A four-dimensional counterexample to Borel rigidity at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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A four-dimensional counterexample to Borel rigidity

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Nonhomeomorphic closed aspherical four-manifolds. Constructs closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. This disproves even the homeomorphism-existence formulation of the Borel conjecture in dimension four.

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LEVEL 1 Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type

released 2026-10-04 | 5 theorems · 27 lemmas · 50 proofs · 31,666 words | PLAY LEVEL 1 » (pdf)

We construct closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, disproving the homeomorphism-existence formulation of the Borel conjecture in dimension four. Their common fundamental group is word-hyperbolic. One of the manifolds also has a self-homotopy equivalence not homotopic to a homeomorphism.

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