ProofAtlas

Geometric topology · homogeneous ANRs · homology manifolds

Bing–Borsuk Conjecture

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The Bing–Borsuk conjecture asks whether every finite-dimensional homogeneous metric ANR is a topological manifold. The selected source reports a compact, connected counterexample conditional on an imported Bryant–Ferry existence theorem whose revised construction proof has not been independently line-audited here.

∃U[Ucompact, connected, finite-dimensional, homogeneous, ANR, and not a manifold]
Known results and sources
A dark cellular surface folds through a balanced field of repeating local patches, with one unresolved amber seam interrupting the otherwise homogeneous pattern.
The selected source reports a homogeneous nonmanifold counterexample conditional on an imported Bryant–Ferry existence theorem; this image explains the topology and is not independent evidence for that import.

Research problem

Exact mathematical statement

The Bing–Borsuk conjecture asserts that every finite-dimensional homogeneous metric absolute neighborhood retract is a topological manifold of its dimension. Homogeneity means that for every two points x,yx,y, some self-homeomorphism carries xx to yy. Thus a negative solution requires just one space UU satisfying

Ufinite-dimensional, homogeneous, and a metric ANR,Unot a topological manifold.U\text{ finite-dimensional, homogeneous, and a metric ANR},\qquad U\text{ not a topological manifold}.

The selected v16 source reports a counterexample with the additional properties of compactness and connectedness as a complete deduction from an imported Bryant–Ferry existence theorem, and it reports a stronger fixed-root endpoint from a second root-scoped import. ProofAtlas has not independently line-audited the unavailable revised Bryant–Ferry construction manuscript, so these remain source-reported research claims rather than an accepted ProofAtlas result.

Problem infographic

Problem at a glance

A wide problem-first topology scene contrasts smooth sheet-like neighborhoods with uniformly repeated cellular neighborhoods around a compact object, while a central transition remains visibly unresolved.
Homogeneity makes all points equivalent under homeomorphisms, while manifold status requires Euclidean local neighborhoods. The source reports that its imported index-nine example separates these properties; ProofAtlas has not independently line-audited the imported construction.

Current mathematical picture

Where work on Bing–Borsuk Conjecture stands

Recent proof claim under review

The v16 source reports a complete classical refutation and a complete fixed-root strong endpoint relative to exact imported Bryant–Ferry premises, with explicit normal-map, later-BFMW, and compact-ANR cell-like closures; the revised construction manuscript remains unavailable for independent line audit.

Leading routeTheorem-import route

The source reports the formal classical refutation and strong fixed-root endpoint as complete deductions from the imported Bryant–Ferry statements; no live implication remains after those imports, but the underlying construction is not independently audited.

Route status · Active route
Useful failurefuture-base lookahead scheduler

Its dependency order is choose current errors, create the next stage, discover the following threshold, then retroactively declare the current errors small enough. Finite-stage graph-collar conversion, pointed preservation, conditional parity recognition, and the endpoint nonresolvability argument remain usable only within their stated interfaces.

Route status · Eliminated route
Main reductionNonresolvable endpoint reduction

The current work source-reports that a manifold resolution of the constructed U would compose with its cell-like map to resolve the fixed nonresolvable root, so a valid endpoint U would not be a manifold.

Evidence posture · Source-reported route statement · dependencies incomplete
Completed special caseRoot-scoped imported Bryant–Ferry source theorem

For the fixed compact connected normally prepared ANR homology 7-manifold X_0 with X_0 homotopy equivalent to S^7 and Quinn obstruction 9, the source imports T-BF-UNIVERSAL-X0 only at that fixed root: a compact outstanding DDP ENR homology 7-manifold U_hat and a cell-like surjection c:U_hat -> X_0.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeObtain and independently line-audit the revised Bryant–Ferry manuscript

Obtain a stable exact copy of An alpha approximation theorem for homology manifolds and audit its hypotheses, fine-web construction, outstanding control improvement, alpha approximation, and parameterized conclusions.

Task status · Ready to work on
Later mathematical updatev16 separates the classical deduction from its imported construction

The source reports a classical refutation and fixed-root strong endpoint relative to imported Bryant–Ferry premises, records the closed downstream interfaces, and makes the independent construction boundary explicit.

v16 cumulative source order; not independent priority evidence

Work mapped so far

Bing–Borsuk Conjecture in numbers

2.1kretained lines of mathematical investigation1,068 in the current working snapshot
Argument development
1,742 · 82%
Explored or eliminated routes
95 · 4%
Computational analysis
18 · 1%
Open obligations
125 · 6%
Definitions and setup
146 · 7%
13selected mapped statements5routes investigated4open questions4contribution-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 byR. H. Bing and Karol BorsukProposed1965Status checkedAug 26, 2026
Source recordedAug 30, 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

19 selected steps

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

19 selected steps

Scroll horizontally to explore the route

Working route overview for Bing–Borsuk ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Imported Bryant–Ferry homogeneous sphere-like existence theorem — Depends on missing premiseImported Bryant–Ferryhomogeneous sphere-likeexistence…Root-scoped imported Bryant–Ferry source theorem — Depends on missing premiseRoot-scoped importedBryant–Ferry source theoremSource-reported compact-ANR cell-like homotopy closure — Depends on missing premiseSource-reported compact-ANRcell-like homotopy closureSource-reported post-erratum BFMW excellence theorem — ActiveSource-reported post-erratumBFMW excellence theoremIndependent construction remains unavailable for line audit — Depends on missing premiseIndependent constructionremains unavailable for lineauditSource-reported classical Bing–Borsuk refutation from an imported theorem — Depends on missing premiseSource-reported classicalBing–Borsuk refutation froman…Source-reported complete strong endpoint at the fixed root — Depends on missing premiseSource-reported completestrong endpoint at the fixedrootSource-reported explicit normal-map closure — ActiveSource-reported explicitnormal-map closureNonresolvable endpoint reduction — Depends on missing premiseNonresolvable endpointreductionConditional parity recognition — Depends on missing premiseConditional parityrecognitionFinite-stage graph-collar conversion — Depends on missing premiseFinite-stage graph-collarconversionFuture-base lookahead failure — Depends on missing premiseFuture-base lookaheadfailureTheorem-import route — activeTheorem-import routeBryant–Ferry source-audit route — activeBryant–Ferry source-auditrouteOptional independent Bryant–Ferry reconstruction route — activeOptional independentBryant–Ferry reconstructionrouteObtain and independently line-audit the revised Bryant–Ferry manuscript — OpenObtain and independentlyline-audit the revisedBryant–Ferry…Determine the exact post-erratum universal-source scope — OpenDetermine the exactpost-erratumuniversal-source…Optional independent finite-web successor reconstruction — OpenOptional independentfinite-web successorreconstructionOptional alpha completion after the web theorem — OpenOptional alpha completionafter the web theorem
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 routeTheorem-import route

The source reports the formal classical refutation and strong fixed-root endpoint as complete deductions from the imported Bryant–Ferry statements; no live implication remains after those imports, but the underlying construction is not independently audited.

Route status · Active route
Active routeBryant–Ferry source-audit route

The primary next route is to obtain the revised manuscript or a later publication and audit exact hypotheses and all construction interfaces, without confusing author statements or secondary corroboration with a line-audited proof.

Route status · Active route
Active routeOptional independent Bryant–Ferry reconstruction route

When the revised manuscript cannot be obtained, the source keeps the fine-web and alpha reconstruction as a primary alternate route, with exact-fiber descent, carrier, and dipole work demoted to retained subroutes or subordinate infrastructure rather than current theorem-import prerequisites.

Route status · Active route

Explored alternatives

Other routes

2 recorded
Eliminated routefuture-base lookahead scheduler

Its dependency order is choose current errors, create the next stage, discover the following threshold, then retroactively declare the current errors small enough. Finite-stage graph-collar conversion, pointed preservation, conditional parity recognition, and the endpoint nonresolvability argument remain usable only within their stated interfaces.

Route status · Eliminated route
Eliminated routeannounced alpha-theorem route

The current work reports that no complete public manuscript was found as of 2026-08-26. The announced route remains source-dependent context, not current proof evidence.

Route status · Eliminated route

Route statements and reductions

Statements the next route can inspect and build on

Route statementSource-reported classical Bing–Borsuk refutation from an imported theorem

For U = H_{7,9}, the imported Bryant–Ferry theorem supplies one compact connected finite-dimensional homogeneous metric ANR with Quinn obstruction 9; the identity-resolution contradiction shows that U is not a topological manifold, so the universal Bing–Borsuk statement is false, conditional on that imported existence theorem.

Source-reported route statement · dependencies incomplete
Route statementImported Bryant–Ferry homogeneous sphere-like existence theorem

For every integer n >= 6 and every integer k in 1+8Z, the source reports that Bryant and Ferry state the existence of a topologically homogeneous DDP homology n-manifold homotopy equivalent to S^n with Quinn obstruction k, nonresolvable when k != 1.

Source-reported route statement · dependencies incomplete
Route statementIndependent construction remains unavailable for line audit

The source explicitly reports that the revised Bryant–Ferry proof manuscript has not been obtained, so the fine singular web, outstanding control improvement, alpha approximation, and homogeneous-example construction remain external imports rather than independently audited mathematics.

Source-reported route statement · dependencies incomplete
Route statementRoot-scoped imported Bryant–Ferry source theorem

For the fixed compact connected normally prepared ANR homology 7-manifold X_0 with X_0 homotopy equivalent to S^7 and Quinn obstruction 9, the source imports T-BF-UNIVERSAL-X0 only at that fixed root: a compact outstanding DDP ENR homology 7-manifold U_hat and a cell-like surjection c:U_hat -> X_0.

Source-reported route statement · dependencies incomplete
Route statementSource-reported complete strong endpoint at the fixed root

From the root-scoped imported theorem, the cell-like compact-ANR theorem, and the post-erratum excellence theorem, the source reports a compact connected seven-dimensional homogeneous DDP metric ANR homology 7-manifold U_hat homotopy equivalent to S^7, BFMW-excellent, cell-like over X_0, and not a manifold or locally Euclidean anywhere.

Source-reported route statement · dependencies incomplete
Route statementSource-reported explicit normal-map closure

For Y homotopy equivalent to a closed topological n-manifold M, with M oriented so h has degree one, the source constructs xi=g^*(nu_M), identifies its spherical fibration with the Spivak normal fibration of Y, and uses h^*xi equivalent to nu_M to equip h:M->Y as a degree-one normal map; this is not inferred from a total-surgery obstruction.

Source-reported route statement
Route statementSource-reported compact-ANR cell-like homotopy closure

The source reports from PUB-LACHER68 that a cell-like map between compact ANRs is a homotopy equivalence, and applies this to c:U_hat->X_0 to obtain U_hat homotopy equivalent to X_0 and hence to S^7.

Source-reported route statement · dependencies incomplete
Route statementSource-reported post-erratum BFMW excellence theorem

The source reports that a compact DDP ANR homology n-manifold with n>=6 and closed-manifold homotopy type is BFMW-excellent: the explicit normal map repairs Lemma 7.2, DDP supplies UV^1, and the inspected later reverse proof uses no second unrestricted normal-reduction premise.

Source-reported route statement

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Obtain and independently line-audit the revised Bryant–Ferry manuscript

Obtain a stable exact copy of An alpha approximation theorem for homology manifolds and audit its hypotheses, fine-web construction, outstanding control improvement, alpha approximation, and parameterized conclusions.

Suggested move: Use repository, library, author-contact, or later-title channels; freeze the exact file digest and version before any proof audit.
Ready to work on
02
Determine the exact post-erratum universal-source scope

Determine whether the imported universal source theorem applies to every ANR homology manifold or only to represented or oriented roots after the correction; this affects breadth but not the fixed-root classical deduction.

Suggested move: Read the exact revised theorem normal and orientation hypotheses if the manuscript becomes available.
Ready to work on
03
Optional independent finite-web successor reconstruction

If the revised preprint remains inaccessible, construct the finite carrier web with intersection-relative UV^1, local stabilization sites, and carrier-controlled terminal cancellation, beginning with one buffered gate and the relative Whitney/handle-support theorem.

Suggested move: Treat this as the primary alternate reconstruction route only after preserving preprint acquisition as the source-level priority.
Ready to work on
04
Optional alpha completion after the web theorem

Given the finite-web theorem, construct point-moving controlled self-equivalences and use the authors' alpha mechanism or an independently proved analogue; a contracting sequence with summable bi-uniform movement is sufficient.

Suggested move: Do not treat alpha completion as a current prerequisite for the theorem-import endpoint; resume it only inside the optional independent reconstruction route.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 26, 2026
Current statusRecent proof claim under review

The bounded external review retains the conjecture as open with an unverified resolution claim. An official 2018 seminar page announces counterexamples in all dimensions above five, while a peer-reviewed 2020 survey still presents the conjecture as open; no complete public proof of the announcement was located.

[3][4]
External progress

What the literature has established

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

  1. Peer reviewedThe BFMW erratum restricts a normal-map existence claim while stating that the original examples, including simply connected examples, remain unchanged.[5]
  2. Peer reviewedValov's survey describes the Bing–Borsuk conjecture and related generalized-manifold problems as open, creating a status tension with the later announcement rather than resolving it.[4]
  3. Authoritative summaryAn official Florida State University seminar announcement states that counterexamples exist in every dimension greater than five; the bounded review did not locate a complete public proof to verify that…[3]
  4. Peer reviewedBryant, Ferry, Mio, and Weinberger developed desingularization results for homology manifolds, providing central machinery for proposed counterexample routes without establishing homogeneity by itself.[2]
5 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusBing–Borsuk conjecture
Stronger or generalized formgeneralized manifold recognition problems

Generalized-manifold recognition questions ask when homology-manifold or resolvability hypotheses force genuine manifold structure; they provide context but are not identical to the homogeneous-ANR assertion.

[4]
Dependency or reductionnonresolvable homology manifolds and desingularization

Desingularizing a corrected nonresolvable homology-manifold root can supply a nonmanifold endpoint, but a Bing–Borsuk counterexample additionally requires a rigorously established homogeneous total space.

[2][5]

Formalization opportunities

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

  • Formalization targetA statement-aligned formalization would require libraries for compact metric ANRs, topological homogeneity, finite-dimensionality, and topological manifolds.
  • Formalization targetThe proposed homology-manifold route would additionally need formal treatments of DDP, cell-like maps, resolvability, controlled topology, and inverse-limit recognition.
  • Formalization targetThe announced high-dimensional counterexamples require a complete public mathematical proof before they can be represented as checked or resolved evidence.

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.

v16 separates the classical deduction from its imported constructionThe source reports a classical refutation and fixed-root strong endpoint relative to imported Bryant–Ferry premises, records the closed downstream interfaces, and makes the independent construction boundary explicit.

Changed the research frontierLater mathematical revision

v16 cumulative source order; not independent priority evidence

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.

12 standing statements1 proposed statements4 open questions3 conditional results2 completed special cases
Statements by mathematical role13 selected mapped statements
  • reduction1 of 131
  • lemma4 of 134
  • negative result2 of 132
  • counterexample1 of 131
  • theorem candidate4 of 134
  • special case1 of 131
Selected mathematical clusters1 mathematical clusters
Current research mapThe current source-reported map separates the exact classical deduction from its imported Bryant–Ferry existence premise, preserves the unavailable-manuscript boundary, retains prior failures and reconstruction tools as optional history, and keeps source acquisition and scope audit open.24 displayed rows · 3 routes included
  • retained route statementImported Bryant–Ferry homogeneous sphere-like existence theorem
  • retained route statementSource-reported compact-ANR cell-like homotopy closureconditional
  • retained route statementConditional parity recognitionintermediate
  • retained route statementSource-reported classical Bing–Borsuk refutation from an imported theorem
  • retained route statementFinite-stage graph-collar conversionintermediate
  • retained route statementFuture-base lookahead failureintermediate
  • retained route statementIndependent construction remains unavailable for line auditintermediate
  • retained route statementNonresolvable endpoint reductionintermediate
  • retained route statementSource-reported explicit normal-map closureconditional
  • retained route statementPointed pleasant-stage interfaceintermediate
  • retained route statementSource-reported post-erratum BFMW excellence theoremconditional
  • retained route statementRoot-scoped imported Bryant–Ferry source theoremspecial case
  • retained route statementSource-reported complete strong endpoint at the fixed rootspecial case
  • DerivationChoose the imported example at n=7 and k=9; sphere homotopy type gives connectedness, nonresolvability contradicts the identity map being a manifold resolution, and one counterexample refutes the universal assertion. The deduction is complete only relative to the imported existence theorem.active reported
  • DerivationClosed-manifold homotopy type supplies the explicit degree-one normal map, DDP in dimension at least six supplies UV^1, and the source reports that the later BFMW reverse construction uses Lemma 7.2, Lemma 2.5, summable controls, and inverse-limit recognition without a second unrestricted normal-reduction appeal.active reported
  • DerivationFor the fixed nonresolvable normally prepared root, the imported cell-like source is connected and homogeneous, cannot be a manifold because its map would resolve X_0, has sphere homotopy type by the compact-ANR cell-like theorem, and is BFMW-excellent by the explicit-normal-map post-erratum branch theorem.active reported
  • Recorded relationshipThe exact source-reported Bryant–Ferry existence statement supplies the sole external existence premise used by the internal identity-resolution deduction; this edge records source semantics and is not independent verification.supports · reported by source
  • Research targetOptional alpha completion after the web theoremopen
  • Research targetObtain and independently line-audit the revised Bryant–Ferry manuscriptopen
  • Research targetDetermine the exact post-erratum universal-source scopeopen
  • Research targetOptional independent finite-web successor reconstructionopen
  • Active routeOptional independent Bryant–Ferry reconstruction routeWhen the revised manuscript cannot be obtained, the source keeps the fine-web and alpha reconstruction as a primary alternate route, with exact-fiber descent, carrier, and dipole work demoted to retained subroutes or subordinate infrastructure rather than current theorem-import prerequisites.
  • Active routeBryant–Ferry source-audit routeThe primary next route is to obtain the revised manuscript or a later publication and audit exact hypotheses and all construction interfaces, without confusing author statements or secondary corroboration with a line-audited proof.
  • Active routeTheorem-import routeThe source reports the formal classical refutation and strong fixed-root endpoint as complete deductions from the imported Bryant–Ferry statements; no live implication remains after those imports, but the underlying construction is not independently audited.
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 bridgeObtain a stable exact copy of An alpha approximation theorem for homology manifolds and audit its hypotheses, fine-web construction, outstanding control improvement, alpha approximation, and parameterized conclusions.

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.

  • Retain one exact stable manuscript or publication with version identity.
  • Independently audit every load-bearing theorem hypothesis and construction step.

Continue the mathematics

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Prepared starting pointObtain and independently line-audit the revised Bryant–Ferry manuscript

Bing–Borsuk Conjecture · ready to start

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Research contextPrepared context for any AI agent

The Bing–Borsuk conjecture asks whether every finite-dimensional homogeneous metric ANR is a topological manifold. The selected source reports a compact, connected counterexample conditional on an imported Bryant–Ferry existence theorem whose revised construction proof has not been independently line-audited here.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references5 cited works · next context review by Nov 26, 2026

The mathematical context was checked on Aug 26, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Some Remarks Concerning Topologically Homogeneous Spacesoriginal source · R. H. Bing, Karol Borsuk · Annals of Mathematics · 1965 · DOI 10.2307/1970385 · accessed Aug 26, 2026
  2. 2
    Desingularizing homology manifoldspeer reviewed result · J. L. Bryant, S. C. Ferry, W. Mio, S. Weinberger · Geometry & Topology · 2007 · DOI 10.2140/gt.2007.11.1289 · accessed Aug 26, 2026
  3. 3
    The Bing-Borsuk Conjectureauthoritative webpage · John Bryant · Florida State University Department of Mathematics · 2018-04-18 · accessed Aug 26, 2026
  4. 4
    Homogeneous ANR-spaces and the Bing–Borsuk conjecturesurvey or monograph · Vesko Valov · Serdica Mathematical Journal · 2020 · ARXIV 2003.06907 · accessed Aug 26, 2026
  5. 5
    Erratum: ‘Topology of homology manifolds’peer reviewed result · J. L. Bryant, S. C. Ferry, W. Mio, S. Weinberger · Annals of Mathematics · 2024 · DOI 10.4007/annals.2024.200.2.8 · accessed Aug 26, 2026

Important qualifications

  • Open-with-unverified-claim is a conservative status: an official 2018 seminar page announces counterexamples in dimensions above five, while a 2020 peer-reviewed survey still describes the Bing–Borsuk conjecture as open and no complete public proof was located in the bounded review.
  • The 2024 BFMW erratum corrects a normal-map existence claim and says the original examples are unchanged; it does not itself prove a homogeneous nonmanifold or the Bing–Borsuk counterexample.
  • The 2007 desingularization theorem supplies homology-manifold and cell-like-resolution machinery, not by itself the homogeneity needed for the conjecture.
  • No packet attachment, submitted URL, internal project theorem, source-reported audit, or model-generated summary was treated as independent external status authority.
  • No statement-aligned formalization or independently checked construction was established by the scoped search; an empty formalization list does not assert nonexistence.

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