Theorem · version 1
If Mⁿ is a differentiable homotopy sphere of dimension n ≥ 5, then Mⁿ is homeomorphic to Sⁿ. In fact, Mⁿ is diffeomorphic to a manifold obtained by gluing together the boundaries of two closed n-balls under a suitable diffeomorphism.
The conclusion is a homeomorphism to Sⁿ. The smooth conclusion is a diffeomorphism to two balls glued along their boundaries, not necessarily to Sⁿ.
Proved
Only the exact statement and passage cited here; not an independent OpenMath assessment.
The Poincaré Conjecture · official description · §3 Higher Dimensions, p. 4, Smale Theorem →
Observed Oct 21, 2026
In this problem · publication v13 · Selected context
Different statement · related variant · Proved
A different dimension: dimension five and above were settled before dimensions four and three.
Relationship not independently reviewed. This concerns its use in this problem.
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.
paraphrase · Follow the original source for authoritative statements and reuse terms.
John Milnor · Published
Source-backed selection; publication and proof assessment are separate.
Source revision: clay-description-observed-2026-10-12. Observed Oct 12, 2026.
AI use is not reported in the selected source.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026 · Source passage →