Proved
Established result reported in the author’s lecture ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Source dated Aug 15, 2014 · Observed Oct 21, 2026
Theorem · version 1
Dimensions where every smooth sphere is standard
Statement
For n = 1, 2, 3, 5, 6, 12 and 61, every smooth manifold homeomorphic to Sⁿ is diffeomorphic to the standard smooth Sⁿ.
Assumptions
- Closed smooth manifolds homeomorphic to the n-sphere.
- Reported by Milnor from Kervaire–Milnor, with Perelman for n = 3.
Scope
The source states that the group of oriented diffeomorphism classes of manifolds homeomorphic to Sⁿ is trivial for these n. Dimension four is excluded: there the question is open.
Evidence
Source reports
Connections
In this problem · publication v13 · Selected context
Poincaré conjecture →
Different statement · related variant · Proved
A different question: smooth spheres in other dimensions, where diffeomorphism and homeomorphism can differ.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Topology through Four Centuries: Low Dimensional Manifolds →
- Slide 33, “To Conclude: An Open Problem” → · Original passage ↗
Recorded wording
For each n, the oriented diffeomorphism classes of manifolds homeomorphic to Sⁿ form a semigroup under connected sum. For n ≠ 4 it is a finite abelian group (Kervaire–Milnor, with Perelman); it is trivial for n = 1, 2, 3, 5, 6, 12, 61 and non-trivial for every other n ≤ 63 except 4. The smooth analogue in dimension four remains open.
paraphrase · Follow the original source for authoritative statements and reuse terms.
John Milnor · Published · Publication date Aug 15, 2014
Slides of the author's lecture at the International Congress of Mathematicians, Seoul, 15 August 2014, posted on his Stony Brook page; not a refereed paper.
Revision & attribution
Source revision: icm-2014-slides-sha256-89cf5c46. Observed Oct 21, 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 reported in the author’s lecture ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Source dated Aug 15, 2014 · Observed Oct 21, 2026 · Source passage →
Record details
- Claim ID
- claim-poincare-smooth-spheres-standard
- Statement version
- 1
- SHA-256
- 00eeccdf162fb4031874ddf1ad4a641f297698aec060311208d9c5f82c5dfd08