The Poincaré Conjecture · official description
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2 cited passages ↓1 problem ↓Record clay-description-observed-2026-10-12 ↓
The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
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.
Passage ID: poincare-conjecture:entry:dimension-scope:v1
Source item: §§1–3; dimension-three comparison on page 5
Source record: milnor-poincare-description@clay-description-observed-2026-10-12
Smale Theorem: a differentiable homotopy sphere of dimension n ≥ 5 is homeomorphic to the n-sphere, and diffeomorphic to a manifold obtained by gluing two closed n-balls along their boundaries.
Passage ID: poincare-conjecture:entry:milnor-smale:v1
Source item: Smale Theorem, page 4
Source record: milnor-poincare-description@clay-description-observed-2026-10-12
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v13
Scope
The smooth three-dimensional theorem
Source assessment
Proved
Established result reported in the official description
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Connection
The four-dimensional smooth question concerns a different dimension and equivalence notion.
Question
Poincaré conjecture
Further reading
Read the setting and notation before comparing the results.
Recorded authorship
John Milnor
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OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
The Poincaré Conjecture · official description
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