Equivariant maps between complex Stiefel manifolds
The authors consider complex Stiefel manifolds \(V_k(\mathbb C^n)\) with the canonical free \(S^1\)-action \(z(v_1,v_2, \dots,v_k)=(zv_1,zv_2, \dots,zv_k)\). The problem of existence of equivariant maps from \(V_l(\mathbb C^n)\) to \(V_k(\mathbb C^m)\) is investigated. The method is by use of the cohomological ideal-valued index theory of \textit{E. Fadell} and \textit{S. Husseini} [Ergodic Theory Dyn. Syst. (Charles Conley Mem. Vol.) 8, 73--85 (1988; Zbl 0657.55002)]. Some of the results are as follows: There exists an \(S^1\)-map \(V_k(\mathbb C^n)\to V_k(\mathbb C^m)\) if and only if \(n\leq m\). There exists an \(S^1\)-map \(V_k(\mathbb C^n)\to V_l(\mathbb C^n)\) if and only if \(k\geq l\). If there exists an \(S^1\)-map \(V_{n-s}(\mathbb C^n)\to V_{m-s}(\mathbb C^m)\) then \(n\leq m\). If for \(n>m\) there exists an \(S^1\)-map \(V_k(\mathbb C^n)\to V_k(\mathbb C^m)\) then the following conditions must be satisfied: \(n\) is not a power of a prime, \(n\) divides \((k-1)!\), \(n-k<m-l\), if \(k\) is a power of a prime, then it does not divides \(n\).
scientific article; zbMATH DE number 6159574