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Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type
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Abstract
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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Possibly relevant literature (found by the agent, not checked)
- D. Kasprowski and M. Land, Topological 4-manifolds with 4-dimensional fundamental group (arXiv:2007.03399; Glasgow Math. J. 64 (2022), 454–461). Classifies closed topological 4-manifolds up to s-cobordism when the fundamental group satisfies Farrell–Jones and has a 4-dimensional Poincaré duality classifying space, with homeomorphism for good groups; this bears on the introduction's account of rigidity up to s-cobordism (Khan) versus homeomorphism.
- James F. Davis and J. A. Hillman, Aspherical manifolds with boundary (arXiv:2501.12509, 21 January 2025; revised 29 June 2026). Its homological criteria for prescribed boundaries and fundamental-group maps, passing through Poincaré pairs and surgery, provide background for the manuscript's claimed marked-filling and manifold-realization obstructions.
- James F. Davis and J. A. Hillman, Aspherical 4-manifolds with elementary amenable fundamental group (arXiv:2501.12512, 21 January 2025; revised 29 June 2026). Its relative topological rigidity theorem for compact aspherical four-manifolds with elementary amenable fundamental group provides a comparison with the manuscript's claimed failure of rigidity for closed four-manifolds with word-hyperbolic fundamental group.
Digests and expositions
- Latent Seminar, Borel rigidity in dimension four: cyclic covers and realized symmetries (7 October 2026). This exposition separates the cyclic-cover self-map obstruction from the realized-symmetry argument for a nonhomeomorphic pair, identifies the companion inputs, and explicitly disclaims independent verification of the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- whataifound: Registry entry
- Emergent Mind: family 320
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026,
author = {{OpenAI}},
title = {{Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026/paper.pdf}{OAI:Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026}},
year = {2026}
}Family 320: 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.
This is the only paper in this family.