Result 320, Topology

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.

Disproof or counterexample

The bigger picture

Why it matters

The unreviewed manuscript reports four-dimensional spaces that agree up to continuous deformation but cannot be matched by a continuous, invertible map. This challenges the idea that a space's deformation properties determine its full topology.

What changes?

The claimed examples are closed, connected topological four-manifolds: compact spaces without boundary that locally look like four-dimensional Euclidean space. They are aspherical, meaning their universal covering spaces can be contracted to a point. They are homotopy equivalent, so maps between them have inverses up to continuous deformation, but they are not homeomorphic. Their common fundamental group, which records loops, is word-hyperbolic. This disproves the homeomorphism-existence version of the Borel conjecture in dimension four.

What does that help mathematicians do?

Word-hyperbolicity imposes a thin-triangle condition on a group's geometry, yet the reported examples show that even this restriction does not restore four-dimensional rigidity. Researchers therefore cannot infer a homeomorphism merely from homotopy equivalence for these manifolds. The manuscript also reports a self-homotopy equivalence of one example that cannot be continuously deformed into a homeomorphism, exposing an additional obstruction at the level of maps.

Are there practical applications?

The immediate value is foundational: the construction identifies a limit on classifying aspherical four-manifolds using homotopy information alone. Any classification up to homeomorphism must distinguish examples that this information treats as identical. The reported counterexamples also constrain which rigidity statements researchers can reasonably seek in dimension four.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type

October 4, 2026 60 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.