Proof — Milnor: exotic differential structures on the 7-sphere
Statement
The oriented 7-sphere admits at least seven distinct smooth structures (exotic spheres): there exist smooth 7-manifolds homeomorphic but not diffeomorphic to the standard S⁷. This launched the systematic study of Θₙ and high-dimensional differential topology. Proved in ZFC.
Formula
∃ M^7 ≅_{\mathrm{top}} S^7,\ M^7 \not\cong_{\mathrm{diff}} S^7
Source
J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64 (1956), 399–405.
Domains
mathematics
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- Theorem: Milnor: exotic differential structures on the 7-sphere
- Proof: Milnor: exotic differential structures on the 7-sphere