Proved
Established result in cited source ↗
Observed Oct 12, 2026
Theorem · version 1
Smoothing a topological three-manifold
Statement
Every closed topological three-manifold admits a smooth structure compatible with its topology.
Assumptions
- This existence statement supplies the smooth input for the topological theorem; it alone does not classify the manifold.
Evidence
Source reports
Connections
In this problem · publication v8 · Selected context
Poincaré conjecture →
Supporting result · Proved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Poincaré conjecture →
Supporting result · Proved
Relationship not independently reviewed. This concerns its use in this problem.
Sources
A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture →
- §1, Theorems 1–3; §3, release and verification → · Original passage ↗
Recorded wording
The authors report Lean proofs of the smooth theorem and Moise smoothing, yielding the topological theorem. Their release v0.1.3 report names commit 7a48598d35109aa99d1cc678e2724c213cdf4ff3 and audits the endpoint closures for the three standard Lean axioms, without sorryAx or project axioms.
paraphrase · Follow the original source for authoritative statements and reuse terms.
Ziyang Qin · Yuan Liao · Ayush Khaitan · Bennett Chow · Preprint · Publication date Sep 27, 2026
Source-backed selection; publication and proof assessment are separate.
Revision & attribution
Source revision: arxiv-2609.33842v1. Observed Oct 12, 2026.
Authors report automated assistance; OpenMath has not independently assessed that workflow.
The Poincaré Conjecture · official description →
- §§1–3; dimension-three comparison on page 5 → · Original passage ↗
Recorded wording
Milnor distinguishes topological from smooth classification and explains why the distinctions disappear in dimension three. A homology sphere alone does not satisfy the Poincaré hypothesis.
paraphrase · Follow the original source for authoritative statements and reuse terms.
John Milnor · Published
Source-backed selection; publication and proof assessment are separate.
Revision & attribution
Source revision: clay-description-observed-2026-10-12. Observed Oct 12, 2026.
AI use is not reported in the selected source.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Established result in cited source ↗
Observed Oct 12, 2026 · Source passage →
Record details
- Claim ID
- claim-poincare-moise-smoothing
- Statement version
- 1
- SHA-256
- d5194c1d36b8d45326ac332c285fc2047d4d0b33cd58ef1a2c7e23755cb3b423