On the index of approximating sets of periodic points (Q1910149)

Suppose that \(K\) is a compact space, \(f: K\to K\) is a fixed point free map, and \(p\) is a prime number. Then \(f\) induces a free \(\mathbb{Z}/ p\)-action on the fixed point sets \(\text{Fix} (f^p)\) of the \(p\)-iterates of \(f\). Let \(q\) be another prime number. The author studies the question of estimating the topological indices \(i(\text{Fix} (f|U)^p, f)\), where \(U\) is an invariant neighborhood of a sphere contained in the fixed point set of \(f^q\). He constructs an example of a fixed point free map \(f: K\to K\) of a compact space \(K\), with a \((2m- 1)\)-sphere contained in \(\text{Fix} (f^q)\), such that for each invariant neighborhood \(U\) of \(S\), the inequality \(i(\text{Fix} (f|U)^p, f)\geq d_1p- d_2\) \((d_1, d_2\in \mathbb{R}_+)\) holds for all prime numbers \(p\). The topological index of a free \(\mathbb{Z}/ p\)-action is understood in the sense of Yang-Fadell-Husseini; it is an integer assigned to a free \(\mathbb{Z}/ p\)-space, satisfying some properties or axioms.