The Borel conjecture for closed aspherical four-manifolds (homeomorphism form)
Theorem 1.1: there are closed connected aspherical topological four-manifolds and , with the same word-hyperbolic fundamental group, that are homotopy equivalent but not homeomorphic. Theorem 1.2: some such has a self-homotopy equivalence not homotopic to any homeomorphism, so the prescribed-class form also fails. The examples come from Davis reflection applied to a Poincaré chamber whose odd cyclic covers have no manifold model. This disproves the Borel conjecture as an all-dimension statement and in dimension four specifically. Not shown: anything in dimensions five or higher, or smooth-category statements; it also shows that Khan's rigidity up to s-cobordism cannot be upgraded to homeomorphism for such groups (curator's reading of the introduction).
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Occurred: Oct 4, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.
Canonical aliases: The Borel conjecture for closed aspherical four-manifolds (homeomorphism form) · Borel conjecture in dimension four
Confidence: Not scored
Registry verification: unreviewed · announcement · candidate
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Unreleased internal OpenAI model
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The Borel conjecture for closed aspherical four-manifolds (homeomorphism form) parent of this event
Borel's 1953 letter to Serre, with Ranicki's commentary evidence for this event
Input (family 305): A PD4 group without an aspherical manifold model evidence for this event
This event attributed to Unreleased internal OpenAI model
Theorem 1.1: there are closed connected aspherical topological four-manifolds and , with the same word-hyperbolic fundamental group, that are homotopy equivalent but not homeomorphic. Theorem 1.2: some such has a self-homotopy equivalence not homotopic to any homeomorphism, so the prescribed-class form also fails. The examples come from Davis reflection applied to a Poincaré chamber whose odd cyclic covers have no manifold model. This disproves the Borel conjecture as an all-dimension statement and in dimension four specifically. Not shown: anything in dimensions five or higher, or smooth-category statements; it also shows that Khan's rigidity up to s-cobordism cannot be upgraded to homeomorphism for such groups (curator's reading of the introduction). parent of this event
VibeMathed record: The Borel conjecture for closed aspherical four-manifolds (homeomorphism form) evidence for this event
The Borel conjecture for closed aspherical four-manifolds (homeomorphism form) evidence for this event
The disc-embedding counterexample this proof depends on evidence for this event
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