ProofAtlas

Transformation groups · geometric topology · p-adic groups

Hilbert–Smith Conjecture

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Must every locally compact group that acts faithfully and continuously on a connected finite-dimensional manifold be a Lie group?

G↷Mfaithfully and continuously⇒Gis a Lie group
Known results and sources
A connected dark-green manifold lies between cool-blue nested p-adic approximations and a smooth golden Lie-group arc, leaving their relationship visibly unresolved.
Hilbert–Smith asks whether faithful continuous symmetry on a finite-dimensional manifold must always come from a Lie group.

Research problem

Exact mathematical statement

Let G be a locally compact topological group acting faithfully and continuously on a connected finite-dimensional topological manifold M. The Hilbert–Smith Conjecture asserts that G is a Lie group.

By the standard reduction used in the source, it is enough to rule out a faithful continuous action

ℤp↷M\mathbb Z_p\curvearrowright M

of the additive group of p-adic integers on such a manifold. The unrestricted topological case remains open; the source develops an internal route toward this p-adic exclusion and does not claim a proof.

Problem infographic

Problem at a glance

Problem-first diagram of a faithful locally compact group action on a connected manifold, the reduction to nested p-adic subgroups, the open faithful-action question, the Lie-group endpoint, and known cases in dimensions at most three and for Lipschitz actions.
The standard reduction asks whether the p-adic integers can act faithfully on a connected finite-dimensional manifold; the unrestricted topological case remains open.

Current mathematical picture

Where work on Hilbert–Smith Conjecture stands

Open conjecture

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

Useful failureExhausted routes and open replacements

Tangent-microbundle comparison, a bare dimension estimate, and an undecorated wall closure do not suffice to contradict a faithful p-adic action. The finite branch still needs a compact pointed wall core that preserves nonzero localized index data through component mergers; the infinite branch separately needs a manifold-neighborhood obstruction to the compatible unbounded-depth torsor/orientation tower. Neither closing obstruction has been proved.

Route status · Narrowed route
Main reductionConstant-section endpoint

A nonzero constant section supplies the equivariant endpoint comparison at every active depth, so the earlier tangent-microbundle displacement comparison is no longer load-bearing.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeDefine the canonical localized zero-cycle so that its augmentation recovers degree and its clopen coordinates retain the local index labels.Task status · Work already reported in progress

Work mapped so far

Hilbert–Smith Conjecture in numbers

2.8kretained lines of mathematical investigation2,783 in the current working snapshot
Argument development
2,274 · 82%
Explored or eliminated routes
48 · 2%
Computational analysis
4 · 0%
Open obligations
237 · 9%
Definitions and setup
220 · 8%
8selected mapped statements1routes investigated3open questions2contribution-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 7, 2026
Source recordedAug 7, 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 Hilbert–Smith ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Faithful continuous manifold actions should force locally compact groups to be Lie groups. — Depends on missing premiseFaithful continuous manifoldactions should force locallycompact…Constant-section endpoint — Depends on missing premiseConstant-section endpointCurrent reduction — Depends on missing premiseCurrent reductionFinite or infinite branch — Depends on missing premiseFinite or infinite branchUncentered Haar map — Depends on missing premiseUncentered Haar mapClosing target — Depends on missing premiseClosing targetEssential fiber continua — Depends on missing premiseEssential fiber continuaPointed decorated core — Depends on missing premisePointed decorated coreExhausted source-reported routes — stoppedExhausted source-reportedroutesDefine the canonical localized zero-cycle so that its augmentation recovers degree and its clopen coordinates retain the local index labels. — Work reported in progressDefine the canonicallocalized zero-cycle so thatits…Construct the pointed index-decorated essential wall core and verify that it remains compact, target-surjective, and degree-essential. — OpenConstruct the pointedindex-decorated essentialwall…Convert the decorated finite-wall structure or the infinite-depth core into an actual manifold contradiction without reviving the exhausted comparison, dimension-only, or undecorated-closure routes. — OpenConvert the decoratedfinite-wall structure or theinfinite-depth…
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 routeExhausted routes and open replacements

Tangent-microbundle comparison, a bare dimension estimate, and an undecorated wall closure do not suffice to contradict a faithful p-adic action. The finite branch still needs a compact pointed wall core that preserves nonzero localized index data through component mergers; the infinite branch separately needs a manifold-neighborhood obstruction to the compatible unbounded-depth torsor/orientation tower. Neither closing obstruction has been proved.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Construct the pointed index-decorated essential wall core and verify that it remains compact, target-surjective, and degree-essential.Suggested move: Build the pointed compactification from indexed branches, represented components, and wall points, and prove the required hyperspace continuity without losing degree under mergers.
Ready to work on
02
Convert the decorated finite-wall structure or the infinite-depth core into an actual manifold contradiction without reviving the exhausted comparison, dimension-only, or undecorated-closure routes.Suggested move: Use one stabilized primitive Fourier mode to treat the nodal and phase regimes, while developing a separate neighborhood obstruction for the infinite torsor/orientation tower.
Ready to work on
03
Define the canonical localized zero-cycle so that its augmentation recovers degree and its clopen coordinates retain the local index labels.Suggested move: Formalize the localized mod-p zero-cycle and its finite clopen-package coordinates, with exact hypotheses and no claim about the full conjecture.
Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026
Current statusOpen conjecture

The general conjecture for arbitrary faithful continuous actions on connected finite-dimensional topological manifolds remains open. It is proved in dimensions at most three and for important regularity-restricted classes, including Lipschitz and quasiconformal actions.

[2][1][3]
External progress

What the literature has established

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

  1. Peer reviewedPardon's survey records the conjecture in dimension two, originally due to Montgomery and Zippin, and explains the totally disconnected-group viewpoint.[2]
  2. Peer reviewedPardon proved that every locally compact group acting faithfully on a connected three-manifold is a Lie group.[1]
  3. Peer reviewedMartin proved the conjecture for quasiconformal actions, a regularity-restricted setting.[4]
  4. Peer reviewedRepovš and Ščepin proved the conjecture for actions by Lipschitz maps.[3]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusHilbert–Smith Conjecture
Dependency or reductionNo faithful Z_p action on a connected manifold

Known structural reductions make exclusion of a faithful action by the p-adic integers Z_p the central test case.

[1]
Solved special caseHilbert–Smith in dimensions at most three

The conjecture is established for connected manifolds of dimensions at most three.

[1][2]
Solved special caseRegularity-restricted group actions

Additional regularity on the action, including Lipschitz or quasiconformal hypotheses, yields proved cases without resolving arbitrary continuous actions.

[3][4]

Formalization opportunities

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

  • Formalization targetA precise Lean statement of faithful continuous actions of locally compact groups on connected finite-dimensional topological manifolds.
  • Formalization targetReusable formal interfaces for p-adic integers as topological groups, stabilizer towers, compact group actions, Brouwer degree, Čech cohomology, and the reduction from non-Lie locally compact groups to p-adic actions.
  • Formalization targetA separately reviewed alignment between any formal statement and the exact informal Hilbert–Smith formulation; packet lemmas should be formalized as scoped intermediate targets, not presented as a proof of the conjecture.

Detailed research inventory

Claims, milestones, and routes in the current map

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

5 standing statements3 proposed statements3 open questions1 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction4 of 84
  • lemma2 of 82
  • negative result1 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 statementFaithful continuous manifold actions should force locally compact groups to be Lie groups.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementUncentered Haar mapintermediate
  • supersededDepth-zero branch removedintermediate
  • retained route statementConstant-section endpointintermediate
  • retained route statementEssential fiber continuaintermediate
  • retained route statementFinite or infinite branchintermediate
  • retained route statementPointed decorated coreintermediate
  • Recorded relationshipThe source material 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 work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe current work 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 failureExhausted source-reported routesreported failure
  • Research targetDefine the canonical localized zero-cycle so that its augmentation recovers degree and its clopen coordinates retain the local index labels.in progress reported
  • Research targetConstruct the pointed index-decorated essential wall core and verify that it remains compact, target-surjective, and degree-essential.open
  • Research targetConvert the decorated finite-wall structure or the infinite-depth core into an actual manifold contradiction without reviving the exhausted comparison, dimension-only, or undecorated-closure routes.open
  • Narrowed routeExhausted routes and open replacementsTangent-microbundle comparison, a bare dimension estimate, and an undecorated wall closure do not suffice to contradict a faithful p-adic action. The finite branch still needs a compact pointed wall core that preserves nonzero localized index data through component mergers; the infinite branch separately needs a manifold-neighborhood obstruction to the compatible unbounded-depth torsor/orientation tower. Neither closing obstruction has been proved.
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 bridgeDefine the canonical localized zero-cycle so that its augmentation recovers degree and its clopen coordinates retain the local index labels.

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

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 pointConstruct the pointed index-decorated essential wall core and verify that it remains compact, target-surjective, and degree-essential.

Hilbert–Smith 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 locally compact group that acts faithfully and continuously on a connected finite-dimensional manifold be a Lie group?

  • 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
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Sources and references4 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    The Hilbert–Smith conjecture for three-manifoldspeer reviewed result · John Pardon · Journal of the American Mathematical Society · 2013 · ARXIV 1112.2324 · DOI 10.1090/S0894-0347-2013-00766-3 · accessed Aug 7, 2026
  2. 2
    Totally disconnected groups (not) acting on two-manifoldssurvey or monograph · John Pardon · Proceedings of Symposia in Pure Mathematics · 2019 · ARXIV 1811.08748 · DOI 10.1090/pspum/102/13 · accessed Aug 7, 2026
  3. 3
    A proof of the Hilbert–Smith conjecture for actions by Lipschitz mapspeer reviewed result · Dušan Repovš, Evgenij Ščepin · Mathematische Annalen · 1997 · DOI 10.1007/s002080050080 · accessed Aug 7, 2026
  4. 4
    The Hilbert–Smith conjecture for quasiconformal actionspeer reviewed result · Gaven J. Martin · Electronic Research Announcements of the American Mathematical Society · 1999 · DOI 10.1090/S1079-6762-99-00062-1 · MR 1694197 · accessed Aug 7, 2026

Important qualifications

  • The current work explicitly reports no literature-status search; the external status and known-case statements below therefore come from the separately listed sources.
  • No public proof-assistant formalization of the exact general Hilbert–Smith statement was verified in this scoped search. The empty formalization list means none was verified, not that none exists.
  • The current work's internal reductions were not independently proved or executed during metadata collection and remain source-reported working mathematics.

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