Borsuk-Ulam theorem and Stiefel manifolds (Q1322330)

There are several different, but equivalent versions of the classical Borsuk-Ulam theorem, and many authors contributed to generalizing and extending the Borsuk-Ulam theorem in many ways. \textit{E. Fadell} and \textit{S. Husseini} [Ergodic Theory Dyn. Syst. 8, 73-85 (1988; Zbl 0657.55002)] and \textit{J. W. Jaworowski} [Proc. Edinb. Math. Soc., II. Ser. 32, No. 2, 271-279 (1989; Zbl 0647.55006)] extended the theorem to maps of Stiefel manifolds. Let \((\mathbb{R}^ n)^ k\) denote the cartesian product of \(k\) copies of \(\mathbb{R}^ n\). Any point of \((\mathbb{R}^ n)^ k\) is represented by a \((k\times n)\)-matrix. Then the \(k\)-th orthogonal group \(O(k)\) acts on \((\mathbb{R}^ n)^ k\) by matrix multiplication on the left. When \(k\leq n\), the Stiefel manifold \(V_ k(\mathbb{R}^ n)\) of orthonormal \(k\)-frames in \(\mathbb{R}^ n\) can be considered as a subspace of \((\mathbb{R}^ n)^ k\) on which \(O(k)\) acts freely and which the \(O(k)\)-action keeps invariant. Fadell and Husseini considered \(\mathbb{Z}_ 2^ k\)-maps \(f: V_ k(\mathbb{R}^ n)\to (\mathbb{R}^{n-k})^ k\) where \(\mathbb{Z}_ 2^ k=\) \hbox{\(\mathbb{Z}_ 2\times \cdots\times \mathbb{Z}_ 2\)} (\(k\) times) is a subgroup of \(O(k)\) which is diagonally imbedded, and they estimated the cohomological size of \(f^{-1}(O)/ \mathbb{Z}_ 2^ k\) in terms of ideal-valued cohomological index theory, where \(O\) is the origin of \((\mathbb{R}^{n-k})^ k\). Jaworowski considered \(O(2)\)-maps \(f: V_ 2(\mathbb{R}^ n)\to (\mathbb{R}^ l)^ 2\) and estimated the cohomological size of \(f^{-1}(T)/ O(2)\), where \(T= \{A\in (\mathbb{R}^ l)^ 2\mid \text{rank } A<2\}\). In the paper under review the author considers a more general class of maps of Stiefel manifolds and generalizes their results.