Equivariant cross sections of complex Stiefel manifolds
Let \(G\) be a topological group, \(M\) be a unitary representation space of \(G\), and \(S(M)\) be the unit sphere of \(M\). Let \(W_k(M)\) be the Stiefel manifold of unitary \(k\)-frames on \(M\). The projection map \(W_k(M)\rightarrow S(M)\) sending a unitary \(k\)-frame \((u_1,\ldots,u_k)\) to \(u_k\) is \(G\)-equivariant. The author gives some necessary and sufficient conditions for the existence of a \(G\)-equivariant cross section of \(W_k(M)\) for certain finite group actions, under mild fixed point conditions. When \(G\) acts trivially, this is answered in [\textit{M. F. Atiyah} and \textit{J. A. Todd}, Proc. Camb. Philos. Soc. 56, 342-353 (1960; Zbl 0109.16102); \textit{J. F. Adams} and \textit{G. Walker}, ibid. 61, 81-103 (1965; Zbl 0142.40903)].
scientific article; zbMATH DE number 1566037
scientific article