Let M and N be closed topological manifolds that are aspherical, i.e. connected with pi_n(M)=pi_n(N)=0 for all n≥ 2 (equivalently, with contractible universal covers). Then every homotopy equivalence f colon M→ N is homotopic to a homeomorphism. In particular, if M and N are closed aspherical topological manifolds with pi_1(M)cong pi_1(N), then M and N are homeomorphic.

Borel conjecture

About 73 years old · traced to 1953

Let $M$ and $N$ be closed topological manifolds that are aspherical, i.e. connected with $\pi_n(M)=\pi_n(N)=0$ for all $n\ge 2$ (equivalently, with contractible universal covers). Then every homotopy equivalence

f\colon M\to N

is homotopic to a homeomorphism.

In particular, if $M$ and $N$ are closed aspherical topological manifolds with $\pi_1(M)\cong\pi_1(N)$ , then $M$ and $N$ are homeomorphic.

Equivalent formulations

In geometric topology, the Borel conjecture asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence should imply a stronger, topological notion.
source: Wikipedia

References

Wikipedia

Wikipedia, Borel conjecture, the article this problem comes from.

topology unsolved-problem

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Claimed by OpenAI. Claims homotopy-equivalent nonhomeomorphic closed connected aspherical topological four-manifolds with a common word-hyperbolic fundamental group, contradicting the dimension-four Borel conjecture.See full solutionHide full solution

Scope relative to this problem: The source claims homotopy-equivalent nonhomeomorphic closed connected aspherical topological 4-manifolds with a common word-hyperbolic fundamental group. This is a dimension 4 negative instance of the target topological Borel assertion, not only distinct smooth structures or a failure of a stable-homeomorphism theorem.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026/paper.pdf

OpenAI-320-01-Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type.pdf715,828 bytesOpen