1959/10/01 by John Milnor · 2 citations
Mathematics · Computer Science · #Geometric and Algebraic Topology #Topological and Geometric Data Analysis #Mathematical Dynamics and Fractals #SPHERES #Mathematics #Differentiable function #Pure mathematics #Physics #Astronomy
paper · doi:10.2307/2372998
openalex publication_date 1959/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06
According to [5] the sphere S7 can be given several differentiable structures which are essentially distinct. A corresponding result for the 15-sphere has been proved by Shimada [10] and Tamura [12]. The object of this note is to prove corresponding theorems for other dimensions of the form 4/c -1. In ? 1 certain differentiable manifolds M(f', f2) are constructed and studied; where (f,) E 7m(SOn,,,), (f2) C 7rn(SOm+T). In most cases these manifolds are topologically spheres. In ? 2 an invariant A is defined for differentiable (4k 1)-manifolds which are both homology spheres and boundaries. In ? 3 the invariant (M (f1, f2)) is computed. For kc 8 the calculations are carried out explicitly. It is shown that there exist non-standard differentiable structures on S1`-1 for k -2, 4, 5, 6, 7, 8. For example S31 has over sixteen million distinct differentiable structures. It is conjectured that the same argument works for all k > 4; but I have not succeeded in solving the number theoretic problem which arises. For kc = 1, 3 the argument does not work. This is not surprising in the case kc = 1, since J. Munkres, S. Smale, and J. H. C. Whitehead have shown (independently) that two differentiable 3-manifolds which are homeomorphic must necessarily be diffeomorphic. The word manifold will always be used for a compact, oriented manifold, with or without boundary. The symbol Dk will stand for the unit disk in the euclidean space ck.