Bourgin-Yang version of the Borsuk-Ulam theorem for \(\mathbb Z_{p^{k}}\)-equivariant maps
Let \(V\), \(W\) be two orthogonal representations of a cyclic group \(G\) of prime power order \(p^k\) such that \(V^G=W^G=\{0\}\), let \(S(V)\) be the unit sphere of \(V\) and let \(F:S(V)\to W\) be a \(G\)-equivariant map. Let \(Z_f=f^{-1}(\{0\})\). The authors give a lower bound for the (covering) dimension \(\dim Z_f=\dim(Z_f/G)\). Denote by \(\mathcal{A}_{S(V)}\) the set of all \(G\)-orbits of \(S(V)\) and call \(\mathcal{A}_{m,n}:=\{G/H|\, H\subset G: m\leq|H|\leq n\}\) (where \(1\leq m\leq n\leq p^{k-1}\) are two powers of \(p\)) and assume that both \(\mathcal{A}_{S(V)}\) and \(\mathcal{A}_{S(W)}\) are subsets of \(\mathcal{A}_{m,n}\). In this situation, the authors prove that \(\dim Z_f\geq2\bigl(\bigl[\frac{(\dim(V)-1)m}{n}\bigr]-\dim W\bigr)\).
scientific article; zbMATH DE number 6122386
scientific article