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Fibrations of spheres by great spheres over division algebras and their differentiability. (English) Zbl 0695.57016
The authors characterize the classical Hopf fibrations associated with the four classical division algebras $${\mathbb{R}}$$, $${\mathbb{C}}$$, $${\mathbb{H}}$$, $${\mathbb{O}}$$. Every n-dimensional real division algebra D leads to a fibration of the sphere $$S={\mathbb{S}}_{2n-1}$$ by great (n-1)-spheres in the following way. S is considered as the unit sphere in $$D\times D$$, and the fibers are the intersections of S with the lines through the origin of the affine plane over D. The authors prove that this fibration is differentiable if and only if D is one of $${\mathbb{R}}$$, $${\mathbb{C}}$$, $${\mathbb{H}}$$, $${\mathbb{O}}$$. The proof rests on the identity $$z=(1+1/x)\setminus z+(x+1)\setminus z$$ which is shown to hold in D if the associated fibration is differentiable.
The motivation comes from the theory of Blaschke manifolds. H. Gluck, F. Warner and C. T. Yang [Duke Math. J. 50, 1041-1076 (1983; Zbl 0534.53039)] proved that to any differentiable fibration $$\xi$$ of $${\mathbb{S}}_{2n-1}$$ by great (n-1)-spheres, via some kind of linearization one can associate a fibration $$\xi '$$ coming from a division algebra $$D_{\xi}$$, and that $$\xi$$ and $$\xi '$$ are topologically equivalent. There are examples where $$D_{\xi}$$ is not classical, so from the characterization theorem above it follows that the equivalence of $$\xi$$ and $$\xi '$$ is not differentiable in general. Yet the authors conjecture that a differentiable $$\xi$$ is always differentiably equivalent to a Hopf fibration, which would mean that $$D_{\xi}$$ does not provide the optimal description of $$\xi$$.
Reviewer: R.Löwen

##### MSC:
 57R22 Topology of vector bundles and fiber bundles 51H10 Topological linear incidence structures 53C20 Global Riemannian geometry, including pinching
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