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Math-2201 · mathematics · Topology

Borel Conjecture

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $M$ and $N$ be closed aspherical manifolds, meaning that their universal covers are contractible. If $\pi_1(M)$ and $\pi_1(N)$ are isomorphic, must $M$ and $N$ be homeomorphic? This is the topological rigidity question; no diffeomorphism is demanded.

Formulation references: [S1], [S2].

Short English statement

Let $M$ and $N$ be closed aspherical manifolds, meaning that their universal covers are contractible. If $\pi_1(M)$ and $\pi_1(N)$ are isomorphic, must $M$ and $N$ be homeomorphic? This is the topological rigidity question; no diffeomorphism is demanded.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2201-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2201

Manuscript 1 (family 320, 2026-10-04): The homotopy-equivalent closed aspherical topological four-manifolds have isomorphic fundamental groups yet no homeomorphism at all, refuting both homeomorphism-existence and specific-homotopy-class Borel assertions.

Borel Conjecture

OpenAI's claim

Two closed aspherical topological four-manifolds are claimed homotopy equivalent but not homeomorphic, sharing a word-hyperbolic fundamental group; one also admits a self-equivalence outside every homeomorphism homotopy class. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims closed connected aspherical topological four-manifolds $M,N$ with a common word-hyperbolic fundamental group such that \[ M\simeq N\quad\text{but}\quad M\not\cong N\text{ by any homeomorphism}. \] Aspherical means that the universal cover is contractible; $\simeq$ denotes homotopy equivalence. A second theorem claims a topological four-manifold that is aspherical, closed and connected $M$ and a self homotopy equivalence $f:M\to M$ for which no homeomorphism is homotopic to $f$. These are four-dimensional topological counterexamples; the first excludes homeomorphism existence in any homotopy class.

Outline of the argument

Manuscript 1 (family 320, 2026-10-04): Reflected marked chambers retain an odd-cover filling obstruction, and carefully chosen binary label relations eliminate unwanted chamber symmetries. Hyperbolicity and Poincare duality make the outer automorphism group finite, with a controlled normalizer of the realized chamber action. A hypothetical homeomorphism would transport another finite action into that realized subgroup and force an excluded conjugacy.

Formal verification

Family 320 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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