Deformation and Extension of Fibrations of Spheres by Great Circles

Deformation and Extension of Fibrations of Spheres by Great Circles

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Herman Gluck, University of Pennsylvania
Fine Hall 314

In a 1983 paper with Frank Warner, we proved that the space of all great circle fibrations of the 3-sphere S3 deformation retracts to the subspace of Hopf fibrations, and so has the homotopy type of a pair of disjoint two-spheres. Since that time, no generalization of this result to higher dimensions has been found, and so we narrow our sights here and show that in an infinitesimal sense explained below, the space of all smooth oriented great circle fibrations of the 2n+1 sphere S2n+1 deformation retracts to its subspace of Hopf fibrations. The tools gathered to prove this also serve to show that every germ of a smooth great circle fibration of S2n+1 extends to such a fibration of all of S2n+1, a result previously known only for S3 .  Joint work with Patricia Cahn (UPenn) and Haggai Nuchi (Univ. of Toronto)