On the Cohen-Lusk theorem (Q843621)

Let \(G\) be a finite group and \(X\) a \(G\)-space. If \(f:X\to Y\) is continuous and \(2\leq k\leq|G|\) denote by \(A(f,k)\) the set of \(x\in X\) such that there are distinct \(g_1,\dots,g_k\in G\) with \(f(g_1x)=\dots=f(g_kx)\). Using his cohomological index [Sb. Math. 191, No.~9, 1259--1277 (2000); translation from Mat. Sb. 191, No.~9, 3--22 (2000; Zbl 0987.57016)], the author proves the following theorem: Assume that \(G\) is a \(p\)-torus, \(G=\mathbb{Z}_p^n\), and \(2\leq k\leq p^n\) but \(p\not=3\) and that \(X\) is connected and acyclic over \(\mathbb{Z}_p\) in dimensions up to \((m-1)(p^n-1)+k-2\) then \(A(f,k)\not=\emptyset\) if \(Y=\mathbb{R}^m\).