Borsuk-Ulam theorems for products of spheres and Stiefel manifolds revisited (Q2194623)

Let \(G\) be a compact group, \(X\) a compact \(CW\) complex equipped with a \(G\)-action and \(V\) a finite dimensional real representation of \(G\). A well-studied question with a multitude of applications in topological combinatorics is to decide whether a \(G\)-equivariant map \(X \to V\) has \(0\) in its image. Equivalently, whether there exists a \(G\)-equivariant map \(X \to S(V)\) into the unit sphere \(S(V)\) of \(V\). The idea has applications in hyperplane mass partitions, the square-peg problem, Tverberg-type theorems and chromatic numbers of hypergraphs, to name a few. There is a well-developed obstruction theory pioneered by \textit{E. Fadell} and \textit{S. Husseini} [Ergodic Theory Dyn. Syst. 8, 73--85 (1988; Zbl 0657.55002)] to detemine the existence of equivariant maps. A special case of the problem is when \(X=S^{n_1} \times \cdots \times S^{n_k}\) is a product of spheres equipped with an action of the elementary abelian 2-group \(\mathbb{Z}_2^k\) with the \(i\)-th copy of \(\mathbb{Z}_2\) acting non-trivially on the \(i\)-th sphere. The product of \(k\) equidimensional spheres of dimension \(n\) contains the Stiefel manifold \(V_{n,k}\) of \(k\)-frames of mutually orthonormal vectors in \(\mathbb{R}^n\). The main result of the paper is the following Theorem. Every \(\mathbb{Z}_2^k\)-equivariant map \[V_{n,k} \to \mathbb{R}^{n-1}\times \mathbb{R}^{n-2} \times \cdots \times \mathbb{R}^{n-k}\] has a zero, where the \(i\)-th copy of \(\mathbb{Z}_2\) acts non-trivially on the \(i\)-th factor \(\mathbb{R}^{n-i}\) and acts on \(V_{n,k}\) by \((x_1, \ldots, x_i, \ldots, x_n) \mapsto (x_1, \ldots, -x_i, \ldots, x_n)\). The result is established by first proving the non-existence of \(\mathbb{Z}_2^k\)-equivariant maps from a product of \(k\) spheres to the unit sphere in a real \(\mathbb{Z}_2^k\)-representation of the same dimension, a result originally due to \textit{E. A. Ramos} [Discrete Comput. Geom. 15, No. 2, 147--167 (1996; Zbl 0843.68120)]. Interestingly, the main theorem also leads to alternative proofs and extensions of some results of \textit{E. Fadell} and \textit{S. Husseini} [loc. cit.].