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Algebraic topology · torus actions · rational homotopy theory · free differential modules

Halperin–Carlsson Toral Rank Conjecture

Collaboration beta

An almost-free r-dimensional torus action should force at least 2^r total rational cohomology classes. The packet narrows one rank-four algebraic configuration but does not prove rank four or the conjecture in arbitrary rank.

∑idimℚHi(X;ℚ)≥2r
Known results and sources
Landscape illustration of an almost-free rank-r torus action on a connected finite CW complex, with rational cohomology tiles doubling by rank and an OPEN marker beside the 2^r threshold.
For a connected finite CW complex, the toral-rank conjecture asks whether an almost-free rank-r torus action forces at least 2^r total rational cohomology; the threshold remains open in general.

Research problem

Exact mathematical statement

Let a torus of rank r act almost freely on a connected finite CW complex X. The Halperin–Carlsson toral-rank conjecture predicts

∑idimℚHi(X;ℚ)≥2r.\sum_i \dim_{\mathbb Q} H^i(X;\mathbb Q) \ge 2^r.

The governing source concentrates on a source-reported four-variable reduction. Its numerical enumeration and its geometric eliminations have different evidence postures and must not be conflated.

Problem infographic

Problem at a glance

Landscape problem-first explainer showing an almost-free torus action on a connected finite CW complex, the 2^r rational Betti-number question, the minimal free differential module translation, and a separately marked rank-four audit frontier.
The exact conjecture concerns connected finite CW complexes and is universal in r; the packet's audit-pending rank-four configuration cannot stand in for an arbitrary-rank proof.

Current mathematical picture

Where work on Halperin–Carlsson Toral Rank Conjecture stands

Open conjecture

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

Useful failureTreating the historical s=27 branch count as reproducible

The literal documented filters give nineteen preliminary open branches across seventeen signatures, while the current work says neither count is publication-certified. Use all ninety F8 rows and formally specify every stable and unstable geometric filter before assigning a branch count.

Route status · Narrowed route
Main reductionCurrent reduction

The source uses the minimal Hirsch–Brown free differential module and projective sheafification to isolate a four-variable central configuration. A uniform incompatibility theorem for a non-split globally generated rank-two core with complementary split presentations would eliminate that configuration.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeAudit the geometry-dependent eliminations through s=26.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

Halperin–Carlsson Toral Rank Conjecture in numbers

2.6kretained lines of mathematical investigation2,645 in the current working snapshot
Argument development
2,284 · 86%
Explored or eliminated routes
96 · 4%
Computational analysis
68 · 3%
Open obligations
78 · 3%
Definitions and setup
119 · 4%
8selected mapped statements2routes 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 byStephen Halperin for almost-free torus actions, Gunnar Carlsson for finite elementary-abelian actionsProposed1985Status 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

14 selected steps

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

14 selected steps

Scroll horizontally to explore the route

Working route overview for Halperin–Carlsson Toral Rank ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Must almost-free torus symmetry force exponentially large cohomology? — Depends on missing premiseMust almost-free torussymmetry force exponentiallylarge…Central four-variable configuration — Depends on missing premiseCentral four-variableconfigurationCurrent reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetExact arithmetic frontier — Depends on missing premiseExact arithmetic frontierExact toral-rank target — Depends on missing premiseExact toral-rank targetHistorical branch count unresolved — Depends on missing premiseHistorical branch countunresolvedNon-ACM cohomology defect — Depends on missing premiseNon-ACM cohomology defectTreating the historical s=27 branch count as reproducible — stoppedTreating the historical s=27branch count as reproduciblePure arithmetic enumeration as a geometric proof — stoppedPure arithmetic enumerationas a geometric proofAudit the geometry-dependent eliminations through s=26. — OpenAudit the geometry-dependenteliminations through s=26.Formally specify and solve the full s=27 frontier. — OpenFormally specify and solvethe full s=27 frontier.Provide a bridge from the central rank-four case to arbitrary rank. — OpenProvide a bridge from thecentral rank-four case toarbitrary…Uniform rank-two incompatibility — OpenUniform rank-twoincompatibility
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

2 recorded
Narrowed routeTreating the historical s=27 branch count as reproducible

The literal documented filters give nineteen preliminary open branches across seventeen signatures, while the current work says neither count is publication-certified. Use all ninety F8 rows and formally specify every stable and unstable geometric filter before assigning a branch count.

Route status · Narrowed route
Narrowed routePure arithmetic enumeration as a geometric proof

The current work explicitly separates exact arithmetic reproduction from classification, curve-adjunction, and equality-to-splitting steps. A publication-quality audit or a uniform rank-two incompatibility theorem can close the gap.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Audit the geometry-dependent eliminations through s=26.Suggested move: Reconstruct each stable and unstable bundle/curve argument from the complete F8 rows, with exact hypotheses for saturation, Bezout, adjunction, and splitting.
Ready to work on
02
Formally specify and solve the full s=27 frontier.Suggested move: Implement a declarative F9–F10 geometry specification and work all ninety Riemann–Roch-admissible signatures rather than selecting a historical branch count.
Ready to work on
03
Uniform rank-two incompatibility

A uniform no-go theorem for the non-split globally generated rank-two core is the current work's primary mathematical target.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Provide a bridge from the central rank-four case to arbitrary rank.Suggested move: Prove a product-amplifiable asymptotic bound, sharp cyclic theorem, or unit-anchored syzygy-cost theorem that retains the full Hirsch–Brown structure.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The rational almost-free torus-action conjecture remains highly open in general. Structured cases, including MOD-formal/formal-core actions and some low-rank formal-quotient cases, are proved and should not be generalized beyond their hypotheses.

[3][5]
External progress

What the literature has established

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

  1. Peer reviewedAmann and Zoller proved the conjecture for MOD-formal actions and actions of formal core, encompassing formal orbit spaces, while describing the general conjecture as highly open.[5]
  2. PreprintAmann surveyed cohomological consequences and improved general bounds, while Ustinovsky proved the conjecture for formal quotients with torus rank at most five.[3][4]
  3. Historical sourceHalperin's printed formulation asks for the exponential rational Betti-number lower bound under almost-free torus actions.[2]
  4. Historical sourceCarlsson developed the finite elementary-abelian action formulation that forms the finite-coefficient side of the Halperin–Carlsson family.[1]
5 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHalperin–Carlsson Toral Rank Conjecture
Solved special casealmost-free torus actions with MOD-formality or formal core

The 2023 theorem proves the exponential bound for these structured actions, not for all almost-free torus actions.

[5]
Solved special caseformal quotient with torus rank at most five

This is a bounded-rank structured case and does not settle arbitrary spaces or ranks.

[4]
Related problemfinite elementary-abelian Halperin–Carlsson conjecture

The finite-group version uses mod-p cohomology and free actions; it is related but not identical to the rational almost-free torus statement staged here.

[1][3]

Formalization opportunities

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

  • Formalization targetA formal target must fix the topological category, finite-CW or compactness hypotheses, almost-free action definition, rational coefficients, and total cohomology finiteness.
  • Formalization targetThe minimal Hirsch–Brown model reduction and unit-anchor hypotheses require exact formal alignment before an algebraic rank theorem can be advertised as the topological conjecture.
  • Formalization targetNo checked proof-assistant proof of the universal statement was located in the scoped review; this is not evidence that none exists.

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.

3 standing statements5 proposed statements4 open questions2 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction2 of 82
  • lemma3 of 83
  • computational claim1 of 81
  • negative result1 of 81
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included
  • retained route statementMust almost-free torus symmetry force exponentially large cohomology?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact toral-rank targetintermediate
  • retained route statementCentral four-variable configurationintermediate
  • retained route statementNon-ACM cohomology defectintermediate
  • retained route statementExact arithmetic frontierintermediate
  • retained route statementHistorical branch count unresolvedintermediate
  • 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 failureTreating the historical s=27 branch count as reproduciblereported failure
  • Useful failurePure arithmetic enumeration as a geometric proofreported failure
  • Research targetAudit the geometry-dependent eliminations through s=26.open
  • Research targetFormally specify and solve the full s=27 frontier.open
  • Research targetProvide a bridge from the central rank-four case to arbitrary rank.open
  • Research targetUniform rank-two incompatibilityopen
  • ComputationA dependency-free source-reported enumerator applies ten staged moment, integrality, Chern, support, rank-two, Schur, Riemann–Roch, stability, and saturation filters.The current work reports exact F7/F8 counts through s=27, including 333 and 90 respectively at s=27; ProofAtlas did not execute the attached enumerator under this intake task. · reported unreproduced
  • Narrowed routeTreating the historical s=27 branch count as reproducibleThe literal documented filters give nineteen preliminary open branches across seventeen signatures, while the current work says neither count is publication-certified. Use all ninety F8 rows and formally specify every stable and unstable geometric filter before assigning a branch count.
  • Narrowed routePure arithmetic enumeration as a geometric proofThe current work explicitly separates exact arithmetic reproduction from classification, curve-adjunction, and equality-to-splitting steps. A publication-quality audit or a uniform rank-two incompatibility theorem can close the gap.
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 geometry-dependent eliminations through s=26.

2 approaches have 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 pointAudit the geometry-dependent eliminations through s=26.

Halperin–Carlsson Toral Rank Conjecture · ready to start

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Research contextPrepared context for any AI agent

An almost-free r-dimensional torus action should force at least 2^r total rational cohomology classes. The current work narrows one rank-four algebraic configuration but does not prove rank four or the conjecture in arbitrary rank.

  • 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.

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Sources and references5 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
    On the Homology of Finite Free (Z/2)^n-Complexesoriginal source · Gunnar Carlsson · Inventiones Mathematicae 74, 139–148 · 1983 · accessed Aug 14, 2026
  2. 2
    Rational homotopy and torus actionsoriginal source · Stephen Halperin · Aspects of Topology, London Mathematical Society Lecture Note Series 93, 293–306 · 1985 · accessed Aug 14, 2026
  3. 3
    Cohomological consequences of (almost) free torus actionspreprint · Manuel Amann · arXiv · 2012 · ARXIV 1204.6276 · accessed Aug 14, 2026
  4. 4
    On almost free torus actions and Horrocks conjecturepreprint · Yury Ustinovsky · arXiv · 2012 · ARXIV 1203.3685 · accessed Aug 14, 2026
  5. 5
    The Toral Rank Conjecture and variants of equivariant formalitypeer reviewed result · Manuel Amann, Leopold Zoller · Journal de Mathématiques Pures et Appliquées 173, 43–95 · 2023 · ARXIV 1910.04746 · DOI 10.1016/j.matpur.2023.02.002 · accessed Aug 14, 2026

Important qualifications

  • This metadata is external administrative context and grants no proof, review, credit, publication, or deployment authority.
  • The rational torus-action statement and finite elementary-abelian coefficient variants are related but not identical; this record stages the rational statement.
  • Special-case theorems retain their formality, quotient, rank, and coefficient hypotheses.
  • Scoped searches found no checked formal proof of the universal statement; negative search results do not establish nonexistence.

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