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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ G$-structures on spheres

Author: Peter Leonard
Journal: Trans. Amer. Math. Soc. 157 (1971), 311-327
MSC: Primary 57.40; Secondary 53.00
MathSciNet review: 0275468
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Abstract: $ {G_n}$ denotes one of the classical groups $ SO(n),SU(n)$ or $ Sp(n)$ and $ H$ a closed connected subgroup of $ {G_n}$. We ask whether the principal bundle $ {G_n} \to {G_{n + 1}} \to {G_{n + 1}}/{G_n}$ admits a reduction of structure group to $ H$. If $ n$ is even and $ {G_n}$ is $ SO(n)$ or $ SU(n)$ or if $ n \not\equiv 11\bmod 12$ and $ {G_n}$ is $ Sp(n)$, we prove that there are no such reductions unless $ n = 6,{G_6} = SO(6)$ and $ H = SU(3)$ or $ U(3)$. In the remaining cases we consider the problem for $ H$ maximal. We divide the maximal subgroups into three main classes: reducible, nonsimple irreducible and simple irreducible. We find a necessary and sufficient condition for reduction to a reducible maximal subgroup and prove that there are no reductions to the nonsimple irreducible maximal subgroups. The remaining case is unanswered.

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Keywords: $ n$-sphere, $ G$-structure, reduction of structure group, special orthogonal group, special unitary group, symplectic group, reducible subgroup, irreducible subgroup, maximal subgroup, homotopy exact sequence, fibration
Article copyright: © Copyright 1971 American Mathematical Society

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