The Bing-Borsuk Conjecture

The Bing-Borsuk Conjecture: every n-dimensional homogeneous absolute neighborhood retract is a topological n-manifold. A topological space X is an n-dimensional manifold when T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X).

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-bing-borsuk,
  title        = {The Bing-Borsuk Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/bing-borsuk}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

The Bing-Borsuk Conjecture: every $n$ -dimensional homogeneous absolute neighborhood retract is a topological $n$ -manifold. A topological space $X$ is an $n$ -dimensional manifold when T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X). The hypothesis [MetrizableSpace X] implies T2Space X so this does not appear in the conclusion.

The Bing-Borsuk conjecture states that every $n$ -dimensional homogeneous absolute neighborhood retract is a topological $n$ -manifold.

The conjecture has been verified in dimensions $1$ and $2$ but remains open in higher dimensions. A notable consequence is that if the $3$ -dimensional case is true, it implies the Poincaré conjecture.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.BingBorsuk.

theorem bing_borsuk_conjecture : ∀ n : ℕ, ∀ (X : Type) [TopologicalSpace X] [MetrizableSpace X] [HomogeneousSpace X] [IsAbsoluteNeighborhoodRetract X],
    HasLebesgueCoveringDimensionEq X n → Nonempty (ChartedSpace (Fin n → ℝ) X)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
[HR2008] Halverson, Denise M., and Dušan Repovš. "The Bing-Borsuk and the Busemann conjectures." Mathematical Communications 13.2 (2008): 163-184. https://arxiv.org/abs/0811.0886

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.