with exact reductions reported for several structured supports.

Evidence posture · Reported reduction
Completed special casePure BS(1,2) five-point cell isolated

An exact table presentation, quotient-fiber counts, and a degree-one crossed-product reduction are given for a specific five-point support.

Evidence posture · Reported special case
Priority open bridgeClose the current load-bearing frontier

Independently reproduce the finite certificates, close the two common-square descendant trees, and then prove the reciprocal-support approximation and relative finite-Rips realization interfaces needed to leave the discrete table layer.

Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Baum–Connes Conjecture Without Coefficients in numbers

2.9kretained lines of mathematical investigation2,921 in the current working snapshot
Argument development
2,441 · 84%
Explored or eliminated routes
37 · 1%
Computational analysis
155 · 5%
Open obligations
141 · 5%
Definitions and setup
147 · 5%
11selected mapped statements6routes investigated6reported milestones6open questions5contribution-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 byPaul Baum, Alain ConnesProposed1982Status checkedAug 6, 2026
Source recordedAug 7, 2026

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

25 selected steps

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

25 selected steps

Scroll horizontally to explore the route

Working route overview for Baum–Connes Conjecture Without CoefficientsA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Coefficient-free Baum–Connes conjecture — ChallengedCoefficient-free Baum–ConnesconjectureCommon-square first-stage orbit reduction — Depends on missing premiseCommon-square first-stageorbit reductionDimension-free square-root table lifting — Depends on missing premiseDimension-free square-roottable liftingFinite Fourier representative reduction — Depends on missing premiseFinite Fourierrepresentative reductionFull-to-reduced assembly-image factorization — Depends on missing premiseFull-to-reducedassembly-image factorizationComplete binary-rigidity census — Depends on missing premiseComplete binary-rigiditycensusExact and quantitative quotient-table lifting — ActiveExact and quantitativequotient-table liftingFixed-table compact-family protocol — Depends on missing premiseFixed-table compact-familyprotocolFour-point formal-subset theorem — Depends on missing premiseFour-point formal-subsettheoremPure five-point BS(1,2) table cell — Depends on missing premisePure five-point BS(1,2)table cellSupport-size-at-most-three reduction — ActiveSupport-size-at-most-threereductionCommon-square descendant closure — activeCommon-square descendantclosureSupport-versus-accuracy control — activeSupport-versus-accuracycontrolResidual five-point table census — activeResidual five-point tablecensusRelative finite-Rips realization — activeRelative finite-RipsrealizationUse internal Gaussian matrix units to shrink geometric propagation — stoppedUse internal Gaussian matrixunits to shrink geometricpropagationUse finite-collar compression with boundary-to-volume estimates for arbitrary groups — stoppedUse finite-collarcompression withboundary-to-volume…Realize index data with rooted append trees or star–Fock tails — stoppedRealize index data withrooted append trees orstar–Fock…Multiply local isometries and infer global compatibility automatically — stoppedMultiply local isometriesand infer globalcompatibility…Break the support-versus-accuracy barrier — OpenBreak thesupport-versus-accuracybarrierClassify the smallest residual five-point table — OpenClassify the smallestresidual five-point tableConstruct the relative finite-Rips realization — OpenConstruct the relativefinite-Rips realizationExtend the discrete theorem to locally compact groups — BlockedExtend the discrete theoremto locally compact groupsAudit the standard analytic and K-theory inputs — OpenAudit the standard analyticand K-theory inputsClose the current load-bearing frontier — OpenClose the currentload-bearing frontier
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.

Active routeCommon-square descendant closure

Reproduce the governing finite fixtures and close the two marked common-square descendant trees before moving to the size-two, ERR, and ETR families or returning to the analytic transfer steps.

Route status · Active route
Active routeSupport-versus-accuracy control

Meet the exact reciprocal-support approximation threshold implied by the dimension-free square-root estimate, or improve it through weighted defect energy, a pre-lifting decomposition, or another genuinely scalar mechanism.

Route status · Active route
Active routeResidual five-point table census

Remove all already solved cells and factors, then classify the smallest realizable table whose particular lifted class survives the known retractions and PV boundaries.

Route status · Active route
Active routeRelative finite-Rips realization

Convert surviving analytic class reductions into continuous relative equivariant topological representatives at one finite Rips scale, preserving prescribed boundary data and proving index equality.

Route status · Active route

Explored alternatives

Other routes

2 recorded
Eliminated routeSuperseded four-point closure census

This v8 prerequisite is inactive: the governing formal-subset argument bypasses simultaneous closure and realizability for the structural theorem. Retain the census only as optional independent checking material.

Route status · Eliminated route
Route held in reserveLocally compact extension

After the discrete relative theorem, rebuild the construction with proper spaces, Haar kernels, compact stabilizers, modular corrections, and compactly generated open-subgroup continuity.

Route status · Route held in reserve

Route statements and reductions

Statements the next route can inspect and build on

Route statementFinite Fourier representative reduction

Individual even and odd target K-classes are reduced to finite-support almost projections or almost unitaries followed by spectral or polar correction. Compact parameter families can use common finite support and a common gap, but the relative construction required for injectivity is still open.

Source-reported route statement · dependencies incomplete
Route statementExact and quantitative quotient-table lifting

For a finite support S, the two-sided quotient-table group H_S records all exact left and right quotient relations. Exact unitary and projection relations lift to its full group C*-algebra, while an almost-unitary lift is invertible under the support-dependent condition kappa(S) epsilon < 1.

Source-reported route statement
Route statementPure five-point BS(1,2) table cell

For S={e,a,b,ab,b^2} with aba^{-1}=b^2, the exact table group is identified as BS(1,2), with reported quotient-fiber counts r(S)=15 and l(S)=13 and a reduction of the degree-one class through crossed-product K-theory. The algebraic part is marked [A] and the Pimsner–Voiculescu input [B] by the source.

Source-reported route statement · dependencies incomplete
Route statementFixed-table compact-family protocol

For a compact family with one ambient finite support, a continuous table lift and correction protocol is given under a uniform gap. It is sufficient for a local relative table calculation but is not the finite-Rips relative construction required for injectivity.

Source-reported route statement · dependencies incomplete
Route statementComplete binary-rigidity census

Berge’s incremental minimal-transversal enumeration replaces the earlier size-bounded search and reports 45, 2360, and 715 minimal covers of sizes two, three, and four, with no other sizes.

Source-reported route statement · dependencies incomplete
Route statementCommon-square first-stage orbit reduction

The first common-square audit reduces seventy raw one-step relators to eleven marked-symmetry orbits; only two nonamenable descendant families remain for iterative closure analysis.

Source-reported route statement · dependencies incomplete
Route statementFour-point formal-subset theorem

For each certified four-point normal form, every formal subset of its certified atomic relations presents a free product of virtually abelian groups.

Source-reported route statement · dependencies incomplete
Route statementDimension-free square-root table lifting

The current work records a compact-family square-root lifting interface whose loss is independent of matrix size, sharpening the analytic bridge required by the finite-table route.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks
01
Close the current load-bearing frontier

Independently reproduce the finite certificates, close the two common-square descendant trees, and then prove the reciprocal-support approximation and relative finite-Rips realization interfaces needed to leave the discrete table layer.

Suggested move: Run the retained finite auditors first, then encode the two surviving common-square families in one generic closure checker before returning to the analytic interfaces.
Ready to work on
02
Break the support-versus-accuracy barrier

Produce class-preserving finite Fourier approximations whose error beats the reciprocal support-size threshold implied by the square-root lifting criterion, or justify a weighted defect-energy or pre-lifting decomposition that improves it.

Suggested move: Test weighted defect energy and pre-lifting decompositions against the exact reciprocal-support threshold while retaining the particular K-theory class.
Ready to work on
03
Construct the relative finite-Rips realization

For each surviving analytic reduction, construct a continuous relative equivariant topological representative at one finite Rips scale and prove equality of analytic indices.

Suggested move: Bind one solved analytic branch to explicit KK-data over one finite Rips complex, continuously over compact parameters and relative to closed subsets.
Ready to work on
04
Classify the smallest residual five-point table

After removing solved factors and cell types, identify the smallest realizable five-point table whose particular lifted class survives the known retractions and PV boundaries.

Suggested move: Record each residual table's two-sided quotient partition, exact presentation, Grushko/HNN structure, retractions, PV boundaries, and the image of the particular class.
Ready to work on
05
Audit the standard analytic and K-theory inputs

Supply precise convention-compatible references or model checks for subgroup assembly naturality, amalgamated free-product Mayer–Vietoris, PV naturality, BS(1,2) K-theory, torus Bott generators, and topological continuity.

Suggested move: Fix one model of topological K-theory and verify every cited naturality and induction diagram in that model.
Ready to work on
06
Extend the discrete theorem to locally compact groups

After a discrete relative theorem exists, replace discrete Rips, finite-sum, finite-subgroup, and translation machinery with proper-space, Haar-kernel, compact-stabilizer, modular-function, and open-subgroup-continuity analogues.

Suggested move: Keep this phase separate until a complete discrete relative theorem is available.
Blocked by the current route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 6, 2026
Current statusOpen conjecture

The Baum–Connes conjecture without coefficients remains open for arbitrary second-countable locally compact groups. It is proved for many major classes, including a-T-menable groups, hyperbolic groups, and almost connected groups. Counterexamples to the stronger conjecture with coefficients do not settle or refute this coefficient-free statement.

[2][7][4]
External progress

What the literature has established

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

  1. Authoritative summaryThe universal coefficient-free conjecture remains open. Many major classes satisfy it, but no accepted general proof or coefficient-free counterexample was identified in the current status sources checked for…[2][7]
  2. Peer reviewedLafforgue proved Baum–Connes with coefficients for hyperbolic groups, hence also the coefficient-free conjecture for that broad class.[6][2]
  3. Peer reviewedChabert, Echterhoff, and Nest proved the coefficient-free assembly isomorphism for second-countable almost connected groups and for rational points of linear algebraic groups over characteristic-zero local…[5]
  4. Peer reviewedHigson, Lafforgue, and Skandalis constructed counterexamples to Baum–Connes with coefficients. Their result is a decisive boundary: it does not supply a counterexample to the coefficient-free conjecture.[4][2]
11 cited sources7 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusBaum–Connes conjecture without coefficients
Stronger or generalized formBaum–Connes conjecture with coefficients

The coefficient version replaces the trivial coefficient algebra by arbitrary G-C*-algebras. It implies the coefficient-free case when the coefficient is C, but it is false in full generality.

[1][4]
Logical consequenceNovikov conjecture

Injectivity of the Baum–Connes assembly map yields the higher-signature consequences associated with the Novikov conjecture for the relevant discrete groups.

[2]
Logical consequenceKadison–Kaplansky conjecture

For torsion-free discrete groups, surjectivity of the coefficient-free assembly map implies that the reduced group C*-algebra has no nontrivial idempotents.

[2][7]
Solved special caseConnes–Kasparov conjecture for almost connected groups

For almost connected groups, the assembly-map problem is the Connes–Kasparov setting; Chabert, Echterhoff, and Nest established this coefficient-free case.

[5]
Solved special caseBaum–Connes for a-T-menable groups

Groups with the Haagerup property satisfy the stronger conjecture with coefficients, providing a large solved family that includes amenable groups.

[3]
Solved special caseBaum–Connes for hyperbolic groups

Hyperbolic groups satisfy Baum–Connes with coefficients and therefore also the coefficient-free conjecture.

[6]
Related problemcoarse Baum–Connes conjecture

The coarse conjecture uses Roe algebras and metric-space assembly maps. It is a major source of techniques and counterexamples but is not the same statement as the group conjecture without coefficients.

[2]

Formalization opportunities

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

  • Formalization targetNo problem-level Lean statement or proof of the coefficient-free Baum–Connes conjecture was identified in the scoped current Mathlib and Google DeepMind Formal Conjectures search.
  • Formalization targetA faithful formal statement needs reduced group C*-algebras and crossed products for second-countable locally compact groups, operator K-theory, equivariant K-homology or KK-theory, universal proper G-spaces with compact-support conventions, descent, and the analytic assembly map.
  • Formalization targetThe discrete torsion-free formulation is not a substitute for the full locally compact statement; torsion requires proper-action equivariance and general locally compact groups require Haar-measure and continuity infrastructure.
  • Formalization targetLibrary support for groups, topology, Hilbert spaces, or ordinary algebraic K-theory would remain only prerequisite infrastructure until statement alignment to the operator-algebraic assembly map is checked.
  • Formalization targetAny formal statement would still be statement-only; no proof status follows from the extensive solved-class literature or from the submitted packet's algebraic reductions.

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Four-point route strengthened and finite frontier narrowedThe current work supersedes the v8 closure prerequisite with the formal-subset route, retains dimension-free lifting, repairs the rigidity census, and reduces the common-square first stage to eleven orbits with two unresolved descendants.

Changed the research frontierLater mathematical revision

Research stage 6

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

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 updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. 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.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

5 mapped milestonesretained argument map

Browse all 5 mapped stages

  1. stage 1Finite-support analytic model established
  2. stage 2Exact quotient-table lifting developed
  3. stage 3Small and structured table cells reduced
  4. stage 4Four-point claim narrowed to a finite certificate
  5. stage 5Completion frontier separated into five routes
Finite-support analytic model establishedTarget K-classes are reduced to finite-support controlled convolution matrices, with the relative finite-scale requirement for injectivity kept separate.

Mapped research milestoneInitial research sequence

Research stage 1
Exact quotient-table lifting developedThe exact two-sided table group lifts exact relations to its full group algebra and gives a support-dependent quantitative bound.

Mapped research milestoneInitial research sequence

Research stage 2
Small and structured table cells reducedA complete support-size-at-most-three reduction and a pure five-point BS(1,2) cell provide two concrete solved branches within the broader route.

Mapped research milestoneInitial research sequence

Research stage 3
Four-point claim narrowed to a finite certificateThe reported single-relation enumeration closes one finite subtask while leaving simultaneous closure, realizability, and table-group classification explicitly open.

Mapped research milestoneInitial research sequence

Research stage 4
Completion frontier separated into five routesThe remaining work separates into four-point certification, support-stable approximation, residual five-point tables, relative finite-Rips realization, and the locally compact extension.

Mapped research milestoneInitial research sequence

Research stage 5

Detailed research inventory

Claims, milestones, and routes in the current map

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

10 standing statements1 proposed statements6 mathematical milestones6 open questions1 conditional results3 completed special cases
Statements by mathematical role11 selected mapped statements
  • theorem candidate1 of 111
  • reduction4 of 114
  • lemma5 of 115
  • computational claim1 of 111
Selected mathematical clusters4 mathematical clusters
Conjecture and finite-support reductionThe coefficient-free assembly question, its discrete technical core, and the reduction of target K-classes to finite-support controlled matrices.3 displayed rows
  • retained route statementCoefficient-free Baum–Connes conjecture
  • retained route statementFinite Fourier representative reductionintermediate
  • ChallengeAnalytic target-class generation does not establish injectivity without a relative construction at one finite Rips scale.unsupported step · open
Exact finite-support reductionsSmall-support classification, quotient-table lifting, assembly-image factorization, and the pure BS(1,2) cell.5 displayed rows
  • retained route statementSupport-size-at-most-three reductionspecial case
  • retained route statementExact and quantitative quotient-table liftingintermediate
  • retained route statementFull-to-reduced assembly-image factorizationintermediate
  • retained route statementPure five-point BS(1,2) table cellspecial case
  • DerivationExposed coefficients are separated, while the remaining three-point quotient collisions are classified into cyclic, dihedral, or virtually cyclic cases.active reported
Current finite-table frontierThe source-reported four-point formal-subset and dimension-free lifting layers, the complete rigidity census, the eleven-orbit common-square reduction, two unresolved common-square descendants, and the remaining five-point families.14 displayed rows · 3 routes included
  • retained route statementAll-four-term assembly claimconditional
  • retained route statementFour-point formal-subset theoremspecial case
  • retained route statementDimension-free square-root table liftingintermediate
  • retained route statementComplete binary-rigidity censuscomputational
  • retained route statementCommon-square first-stage orbit reductionintermediate
  • DerivationThis v8 derivation required exhaustive simultaneous table classification and is superseded as a prerequisite by the v10 formal-subset argument.invalidated
  • ChallengeSingle quotient-equality lists do not establish all simultaneous equality closures, realizability, exact presentations, Grushko decompositions, or factor generation in both K_0 and K_1.unsupported step · reported resolved
  • Research targetSuperseded four-point closure prerequisitecompleted reported
  • Research targetClose the current load-bearing frontieropen
  • Research targetClassify the smallest residual five-point tableopen
  • ComputationEmbedded Python source reports exhaustive free-word enumeration of single quotient-equality relators in primitive, cyclic-progression, and rectangular four-point normal forms.The source reports lists of 9, 17, and 15 relators and says all assertions passed. The embedded code was not independently executed. The result covers individual relations only, not simultaneous closure, realizability, or table-group classification. · reported unreproduced
  • Active routeCommon-square descendant closureReproduce the governing finite fixtures and close the two marked common-square descendant trees before moving to the size-two, ERR, and ETR families or returning to the analytic transfer steps.
  • Eliminated routeSuperseded four-point closure censusThis v8 prerequisite is inactive: the governing formal-subset argument bypasses simultaneous closure and realizability for the structural theorem. Retain the census only as optional independent checking material.
  • Active routeResidual five-point table censusRemove all already solved cells and factors, then classify the smallest realizable table whose particular lifted class survives the known retractions and PV boundaries.
Global completion barriersSupport-stable approximation, relative finite-Rips realization, exact cited inputs, scalar-only safeguards, and the separate locally compact extension.13 displayed rows · 3 routes included
  • retained route statementFixed-table compact-family protocolconditional
  • Research targetBreak the support-versus-accuracy barrieropen
  • Research targetConstruct the relative finite-Rips realizationopen
  • Research targetExtend the discrete theorem to locally compact groupsblocked
  • Research targetAudit the standard analytic and K-theory inputsopen
  • Useful failureUse internal Gaussian matrix units to shrink geometric propagationreported failure
  • Useful failureUse finite-collar compression with boundary-to-volume estimates for arbitrary groupsreported failure
  • Useful failureRealize index data with rooted append trees or star–Fock tailsreported failure
  • Useful failureMultiply local isometries and infer global compatibility automaticallyreported failure
  • Useful failureApply a universal arbitrary labelled-graph cell theoremreported failure
  • Active routeSupport-versus-accuracy controlMeet the exact reciprocal-support approximation threshold implied by the dimension-free square-root estimate, or improve it through weighted defect energy, a pre-lifting decomposition, or another genuinely scalar mechanism.
  • Active routeRelative finite-Rips realizationConvert surviving analytic class reductions into continuous relative equivariant topological representatives at one finite Rips scale, preserving prescribed boundary data and proving index equality.
  • Route held in reserveLocally compact extensionAfter the discrete relative theorem, rebuild the construction with proper spaces, Haar kernels, compact stabilizers, modular corrections, and compactly generated open-subgroup continuity.
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

Priority open bridgeIndependently reproduce the finite certificates, close the two common-square descendant trees, and then prove the reciprocal-support approximation and relative finite-Rips realization interfaces needed to leave the discrete table layer.

The current research map records this as an open mathematical step.

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.

  • Reproduce the complete minimal-transversal and eleven-orbit outputs from retained code.
  • Close or explicitly branch every descendant of the two remaining common-square families.
  • State and prove the scalar approximation and relative finite-Rips interfaces with exact hypotheses.

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Baum–Connes Conjecture Without Coefficients · ready to start

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Research contextPrepared context for any AI agent

Does the proper geometry of a group recover every K-theory class of its reduced group C*-algebra, and only those classes?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references11 cited works · next context review by Nov 6, 2026

The mathematical context was checked on Aug 6, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Classifying space for proper actions and K-theory of group C*-algebrasoriginal source · Paul Baum, Alain Connes, Nigel Higson · American Mathematical Society · 1994 · DOI 10.1090/conm/167/1292018 · MR MR1292018 · accessed Aug 6, 2026
  2. 2
    The Baum–Connes conjecture: an extended surveysurvey or monograph · Maria Paula Gomez Aparicio, Pierre Julg, Alain Valette · Springer · 2020 · ARXIV 1905.10081 · DOI 10.1007/978-3-030-29597-4_3 · MR MR4300553 · accessed Aug 6, 2026
  3. 3
    E-theory and KK-theory for groups which act properly and isometrically on Hilbert spacepeer reviewed result · Nigel Higson, Gennadi Kasparov · Inventiones Mathematicae · 2001 · DOI 10.1007/s002220000118 · accessed Aug 6, 2026
  4. 4
    Counterexamples to the Baum–Connes conjecturepeer reviewed result · Nigel Higson, Vincent Lafforgue, Georges Skandalis · Geometric and Functional Analysis · 2002 · DOI 10.1007/s00039-002-8249-5 · accessed Aug 6, 2026
  5. 5
    The Connes–Kasparov conjecture for almost connected groups and for linear p-adic groupspeer reviewed result · Jérôme Chabert, Siegfried Echterhoff, Ryszard Nest · Publications Mathématiques de l'IHÉS · 2003 · DOI 10.1007/s10240-003-0014-2 · accessed Aug 6, 2026
  6. 6
    La conjecture de Baum–Connes à coefficients pour les groupes hyperboliquespeer reviewed result · Vincent Lafforgue · Journal of Noncommutative Geometry · 2012-01-16 · ARXIV 1201.4653 · DOI 10.4171/JNCG/89 · accessed Aug 6, 2026
  7. 7
    Baum–Connes conjectureencyclopedia · Wikimedia Foundation · accessed Aug 6, 2026
  8. 8
    List of conjecturesencyclopedia · Wikimedia Foundation · accessed Aug 6, 2026
  9. 9
    List of unsolved problems in mathematicsencyclopedia · Wikimedia Foundation · accessed Aug 6, 2026
  10. 10
    Mathlib4 repositoryformalization · The Mathlib Contributors · Lean community · accessed Aug 6, 2026
  11. 11
    Formal Conjectures repositoryformalization · The Formal Conjectures Authors · Google DeepMind · accessed Aug 6, 2026

Important qualifications

  • This record is scoped to Baum–Connes without coefficients for second-countable locally compact groups. Baum–Connes with coefficients, coarse Baum–Connes, groupoid formulations, maximal variants, and the Bost conjecture are not treated as interchangeable statements.
  • The coefficient-free conjecture remains open in general; the Higson–Lafforgue–Skandalis counterexamples concern the stronger conjecture with coefficients and do not disprove the coefficient-free statement.
  • The literature contains many additional group classes, permanence results, and refinements. The milestones below are representative rather than a complete bibliography.
  • The scoped formalization search checked current public Mathlib and Google DeepMind Formal Conjectures surfaces but found no problem-level Baum–Connes statement or proof. This does not establish nonexistence in every proof assistant, branch, private repository, or differently named project.
  • No canonical finite computation, public certificate, or dataset deciding the general conjecture was identified. Individual K-theory computations for specific groups are not automatically evidence for the universal statement.
  • Wikipedia recognition entries are broad reference links only, not selective-list memberships, prizes, or mathematical status authorities.
  • No current Clay Millennium, Hilbert, Smale, Erdős, Epoch/FrontierMath, named-prize, or comparable selective-list designation for this exact coefficient-free statement was verified.
  • The submitted packet's confidence labels, finite-support arguments, embedded computation, and completion estimate were not used as external authority and were not independently reviewed in this lane.

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