Proved
Established result in cited source ↗
From Knots to Four-Manifolds · §3, Conjecture 3.1 and preceding topological classification →
Observed Oct 12, 2026
Theorem · version 1
The topological four-dimensional theorem
Statement
Every closed topological four-manifold homotopy equivalent to the four-sphere is homeomorphic to the four-sphere.
Assumptions
- Freedman’s topological conclusion does not identify a smooth structure or imply the smooth four-dimensional conjecture.
- Dimension
- 4
- Category
- Topological · homeomorphism
Evidence
Source reports
Connections
In this problem · publication v7 · Selected context
Poincaré conjecture →
Different statement · related variant · Proved
Related statements
Homeomorphism and diffeomorphism are different requirements in dimension four.
The proved topological theorem does not settle the smooth conjecture.
Details & sources
The topological four-dimensional theorem →
Proved · version 1
Every closed topological four-manifold homotopy equivalent to the four-sphere is homeomorphic to the four-sphere.
- Freedman’s topological conclusion does not identify a smooth structure or imply the smooth four-dimensional conjecture.
Scope: Freedman’s topological conclusion does not identify a smooth structure or imply the smooth four-dimensional conjecture.
The smooth four-dimensional Poincaré question →
Open · version 1
Every closed smooth four-manifold homotopy equivalent to the four-sphere is diffeomorphic to the standard smooth four-sphere.
- This is a separate conjecture in dimension four; simple connectivity alone is insufficient for its hypothesis.
Scope: This is a separate conjecture in dimension four; simple connectivity alone is insufficient for its hypothesis.
Relationship not independently reviewed.
Connection in Poincaré conjecture →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Poincaré conjecture →
Different statement · related variant · Proved
Related statements
Homeomorphism and diffeomorphism are different requirements in dimension four.
The proved topological theorem does not settle the smooth conjecture.
Details & sources
The topological four-dimensional theorem →
Proved · version 1
Every closed topological four-manifold homotopy equivalent to the four-sphere is homeomorphic to the four-sphere.
- Freedman’s topological conclusion does not identify a smooth structure or imply the smooth four-dimensional conjecture.
Scope: Freedman’s topological conclusion does not identify a smooth structure or imply the smooth four-dimensional conjecture.
The smooth four-dimensional Poincaré question →
Open · version 1
Every closed smooth four-manifold homotopy equivalent to the four-sphere is diffeomorphic to the standard smooth four-sphere.
- This is a separate conjecture in dimension four; simple connectivity alone is insufficient for its hypothesis.
Scope: This is a separate conjecture in dimension four; simple connectivity alone is insufficient for its hypothesis.
Relationship not independently reviewed.
Connection in Poincaré conjecture →Relationship not independently reviewed. This concerns its use in this problem.
Sources
From Knots to Four-Manifolds →
- §3, Conjecture 3.1 and preceding topological classification → · Original passage ↗
Recorded wording
The survey states the smooth four-dimensional Poincaré conjecture as an open question, distinct from Freedman’s topological four-dimensional result.
paraphrase · Follow the original source for authoritative statements and reuse terms.
Ciprian Manolescu · Published · Publication date 2026
Source-backed selection; publication and proof assessment are separate.
Revision & attribution
Source revision: icm-2026-doi-25M1806703. 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-topological-four
- Statement version
- 1
- SHA-256
- 1a98e902010d24ad88090ae961d3cd6877c03634bb80731a6dc9a21a5d353347