ProofAtlas

Riemannian geometry, topology, and positive curvature

Hopf positive-curvature conjecture

Collaboration beta

Must every closed even-dimensional manifold whose every tangent two-plane has positive sectional curvature also have positive Euler characteristic?

secg>0⇒χ(M2n)>0
Known results and sources
A closed sphere-like manifold covered by positive-curvature glyphs is surrounded by unresolved topological cycle motifs.
The conjecture asks whether local positivity in every tangent two-plane forces a positive global Euler characteristic.

Research problem

Exact mathematical statement

For every closed even-dimensional Riemannian manifold (M2n,g)(M^{2n},g), if secg(σ)>0\sec_g(\sigma)>0 for every tangent two-plane σ\sigma, then χ(M)>0\chi(M)>0.

secg>0⇒χ(M2n)>0\sec_g>0\;\Longrightarrow\;\chi(M^{2n})>0

Dimension six is the first open dimension. For a closed orientable positively curved six-manifold, Poincaré duality rewrites the target as b3(M)≤2b2(M)b_3(M)\le 2b_2(M). The source's exact-four-weight program is auxiliary and conditional; it does not establish the unrestricted conjecture.

Problem infographic

Problem at a glance

A three-panel plate contrasts positive local curvature glyphs, an unresolved Euler balance, and paired six-dimensional homology motifs.
The first open dimension is six, where the positive-Euler question becomes the exact inequality b3≤2b2.

Current mathematical picture

Where work on Hopf positive-curvature conjecture stands

Open conjecture

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

Useful failureNaive minimal-counterexample symmetry reduction

The source derives that a least-dimensional counterexample must have finite isometry group, so an intrinsic positive-dimensional symmetry argument cannot be assumed. Manufactured ambient symmetry, universal realization, or a direct nonsymmetric argument may still bridge the auxiliary results to the exact conjecture.

Route status · Narrowed route
Main reductionCurrent reduction

The source reduces the first open six-dimensional target to the Betti-number inequality b3≤2b2 and studies conditional ambient fixed-set and Chern-operator mechanisms inside an exact-four-weight framework.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeResolve the lowest-dimensional all-divisible conditional frontier while preserving every lower-Chern correction in the source's first-spoke model.Task status · Ready to work on
Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Hopf positive-curvature conjecture in numbers

2.4kretained lines of mathematical investigation2,414 in the current working snapshot
Argument development
2,086 · 86%
Explored or eliminated routes
67 · 3%
Computational analysis
14 · 1%
Open obligations
84 · 3%
Definitions and setup
163 · 7%
7selected 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.

Proposed byHeinz HopfStatus 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

12 selected steps

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

12 selected steps

Scroll horizontally to explore the route

Working route overview for Hopf positive-curvature 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 positive sectional curvature force positive Euler characteristic? — Depends on missing premiseDoes positive sectionalcurvature force positiveEuler…Current reduction — Depends on missing premiseCurrent reductionExact-four-weight auxiliary program — Depends on missing premiseExact-four-weight auxiliaryprogramSix-dimensional Betti target — Depends on missing premiseSix-dimensional Betti targetClosing target — Depends on missing premiseClosing targetDimensions two and four — Depends on missing premiseDimensions two and fourRank-ten zero detector — Depends on missing premiseRank-ten zero detectorNaive minimal-counterexample symmetry reduction — stoppedNaive minimal-counterexamplesymmetry reductionResolve the lowest-dimensional all-divisible conditional frontier while preserving every lower-Chern correction in the source's first-spoke model. — OpenResolve thelowest-dimensionalall-divisible…Close the explicit rank-ten zero-sector localized top-Euler and paired Chern matrix-pencil problem on the developed edge. — OpenClose the explicit rank-tenzero-sector localizedtop-Euler…Bridge conditional exact-four-weight closure to every closed positively curved six-manifold or replace it with a direct proof of b3≤2b2. — OpenBridge conditionalexact-four-weight closure toevery…Positive-curvature Euler sign — OpenPositive-curvature Eulersign
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 routeNaive minimal-counterexample symmetry reduction

The source derives that a least-dimensional counterexample must have finite isometry group, so an intrinsic positive-dimensional symmetry argument cannot be assumed. Manufactured ambient symmetry, universal realization, or a direct nonsymmetric argument may still bridge the auxiliary results to the exact conjecture.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Resolve the lowest-dimensional all-divisible conditional frontier while preserving every lower-Chern correction in the source's first-spoke model.Suggested move: Analyze the rank-six support cases for the (12,12,12,12) tuple and audit the eight- and ten-dimensional correction terms.
Ready to work on
02
Close the explicit rank-ten zero-sector localized top-Euler and paired Chern matrix-pencil problem on the developed edge.Suggested move: Prove the required rank bound symbolically and independently replay the bounded finite classification before using it.
Ready to work on
03
Positive-curvature Euler sign

Prove positive Euler characteristic for every closed even-dimensional positively curved Riemannian manifold.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Bridge conditional exact-four-weight closure to every closed positively curved six-manifold or replace it with a direct proof of b3≤2b2.Suggested move: Formulate a universal ambient-realization theorem with exact hypotheses, or identify a nonsymmetric curvature-to-cohomology mechanism that avoids E4W.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

Open in general from dimension six onward. Low dimensions and numerous additional-symmetry regimes are known, including specified torus and discrete abelian actions, but no cited theorem removes those assumptions for arbitrary closed even-dimensional positively curved manifolds.

[2][1][3]
External progress

What the literature has established

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

  1. Authoritative summaryGauss–Bonnet settles dimension two, while the standard low-dimensional argument settles dimension four; dimension six is the first open case.[2][1]
  2. Peer reviewedKennard, Khalili Samani, and Searle advanced rigidity for positively curved manifolds with high-rank elementary abelian two-group symmetry and a fixed point; the added symmetry remains essential.[3]
  3. Authoritative summaryAmann's survey records the positive-curvature Euler-sign statement as a prominent Hopf conjecture and surveys its relation to topology and symmetry results.[2]
  4. Peer reviewedSu and Wang proved positive Euler characteristic under specified high-rank isometric elementary abelian p-group actions, for large enough primes and stated dimension and rank conditions.[1]
3 cited sources4 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHopf positive-curvature conjecture
Solved special caseHopf positive-curvature conjecture in dimensions at most four

The Euler-sign conjecture is known in dimensions two and four; dimension six is the first open dimension.

[2][1]
Solved special casepositive curvature with discrete abelian group actions

Specified large-rank discrete abelian symmetry forces positive Euler characteristic under the theorem's prime, rank, and dimension restrictions.

[1]
Related problempositive curvature and discrete abelian symmetry

High-rank elementary abelian two-group symmetry with a fixed point yields strong topological rigidity; it informs the symmetry program without covering arbitrary positively curved manifolds.

[3]
Related problemBott–Grove–Halperin and positive-curvature formality conjectures

The positive Euler-sign conjecture is related to rational ellipticity and formality conjectures, but those implications retain their own hypotheses and unresolved status.

[2]

Formalization opportunities

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

  • Formalization targetA formal statement needs closed smooth even-dimensional Riemannian manifolds, sectional curvature on every tangent two-plane, and Euler characteristic.
  • Formalization targetThe six-dimensional reduction needs orientability, Synge's theorem, rational cohomology, Poincaré duality, and the skew-symmetric middle-dimensional pairing.
  • Formalization targetFormal special cases require faithful and isometric group actions, fixed-point sets, symmetry-rank inequalities, and exact dimension exceptions.
  • Formalization targetNo scoped problem-level formal statement or proof was recorded in this bounded pass.

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 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 statements3 proposed statements4 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma3 of 73
  • equivalence1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementDoes positive sectional curvature force positive Euler characteristic?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementSix-dimensional Betti targetintermediate
  • retained route statementDimensions two and fourintermediate
  • retained route statementExact-four-weight auxiliary programintermediate
  • retained route statementRank-ten zero detectorintermediate
  • 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
  • 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 failureNaive minimal-counterexample symmetry reductionreported failure
  • Research targetResolve the lowest-dimensional all-divisible conditional frontier while preserving every lower-Chern correction in the source's first-spoke model.open
  • Research targetClose the explicit rank-ten zero-sector localized top-Euler and paired Chern matrix-pencil problem on the developed edge.open
  • Research targetBridge conditional exact-four-weight closure to every closed positively curved six-manifold or replace it with a direct proof of b3≤2b2.open
  • Research targetPositive-curvature Euler signopen
  • Research targetUniversal realization or direct proofsuperseded
  • Narrowed routeNaive minimal-counterexample symmetry reductionThe source derives that a least-dimensional counterexample must have finite isometry group, so an intrinsic positive-dimensional symmetry argument cannot be assumed. Manufactured ambient symmetry, universal realization, or a direct nonsymmetric argument may still bridge the auxiliary results to the exact conjecture.
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 bridgeResolve the lowest-dimensional all-divisible conditional frontier while preserving every lower-Chern correction in the source's first-spoke model.

The highlighted low-dimensional all-divisible item is only a conditional E4W task. It is not the final step for the unrestricted Hopf conjecture, which still needs a universal E4W bridge or direct geometric proof.

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

Contribute

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.

Read-only beta · actions unavailable
Prepared starting pointResolve the lowest-dimensional all-divisible conditional frontier while preserving every lower-Chern correction in the source's first-spoke model.

Hopf positive-curvature conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Must every closed even-dimensional manifold whose every tangent two-plane has positive sectional curvature also have positive Euler characteristic?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references3 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
    The Hopf conjecture for positively curved manifolds with discrete abelian group actionspeer reviewed result · Xiaole Su, Yusheng Wang · Differential Geometry and its Applications · 2008 · DOI 10.1016/j.difgeo.2007.11.023 · accessed Aug 14, 2026
  2. 2
    Bounds on Sectional Curvature and Interactions with Topologysurvey or monograph · Manuel Amann · Jahresbericht der Deutschen Mathematiker-Vereinigung · 2021 · DOI 10.1365/s13291-020-00225-x · accessed Aug 14, 2026
  3. 3
    Positive curvature and discrete abelian symmetrypeer reviewed result · Lee Kennard, Elahe Khalili Samani, Catherine Searle · Mathematische Annalen · 2026-07-16 · DOI 10.1007/s00208-026-03509-2 · accessed Aug 14, 2026

Important qualifications

  • The positive-curvature Euler-characteristic conjecture is distinct from Hopf's product conjecture and from the nonpositive-curvature sign conjecture; this record covers only strictly positive sectional curvature implying positive Euler characteristic.
  • The general conjecture is known in dimensions at most four and is widely open from dimension six onward. Results with torus or discrete abelian symmetry remain special cases.
  • The historical proposal year and original Hopf source are left unresolved because this bounded pass relied on a modern peer-reviewed survey rather than an inspected original publication.
  • The 2026 discrete-symmetry paper is contextual progress in the symmetry program and is not recorded as resolving the general Euler-sign conjecture.
  • No intake-packet claim or embedded computation was used as external status authority, and no proof, review, acceptance, credit, publication, or deployment authority is granted.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image