Smooth topology, exact arithmetic
The Sphere With Twenty-Eight Names
Turn the clutching integer of an S3-bundle over S4 and watch its exact Eells-Kuiper residue identify a smooth 7-sphere. Then compare any two bundle total spaces through homotopy, homeomorphism, and diffeomorphism, with orientation and every convention kept visible.
One integer, one smooth name
Keep the Euler class at n = 1. Every total space below is homeomorphic to the ordinary 7-sphere. Move m. The smooth structure can still change.
Crowley-Escher M(m,1)
computing
Computing the exact residue.
The wheel does not prove the classification theorem. It applies the cited Eells-Kuiper formula using exact integer arithmetic. It also shows the limit of Milnor-type bundle total spaces: the unfilled residues are genuine oriented smooth structures on the topological sphere, but this n = 1 family does not reach them.
The residue wheel was only the doorway
For general n, the total space need not even have the homology of a sphere, and μ alone is not enough. Enter two bundles. The instrument computes the full Crowley-Escher tests, from coarse homotopy type to smooth diffeomorphism type.
| invariant | M(0,1) | M(1,1) |
|---|---|---|
| H4 | ||
| linking on generator | ||
| p1/2 | ||
| μ mod 1 | ||
| s1 = 28μ mod 1 |
Show the exact tests just run
Each yes is a theorem application, not a browser proof. For homeomorphism and diffeomorphism the page enumerates every multiplier α in Z/n, tests whether it preserves or reverses the linking form, then checks the characteristic class and the exact rational invariant. No decimal approximation is used.
The check
The shipped page recomputes its claims with BigInt. The independent Node verifier extracts this page’s own engine, executes it in a separate context, and compares it with a second implementation.
Free choices and conventions: Crowley-Escher notation M(m,n); n > 0; their relation to Milnor’s integers is n = k + l and m = -l; canonical nonnegative residues; the generator pulled back from H4(S4); orientation preserved unless the reversing column says otherwise; m limited only by the browser’s exact input guard |m| ≤ 10^12; n ≤ 5000 is a responsiveness choice, not mathematics.
Uncertainties: none in the modular arithmetic after valid integer input. The uncertainty lies in theorem dependence: this page does not construct a manifold, prove that μ is invariant, or prove the classifications complete. Publication dates and theorem counts are source transcriptions. The verifier can catch arithmetic drift, not an error in a cited theorem.
Run research/exotic-spheres/verify-exotic-spheres.mjs. Its separate arithmetic perturbation checks compare hand-written altered expressions with expected values.
Four dimensions still refuse the last word
STATUS CHECKED 2026-08-01
The topological statement is closed: Freedman proved that every topological 4-manifold homotopy equivalent to S4 is homeomorphic to S4. The smooth statement remains open: nobody knows whether every smooth homotopy 4-sphere is diffeomorphic to the standard S4.
This is narrow. Exotic smooth 4-manifolds, including exotic R4s, are known. What remains unknown is an exotic smooth structure on the 4-sphere itself. The order-28 result here is dimension 7, not a pattern that can be carried into dimension 4.
Sources that carry the theorem weight
- John Milnor, “On Manifolds Homeomorphic to the 7-Sphere,” Annals of Mathematics 64(2), 1956. doi:10.2307/1969983. Milnor’s λ is modulo 7, not the later complete 28-valued invariant.
- James Eells Jr. and Nicolaas H. Kuiper, “An Invariant for Certain Smooth Manifolds,” Annali di Matematica Pura ed Applicata 60, 1962. doi:10.1007/BF02412768.
- Michel A. Kervaire and John W. Milnor, “Groups of Homotopy Spheres: I,” Annals of Mathematics 77(3), 1963. doi:10.2307/1970128.
- Diarmuid Crowley and Christine M. Escher, “A Classification of S3-Bundles over S4,” Differential Geometry and its Applications 18(3), 2003. doi:10.1016/S0926-2245(03)00012-3.
- Michael Hartley Freedman, “The Topology of Four-Dimensional Manifolds,” Journal of Differential Geometry 17, 1982. doi:10.4310/jdg/1214437136.
Corrections
Corrected 2026-10-18: With identical inputs M(1,1), the summary said there was no orientation-reversing self-equivalence. This space admits an orientation-reversing homotopy self-equivalence and self-homeomorphism. It has no orientation-reversing self-diffeomorphism, which is the test the summary now names; found by the assay line, 2026-10-18.