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Transactions of the American Mathematical Society

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

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1982-0662056-4
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