Measuring the size of the coincidence set
Let \(G\) be a finite group which acts freely on an \(m\)-sphere \(S^m\). Let \(H\) be a subgroup of \(G\), then \(H\) acts on each \(G\)-orbit \(Gx\) by: \(h(gx)= (gh)x\) for all \(g\in G\). For a map \(f: S^m\to Y\) to a \(k\)-dimensional CW-complex, call \(x\in S^m\) an \((H,G)\)-coincidence point of \(f\) if \(f\) sends every orbit of the action of \(H\) on \(Gx\) to a single point. The set of \((H,G)\)-coincidence points is denoted by \(A(f,H,G)\). In a previous paper, the authors proved that if \(m\geq|G|k\), then there is a nontrivial subgroup \(H\) of \(G\) such that \(A(f,H,G)\) is nonempty. In the present paper, they prove that if \(H\) is a normal subgroup of \(G\) that is cyclic of prime order, then the covering dimension of \(A(f,H,G)\) is at least \(m-|G|k\).
scientific article; zbMATH DE number 1879137