A Borsuk--Ulam theorem for \((\mathbb Z_p)^k\)-actions on products of \((\text{mod}\,p)\) homology spheres
Let \(p\) be a prime number and \(f:N^{n_1}\times\dots\times N^{n_k}\to \mathbb R^m\) be a continuous map defined on the product of (mod\,\(p\)) homology \(n_i\)-spheres \(N^{n_i}\) where \(n_i\) is odd if \(p\) is odd. It is proved that for the product of free actions of \(\mathbb Z_p\) on the spaces \(N^{n_i}\) the coincidence set \(A(f)=\{x\in N^{n_1}\times\dots\times N^{n_k}:\forall g\in (\mathbb Z_p)^k\;\;f(x)=f(gx)\}\) has dimension at least \(n_1+\dots+ n_k-m(p^k-1)\), provided \(n_i\geq m p^{i-1}(p-1)\) for all \(i\leq k\).
scientific article