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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

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