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Transactions of the American Mathematical Society

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On actions of adjoint type on complex Stiefel manifolds

Author: McKenzie Y. Wang
Journal: Trans. Amer. Math. Soc. 272 (1982), 611-628
MSC: Primary 57S15; Secondary 57S25
MathSciNet review: 662056
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Abstract: Let $ G(m)$ denote $ {\rm {SU}}(m)$ or $ {\rm {Sp}}(m)$. It is shown that when $ m \geq 5\,G(m)$ cannot act smoothly on $ W_{n,2}$, the complex Stiefel manifold of orthonormal $ 2$-frames in $ \mathbf C^n$, for $ n$ odd, with connected principal isotropy type equal to the class of maximal tori in $ G(m)$. This demonstrates an important difference between $ W_{n,2}$, $ n$ odd, and $ S^{2n-3}\times S^{2n-1}$ in the behavior of differentiable transformation groups. Exactly the same holds for $ {\rm {SO}}(m)$ or Spin$ (m)$ if it is further assumed that a maximal $ 2$-torus of $ {\rm {SO}}(m)$ has fixed points.$ ^{2}$

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Keywords: Stiefel manifolds, compact differentiable transformation groups, Steenrod operations, actions of adjoint type
Article copyright: © Copyright 1982 American Mathematical Society

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