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Riemannian geometry · nonnegative curvature · rational homotopy · Morse theory

Bott Conjecture on Rational Ellipticity

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Must every closed simply connected manifold admitting nonnegative sectional curvature have only finitely many nonzero rational homotopy groups?

secM≥0⇒∑k≥2dimQ(πk(M)⊗Q)<∞
Known results and sources
A smooth nonnegatively curved manifold hovers above a finite constellation of rational homotopy spheres while a potential infinite free-Lie cascade remains behind an open boundary.
Bott predicts that nonnegative sectional curvature forces rational ellipticity rather than infinite rational-homotopy growth.

Research problem

Exact mathematical statement

Let MnM^n be a closed, simply connected smooth manifold that admits a Riemannian metric with secM≥0\sec_M\ge0. The Bott rational-ellipticity conjecture asserts

∑k≥2dimQ(πk(M)⊗Q)<∞.\sum_{k\ge2}\dim_{\mathbf Q}(\pi_k(M)\otimes\mathbf Q)<\infty.

Equivalently, the primitive part of H*(ΩM;Q)H_*(\Omega M;\mathbf Q) is finite-dimensional. The governing source also records subexponential cumulative primitive growth pM(N)=eo(N)p_M(N)=e^{o(N)} as a sufficient completion criterion, but it does not prove that bound.

Problem infographic

Problem at a glance

A problem-first diagram starts with a closed simply connected manifold of nonnegative sectional curvature, asks whether its total rational homotopy is finite, and shows the primitive loop-homology equivalence and subexponential primitive-growth criterion without any free-Lie growth claim.
The conjecture connects curvature to finite total rational homotopy; primitive loop homology gives an exact reformulation, while subexponential cumulative primitive growth is a sufficient open target.

Current mathematical picture

Where work on Bott Conjecture on Rational Ellipticity stands

Open conjecture

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

Useful failureUsing local product splitting of a Morse handle to declare its attaching class decomposable

The source notes that the full boundary collapse is a sphere with primitive top relative class and exhibits the two-packet Samelson commutator. An audited robust-index branch or fallback index-density branch may supply packet-rich handles, but only a curvature-specific identity or subexponential center-defect bound can close the topological step.

Route status · Narrowed route
Main reductionConstant-drop criterion

A uniform e^{o(k)} bound on the primitive quotient between length R and R−sigma yields Bott by telescoping O(k) windows.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeAudit the provisional phase and Fredholm determinant statements before using robust spectral purification.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 Conjecture on Rational Ellipticity in numbers

980retained lines of mathematical investigation980 in the current working snapshot
Argument development
878 · 90%
Explored or eliminated routes
18 · 2%
Computational analysis
5 · 1%
Open obligations
18 · 2%
Definitions and setup
61 · 6%
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 BottStatus 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 Bott Conjecture on Rational EllipticityA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Nonnegative sectional curvature forbids rationally hyperbolic topology. — Depends on missing premiseNonnegative sectionalcurvature forbids rationallyhyperbolic…Constant-drop criterion — Depends on missing premiseConstant-drop criterionCurrent reduction — Depends on missing premiseCurrent reductionPrimitive loop-homology form — Depends on missing premisePrimitive loop-homology formSubexponential completion criterion — Depends on missing premiseSubexponential completioncriterionClosing target — Depends on missing premiseClosing targetThom-cell obstruction — Depends on missing premiseThom-cell obstructionTwo-packet Samelson defect — Depends on missing premiseTwo-packet Samelson defectUsing local product splitting of a Morse handle to declare its attaching class decomposable — stoppedUsing local productsplitting of a Morse handleto…Audit the provisional phase and Fredholm determinant statements before using robust spectral purification. — OpenAudit the provisional phaseand Fredholm determinantstatements…Build an exact two-packet relative Morse or Conley pair and identify its center Thom and Samelson class in the fixed-length quotient. — OpenBuild an exact two-packetrelative Morse or Conleypair…Prove a uniform subexponential bound for all surviving packet-center defects and complete the constant-window telescoping argument. — OpenProve a uniformsubexponential bound for allsurviving…
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 routeUsing local product splitting of a Morse handle to declare its attaching class decomposable

The source notes that the full boundary collapse is a sphere with primitive top relative class and exhibits the two-packet Samelson commutator. An audited robust-index branch or fallback index-density branch may supply packet-rich handles, but only a curvature-specific identity or subexponential center-defect bound can close the topological step.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Audit the provisional phase and Fredholm determinant statements before using robust spectral purification.Suggested move: Check monotonicity, trace class, normalization, multiplicities, constant-curvature products, and metric scaling with fixed endpoint conventions.
Ready to work on
02
Build an exact two-packet relative Morse or Conley pair and identify its center Thom and Samelson class in the fixed-length quotient.Suggested move: Write the radial collar and center sphere as relative chains and test whether curvature forces descent or a controlled relation.
Ready to work on
03
Prove a uniform subexponential bound for all surviving packet-center defects and complete the constant-window telescoping argument.Suggested move: Retain every homotopy inside the shorter length filtration and verify the sphere, free-Lie, threshold, endpoint, and terminal-sublevel tests.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The general Bott implication from nonnegative sectional curvature to rational ellipticity remains open. Peer-reviewed work proves substantial special cases under cohomogeneity, group-action, submetry, or quotient hypotheses, none of which covers every closed simply connected nonnegatively curved manifold.

[1][2]
External progress

What the literature has established

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

  1. Peer reviewedKhalili Samani and Radeschi proved rational ellipticity from a zero-entropy quotient condition for compact simply connected G-manifolds and manifold submetries.[2]
  2. Peer reviewedGrove, Wilking, and Yeager proved the almost-nonnegative-curvature extension under an isometric action of cohomogeneity at most two.[1]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusBott conjecture on rational ellipticity
Solved special casecohomogeneity-two case

The conjectural conclusion holds when the manifold also has an isometric action with orbit codimension at most two.

[1]
Solved special casezero-entropy quotient criterion

A quotient isometric to one arising from a zero-topological-entropy space forces rational ellipticity in the stated G-manifold setting.

[2]

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 must fix every ambient field, regularity, genericity, equivalence, and exceptional-set convention appearing in the external statement.
  • Formalization targetIt must formalize the exact prerequisite geometry, analysis, topology, or arithmetic library needed to express the theorem without replacing it by a weaker proxy.
  • Formalization targetNo problem-level formal statement or proof was identified in the bounded current Mathlib documentation and public Formal Conjectures repository search.

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
  • reduction3 of 83
  • lemma1 of 81
  • equivalence1 of 81
  • negative result1 of 81
  • counterexample1 of 81
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
  • retained route statementNonnegative sectional curvature forbids rationally hyperbolic topology.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementPrimitive loop-homology formintermediate
  • retained route statementSubexponential completion criterionintermediate
  • retained route statementConstant-drop criterionintermediate
  • retained route statementThom-cell obstructionintermediate
  • retained route statementTwo-packet Samelson defectintermediate
  • 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 counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · 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 failureUsing local product splitting of a Morse handle to declare its attaching class decomposablereported failure
  • Research targetAudit the provisional phase and Fredholm determinant statements before using robust spectral purification.open
  • Research targetBuild an exact two-packet relative Morse or Conley pair and identify its center Thom and Samelson class in the fixed-length quotient.open
  • Research targetProve a uniform subexponential bound for all surviving packet-center defects and complete the constant-window telescoping argument.open
  • Research targetGlobal center-defect boundsuperseded
  • Narrowed routeUsing local product splitting of a Morse handle to declare its attaching class decomposableThe source notes that the full boundary collapse is a sphere with primitive top relative class and exhibits the two-packet Samelson commutator. An audited robust-index branch or fallback index-density branch may supply packet-rich handles, but only a curvature-specific identity or subexponential center-defect bound can close the topological step.
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 bridgeAudit the provisional phase and Fredholm determinant statements before using robust spectral purification.

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.

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Prepared starting pointAudit the provisional phase and Fredholm determinant statements before using robust spectral purification.

Bott Conjecture on Rational Ellipticity · ready to start

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Research contextPrepared context for any AI agent

Must every closed simply connected manifold admitting nonnegative sectional curvature have only finitely many nonzero rational homotopy groups?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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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. 1
    Almost non-negative curvature and rational ellipticity in cohomogeneity twopeer reviewed result · Karsten Grove, Burkhard Wilking, Joseph Yeager · Annales de l'Institut Fourier · 2019 · DOI 10.5802/aif.3340 · accessed Aug 14, 2026
  2. 2
    Rational ellipticity of G-manifolds from their quotientspeer reviewed result · Elahe Khalili Samani, Marco Radeschi · Compositio Mathematica · 2025 · DOI 10.1112/S0010437X24007656 · accessed Aug 14, 2026

Important qualifications

  • External status was checked only against the listed primary, peer-reviewed, official-publisher, or author-hosted sources; omission is not a global nonexistence claim.
  • Special cases, reductions, preprints, and source-reported claims are kept at their exact scope and do not inherit proof, review, acceptance, publication, or priority authority.
  • The scoped formalization check found no problem-level statement or proof in current Mathlib documentation or the public Formal Conjectures repository; this is not evidence that no formalization exists elsewhere.
  • No source material, attachment, submitted URL, or packet computation was executed, rendered, fetched, or used as external authority.

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