Riemannian geometry · rational homotopy · four-manifolds

Bott Rational Ellipticity in Dimension Four

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Must every closed simply connected four-manifold with nonnegative sectional curvature have second Betti number at most two, equivalently be rationally elliptic?

\sec_{M} \geq 0 \Rightarrow b_{2} (M) \leq 2

Known results and sources

In dimension four, Bott rational ellipticity is equivalent to asking whether nonnegative sectional curvature forces b₂≤2.

Research problem

Exact mathematical statement

Let $M^4$ be a closed simply connected smooth four-manifold admitting a Riemannian metric with nonnegative sectional curvature. Must $M$ be rationally elliptic? In dimension four the source records the exact rational-homotopy equivalence

M^4\text{ is rationally elliptic}\iff b_2(M)\le 2,

so the workspace’s exact geometric target is

\sec_M\ge 0\quad\Longrightarrow\quad b_2(M)\le 2.

This is only the dimension-four statement; no all-dimensional geodesic/Morse variant is silently merged into this workspace.

Problem infographic

Problem at a glance

A four-manifold and its self-dual and anti-self-dual harmonic cohomology feed an open b₂≤2 implication above two hypothetical unresolved configurations. — The source-reported parallel-form branch is only a special case; the definite three-mode and mixed two-plus-one configurations remain the two unresolved branches.

Current mathematical picture

Where work on Bott Rational Ellipticity in Dimension Four stands

Open problem

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

Useful failureSource-reported limitation

The one-negative-channel property of the Weitzenböck operator is insufficient on its own to bound b₂; generic scalar Schrödinger index arguments do not use the nonlinear harmonic-form geometry. Configuration D needs a Gram-field or capacity contradiction for three chiral modes, while Configuration M needs a Fisher-versus-segregation, anti-correlation, or splitting-rigidity theorem; both must close.

Route status · Narrowed route

Main reductionExhaustive D/M split

In the no-parallel branch, a hypothetical b₂≥3 example must fall into either the definite three-mode or mixed two-plus-one configuration.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeClose the definite three-mode configuration without scalar-index shortcuts.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

Bott Rational Ellipticity in Dimension Four in numbers

1.7kretained lines of mathematical investigation1,678 in the current working snapshot

Argument development: 1,424 · 85%
Explored or eliminated routes: 24 · 1%
Computational analysis: 37 · 2%
Open obligations: 55 · 3%
Definitions and setup: 138 · 8%

8selected mapped statements1routes investigated3open questions3contribution-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 byRaoul Bott, Karsten Grove, Stephen HalperinStatus checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Close the definite three-mode configuration without scalar-index shortcuts.

Suggested move: Prove a Gram-field Dirichlet or zero-set-capacity inequality strong enough to contradict the source-reported concentration threshold under projective kernel saturation.

What would count as progress

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

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.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

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

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Close the definite three-mode configuration without scalar-index shortcuts.Suggested move: Prove a Gram-field Dirichlet or zero-set-capacity inequality strong enough to contradict the source-reported concentration threshold under projective kernel saturation.

Ready to work on

Close the mixed two-plus-one configuration with a global energy contradiction.Suggested move: Combine the full-pencil anti-correlation and projective zero saturation, or prove the Fisher-versus-winner-take-all inequality with strictness in the no-parallel branch.

Ready to work on

Complete independent audits and assemble the exhaustive branch conclusion.Suggested move: Verify every standard-external hypothesis, replace the active exact-CAS certificate with a checkable hand derivation where required, resolve the quantitative B-bound, and only then combine the closed branches.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen conjecture

The unrestricted dimension-four implication is conservatively recorded as open. Current primary sources still present the Bott–Grove–Halperin statement as a conjecture and prove only stronger-hypothesis special cases; the reviewed sources contain no general sec≥0 ⇒ b2≤2 theorem.

[1][2]

External progress

What the literature has established

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

2025
Peer reviewedLiu classifies the toric Einstein four-dimensional special case with nonnegative sectional curvature, obtaining locally symmetric metrics in that scope.[2]
2024
Peer reviewedChen proves rational ellipticity under the stronger hypothesis that the compact simply connected real-analytic manifold has an entire Grauert tube.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusBott–Grove–Halperin rational ellipticity conjecture in dimension four

Solved special casemanifolds with entire Grauert tube

Entire Grauert tube is a stronger analytic-geometric assumption under which rational ellipticity is proved; it does not settle arbitrary nonnegative sectional curvature.

[1]

Solved special casetoric Einstein four-manifolds with nonnegative sectional curvature

For closed simply connected four-manifolds with both a T²-invariant Einstein metric and nonnegative sectional curvature, the cited paper proves a sharp classification. The symmetry and Einstein assumptions are essential scope restrictions.

[2]

Formalization opportunities

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

Formalization targetA formal problem statement needs closed simply connected smooth four-manifolds, Riemannian metrics, sectional curvature, rational homotopy groups, Betti numbers, and the dimension-four rational-ellipticity classification.
Formalization targetThe source-reported route would require Hodge theory, Weitzenböck formulas, four-dimensional curvature-operator blocks, characteristic classes, spin and Seiberg–Witten inputs, plus analytic inequalities for harmonic forms.
Formalization targetNo scoped public problem-level formalization was identified.

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.

6 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction2 of 82
lemma4 of 84
equivalence1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 1 route included

retained route statementNonnegative sectional curvature in dimension four should force b₂≤2
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementFour-dimensional reductionintermediate
retained route statementOne negative channelintermediate
retained route statementSame-chirality anisotropyintermediate
retained route statementParallel-form special branchintermediate
retained route statementExhaustive D/M splitintermediate
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
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 failureSource-reported limitationreported failure
Research targetClose the definite three-mode configuration without scalar-index shortcuts.open
Research targetClose the mixed two-plus-one configuration with a global energy contradiction.open
Research targetComplete independent audits and assemble the exhaustive branch conclusion.open
Research targetClose configurations D and Msuperseded
ComputationThe current work reports exact symbolic checks for selected finite-dimensional identities, including a mixed Codazzi-defect certificate.Those exact-CAS checks support only the stated polynomial identities and remain insufficient for the global geometric theorem; a compact hand proof is preferred for publication. · reported unreproduced
Narrowed routeSource-reported limitationThe one-negative-channel property of the Weitzenböck operator is insufficient on its own to bound b₂; generic scalar Schrödinger index arguments do not use the nonlinear harmonic-form geometry. Configuration D needs a Gram-field or capacity contradiction for three chiral modes, while Configuration M needs a Fisher-versus-segregation, anti-correlation, or splitting-rigidity theorem; both must close.

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 bridgeClose the definite three-mode configuration without scalar-index shortcuts.

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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Prepared starting pointClose the definite three-mode configuration without scalar-index shortcuts.

Bott Rational Ellipticity in Dimension Four · ready to start

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Research contextPrepared context for any AI agent

Must every closed simply connected four-manifold with nonnegative sectional curvature have second Betti number at most two, equivalently be rationally elliptic?

Exact question and boundaries
Current routes and known obstacles
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Sources and references2 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
Riemannian manifolds with entire Grauert tube are rationally ellipticpeer reviewed result · Xiaoyang Chen · Geometry & Topology 28(3), 1099–1112 · 2024-05-10 · ARXIV 2101.04368 · DOI 10.2140/gt.2024.28.1099 · accessed Aug 14, 2026
2
Toric Einstein 4-manifolds with non-negative sectional curvaturepeer reviewed result · Tianyue Liu · Annals of Global Analysis and Geometry 67, article 14 · 2025-04-02 · DOI 10.1007/s10455-025-09990-3 · accessed Aug 14, 2026

Important qualifications

The reviewed current sources state the unrestricted Bott–Grove–Halperin conjecture and prove stronger-hypothesis special cases; they do not directly publish the workspace’s exact dimension-four b2 formulation.
Open status for the unrestricted dimension-four implication is conservative: the scoped primary-source review found no resolution, but this is not an exhaustive literature nonexistence claim.
The proposal year is left null because the modern name combines Bott, Grove and Halperin and the reviewed sources do not pin one original formulation date.
The toric Einstein theorem is a narrow special case and must not be generalized to arbitrary nonnegatively curved four-manifolds.
The scoped search found no public problem-level formalization; it was not exhaustive.
The submitted packet and its URLs were excluded from external authority and were not fetched.

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