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Transactions of the American Mathematical Society

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



Equivariant vector fields on spheres

Author: Unni Namboodiri
Journal: Trans. Amer. Math. Soc. 278 (1983), 431-460
MSC: Primary 57S99
MathSciNet review: 701504
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Abstract: We address the following question: If $ G$ is a compact Lie group and $ S(M)$ is the unit sphere of an $ R[G]$-module $ M$, then how many orthonormal $ G$-invariant vector fields can be found on $ S(M)$? We call this number the $ G$-field number of $ M$.

Under reasonable hypotheses on $ M$, we reduce this question to determining when the difference of two $ G$-vector bundles vanishes in a certain subquotient of the $ K{O_G}$-theory of a real projective space. In this general setting, we solve the problem for $ 2$-groups, for odd-order groups, and for abelian groups. If $ M$ also has "enough" orbit types (for example, all of them), then we solve the problem for arbitrary finite groups. We also show that under mild hypotheses on $ M$, the $ G$-field number depends only on the dimensions of the fixed point sets of $ M$.

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Article copyright: © Copyright 1983 American Mathematical Society