BingBorsuk.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

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
[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

Imports (1)

FormalConjecturesUtil

theorem BingBorsuk.bing_borsuk_conjecture The Bing-Borsuk Conjecture: every $n$-dimensional homogeneous absolute neighborhood retract is a topological $n$-manifold sorry line 51 source
Source documentation
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.

depends on sorry what this means
Statement uses: Nat, TopologicalSpace, TopologicalSpace.MetrizableSpace, HomogeneousSpace, IsAbsoluteNeighborhoodRetract, HasLebesgueCoveringDimensionEq, Nonempty, ChartedSpace, Fin, Real
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
Mentions: HomogeneousSpace, IsAbsoluteNeighborhoodRetract, HasLebesgueCoveringDimensionEq