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OP-9973Status unassessedGeometry and topology

The Bing-Borsuk Conjecture

The Bing-Borsuk Conjecture

Specialist source · status unassessedAs of 2026-09-10Source & evidence ↓
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Problem statement

bing_borsuk_conjecture

The Bing-Borsuk Conjecture: every nn-dimensional homogeneous absolute neighborhood retract is a topological nn-manifold. A topological space XX is an nn-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.

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Formal statement

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/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

/-!
# The Bing-Borsuk Conjecture

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.

*References:*
 - [Wikipedia](https://en.wikipedia.org/wiki/Bing%E2%80%93Borsuk_conjecture)
 - [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
-/

namespace BingBorsuk

open scoped Manifold
open TopologicalSpace

/--
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.
-/
@[category research open, AMS 54 57]
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) := by
  sorry

end BingBorsuk

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Background

Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.

Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.

Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.

References
  1. [1]Formal Conjectures — The Bing-Borsuk Conjecture link
  2. [2]Original reference cited in the formalization link
  3. [3]Original reference cited in the formalization link

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