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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on skew-Hopf fibrations


Author: Michael E. Gage
Journal: Proc. Amer. Math. Soc. 93 (1985), 145-150
MSC: Primary 55R25; Secondary 57R30
MathSciNet review: 766545
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Abstract: Each great circle fibration of the unit $ 3$-sphere in $ 4$-space can be identified with a subset of the Grassmann manifold of oriented $ 2$-planes in $ 4$-space by associating each great circle fiber with the $ 2$-plane it lies in. This Grassmann manifold can be identified with the space $ {S^2} \times {S^2}$. H. Gluck and F. Warner, in Great circle fibrations of the three sphere, Duke Math. J. 50 (1983), 107-132, have shown that the subsets of this Grassmann manifold which correspond to great circle fibrations can be interpreted as the graphs of distance decreasing maps from $ {S^2}$ and $ {S^2}$ and that Hopf fibrations correspond to constant maps.

This note characterizes explicitly the maps which correspond to "skew-Hopf" fibrations: those fibrations of the $ 3$-sphere obtained from Hopf fibrations by applying a linear transformation of $ 4$-space followed by projection of the fibers back to the unit $ 3$-sphere.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0766545-1
Keywords: Hopf fibration, great circle fibration, Grassmann manifold
Article copyright: © Copyright 1985 American Mathematical Society