The index of certain Stiefel manifolds (Q6630788)

The Borsuk-Ulam theorem states that for every map \(f:S^n\rightarrow\mathbb R^n\) there exists a pair of antipodal points \(x,-x\in S^n\) such that \(f(x)=f(-x)\). This can be seen as a consequence of nonexistence of \(C_2\)-equivariant map \(S^n\rightarrow S^{n-1}\) (\(C_2\) is the cyclic group of order two, with antipodal action on spheres).\N\NThe present paper considers theorems of this type. The authors establish nonexistence of equivariant maps between certain spaces and derive from it some geometric consequences. One such consequence reads as follows: If \(p\) is an odd prime, \(l\) and \(m\) positive integers such that \(l\ge(\lfloor\frac{m}{2}\rfloor+1)(p-1)+1\) and \(f : S^{l-1}\rightarrow\mathbb R^m\) a (continuous) map, then there exist mutually orthogonal \(v_1,v_2,\ldots,v_p\in S^{l-1}\) with the property \(f(v_1)=f(v_2)=\cdots=f(v_p)\). This proposition (as many others in the paper) closely resembles the Kakutani-Yamabe-Yujobo theorem: If \(f : S^{l-1}\rightarrow\mathbb R\) is a map, then there exists an orthonormal basis \((v_1,v_2,\ldots,v_l)\) of \(\mathbb R^l\) with the property \(f(v_1)=f(v_2)=\cdots=f(v_l)\) (which implies that for any closed bounded convex body in \(\mathbb R^l\) there exists a circumscribing cube around it).\N\NIf \(p\) is a prime number, a finite elementary abelian \(p\)-group must be isomorphic to \(C_p^n\) for some \(n\) (\(C_p\) is the cyclic group of order \(p\)). The morphism into symmetric group \(C_p^n\rightarrow\Sigma_{p^n}\) induced by the multiplication in \(C_p^n\) defines an action of \(C_p^n\) on the Stiefel manifold \(V_{p^n}(\mathbb K^l)\) (where \(\mathbb K\) is \(\mathbb R\), \(\mathbb C\) or \(\mathbb H\)). The authors consider this action and prove nonexistence of \(C_p^n\)-equivariant maps between some Stiefel manifolds, which yields to the mentioned geometric results. The method they apply is the cohomological index of Fadell and Husseini. Spectral sequences are used for computing the Fadell-Husseini index of some Stiefel manifolds \(V_{p^n}(\mathbb K^l)\) (considered as \(C_p^n\)-spaces).