ProofAtlas

Smooth four-manifold topology

Smooth four-dimensional Schoenflies conjecture

Collaboration beta

Does every smoothly embedded three-sphere split the standard four-sphere into two standard smooth four-balls?

S3↪S4⇒U+¯≅diffB4andU-¯≅diffB4
Listed inK3 — A New Problem List in Low-Dimensional Topology
Known results and sources
A luminous three-sphere shell divides a four-dimensional globe-like form into two mirrored chambers, each approaching but not certified as a standard four-ball.
A smooth embedded S³ separates S⁴ into two contractible sides; whether both are standard smooth B⁴s is shown as an unresolved two-sided question.

Research problem

Exact mathematical statement

For every smooth embedding i:S³↪S⁴, the two complementary closures should both be diffeomorphic to the standard four-ball:

S4∖i(S3)=U+⊔U-⇒U+¯≅diffB4andU-¯≅diffB4.S^4\setminus i(S^3)=U_+\sqcup U_-\quad\Longrightarrow\quad \overline{U_+}\cong_{\mathrm{diff}}B^4\;\text{ and }\;\overline{U_-}\cong_{\mathrm{diff}}B^4.

The retained source treats this as open. Its handle-theoretic reductions and cancellation lemmas are source-reported and do not establish the conjecture.

Problem infographic

Problem at a glance

A landscape topology plate shows a smooth embedded three-sphere separating the four-sphere into two closures and marks both diffeomorphism-to-four-ball claims as open.
The smooth Schoenflies question is two-sided: each complementary closure is contractible, but the conjecture asks whether each is diffeomorphic to the standard B⁴.

Current mathematical picture

Where work on Smooth four-dimensional Schoenflies conjecture stands

Open conjecture

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

Useful failureContractibility-only inference

The source isolates an unresolved one-middle-handle barrier and a rank-two handle target, showing that contractibility alone does not close the required smooth classification in its route. A handle-theoretic proof may remain viable if the exact spectator-disjoint cancellation, its marked-core audit, and the relevant filling/core compatibility are established.

Route status · Narrowed route
Main reductionCurrent reduction

The source reformulates the conjecture as triviality of the boundary-sum unit group and relates it to homotopy-sphere units; after the one-middle-handle case, it identifies Rank-Two Unit Cancellation as the first unresolved global barrier and restates the exact two-index-2-handle target.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve the spectator-disjoint cancellation in the exact rank-two local handle configuration with no hidden intersections.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

Smooth four-dimensional Schoenflies conjecture in numbers

932retained lines of mathematical investigation932 in the current working snapshot
Argument development
765 · 82%
Explored or eliminated routes
22 · 2%
Computational analysis
14 · 2%
Open obligations
46 · 5%
Definitions and setup
85 · 9%
6selected mapped statements1routes 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.

Status checkedAug 14, 2026
Source recordedAug 14, 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

11 selected steps

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

11 selected steps

Scroll horizontally to explore the route

Working route overview for Smooth four-dimensional Schoenflies conjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Does every smooth three-sphere in the four-sphere cut off two standard four-balls? — Depends on missing premiseDoes every smooththree-sphere in thefour-sphere…Boundary-monoid unit formulation — Depends on missing premiseBoundary-monoid unitformulationCurrent reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetComplementary sides are contractible — Depends on missing premiseComplementary sides arecontractibleOne-middle-handle barrier — Depends on missing premiseOne-middle-handle barrierContractibility-only inference — stoppedContractibility-onlyinferenceProve the spectator-disjoint cancellation in the exact rank-two local handle configuration with no hidden intersections. — OpenProve the spectator-disjointcancellation in the exactrank-two…Prove the marked-core sphere-isotopy and filling-compatibility conditions required by the proposed cancellations, preserving prescribed surgery-core incidence and the disk-and-annulus filling pieces. — OpenProve the marked-coresphere-isotopy andfilling-compatibility…Prove either Mixed Complexity Descent or Uniform Return-Arc Cancellation to pass from the reduced local branches to arbitrary mixed rank-two units. — OpenProve either MixedComplexity Descent orUniform…Exact complementary-ball statement — OpenExact complementary-ballstatement
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 routeContractibility-only inference

The source isolates an unresolved one-middle-handle barrier and a rank-two handle target, showing that contractibility alone does not close the required smooth classification in its route. A handle-theoretic proof may remain viable if the exact spectator-disjoint cancellation, its marked-core audit, and the relevant filling/core compatibility are established.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Prove the spectator-disjoint cancellation in the exact rank-two local handle configuration with no hidden intersections.Suggested move: Write the complete attaching data and verify every slide, Whitney move, framing, and spectator intersection in the local model.
Ready to work on
02
Prove the marked-core sphere-isotopy and filling-compatibility conditions required by the proposed cancellations, preserving prescribed surgery-core incidence and the disk-and-annulus filling pieces.Suggested move: Supply the marked-core sphere lemma, track the sphere slides relative to the surgery core, and verify that the torus isotopy preserves the disk-and-annulus filling pieces.
Ready to work on
03
Exact complementary-ball statement

Prove that both complementary closures of every smooth S³ embedded in S⁴ are diffeomorphic to B⁴.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Prove either Mixed Complexity Descent or Uniform Return-Arc Cancellation to pass from the reduced local branches to arbitrary mixed rank-two units.Suggested move: Define a well-founded mixed complexity and prove strict descent by allowed slides and isotopies, or prove the lexicographically controlled return-arc exchange across every higher complexity spectrum.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The smooth four-dimensional Schoenflies conjecture remains open. Specialist current context separates it from the solved topological theorem. Gabai gives a pseudo-isotopy approach, while Cha and Kim prove a related candidate family standard and discuss potential Schoenflies counterexamples; neither paper settles every smooth embedding S³↪S⁴.

[1][3][4]
External progress

What the literature has established

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

  1. Authoritative summaryThe K3 specialist problem list retains the surrounding smooth four-dimensional embedding and smoothing problems as open.[1]
  2. PreprintCha and Kim proved Calegari's homotopy four-spheres standard and described related potential Schoenflies counterexamples; this is a candidate-family result.[4]
  3. PreprintGabai developed an approach to the smooth four-dimensional Schoenflies conjecture via pseudo-isotopies of S¹×S³, without claiming a complete resolution.[3]
  4. Peer reviewedBrown proved the generalized Schoenflies theorem in the topological locally flat setting. That result does not decide whether smooth four-dimensional complementary balls have the standard smooth structure.[2]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusSmooth four-dimensional Schoenflies conjecture
Weaker or relaxed formtopological generalized Schoenflies theorem

Topological complementary domains are standard topological balls under the appropriate locally flat hypotheses; the conjecture asks for diffeomorphism in the exceptional smooth dimension four.

[2]
Related problemsmooth four-dimensional Poincaré and exotic 4-ball problems

Smooth homotopy four-spheres and exotic four-balls are tightly related sources of candidate constructions, but standardness of selected spheres does not classify every embedded S³ complement.

[1][4]
Dependency or reductionpseudo-isotopy formulation on S¹×S³

Gabai's exact pseudo-isotopy framework offers a route to the Schoenflies problem; its reduction must be used with the stated hypotheses and is not itself a solution.

[3]

Formalization opportunities

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

  • Formalization targetChecked smooth manifolds with boundary, embeddings, complementary closures, collars, and diffeomorphism to standard balls.
  • Formalization targetFormal four-dimensional handle decompositions, Kirby moves, pseudo-isotopies, and boundary-sensitive cancellation theorems.
  • Formalization targetA statement-alignment proof connecting any chosen handle or pseudo-isotopy criterion to both complementary-ball conclusions.

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
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.

Detailed research inventory

Claims, milestones, and routes in the current map

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

4 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role6 selected mapped statements
  • theorem candidate1 of 61
  • reduction1 of 61
  • lemma2 of 62
  • equivalence1 of 61
  • negative result1 of 61
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
  • retained route statementDoes every smooth three-sphere in the four-sphere cut off two standard four-balls?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementComplementary sides are contractibleintermediate
  • retained route statementBoundary-monoid unit formulationintermediate
  • retained route statementOne-middle-handle barrierintermediate
  • Recorded relationshipThe source 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 source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureContractibility-only inferencereported failure
  • Research targetProve the spectator-disjoint cancellation in the exact rank-two local handle configuration with no hidden intersections.open
  • Research targetProve the marked-core sphere-isotopy and filling-compatibility conditions required by the proposed cancellations, preserving prescribed surgery-core incidence and the disk-and-annulus filling pieces.open
  • Research targetProve either Mixed Complexity Descent or Uniform Return-Arc Cancellation to pass from the reduced local branches to arbitrary mixed rank-two units.open
  • Research targetExact complementary-ball statementopen
  • Research targetExact rank-two targetsuperseded
  • Research targetSpectator-disjoint cancellation auditsuperseded
  • Narrowed routeContractibility-only inferenceThe source isolates an unresolved one-middle-handle barrier and a rank-two handle target, showing that contractibility alone does not close the required smooth classification in its route. A handle-theoretic proof may remain viable if the exact spectator-disjoint cancellation, its marked-core audit, and the relevant filling/core compatibility are established.
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 bridgeProve the spectator-disjoint cancellation in the exact rank-two local handle configuration with no hidden intersections.

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.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

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Prepared starting pointProve the spectator-disjoint cancellation in the exact rank-two local handle configuration with no hidden intersections.

Smooth four-dimensional Schoenflies conjecture · ready to start

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Research contextPrepared context for any AI agent

Does every smoothly embedded three-sphere split the standard four-sphere into two standard smooth four-balls?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    K3 — A New Problem List in Low-Dimensional Topologymaintained problem list · R. İnanç Baykur, Robion C. Kirby, Daniel Ruberman · American Mathematical Society · 2026 · accessed Aug 14, 2026
  2. 2
    A proof of the generalized Schoenflies theorempeer reviewed result · Morton Brown · Bulletin of the American Mathematical Society · 1960 · accessed Aug 14, 2026
  3. 3
    3-Spheres in the 4-Sphere and Pseudo-Isotopies of S¹×S³preprint · David Gabai · arXiv · 2022 · ARXIV 2212.02004 · accessed Aug 14, 2026
  4. 4
    Calegari's homotopy 4-spheres from fibered knots are standardpreprint · Jae Choon Cha, Min Hoon Kim · arXiv · 2024 · ARXIV 2411.10051 · accessed Aug 14, 2026

Important qualifications

  • This record concerns smooth embeddings S³ into S⁴ and diffeomorphism of both complementary closures to B⁴; it does not conflate the statement with topological Schoenflies.
  • Recent pseudo-isotopy and candidate-family papers are approaches or special-case results, not a complete proof or counterexample.
  • No handle diagram, submitted attachment, or external computation was rendered or reproduced.
  • The bounded formalization search does not establish nonexistence of private or unindexed work.

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