Periodic maps on simply connected four-manifolds
Let M be a simply-connected closed 4-manifold, and let \(\phi\) : \(G\times M\to M\) be a topological locally linear finite group-action. \(\phi\) is called pseudo-free if the fixed point set of each element G is a finite set. \(\phi\) is semifree if \(M-M^ G\) has a free G-action. When G is an odd-order cyclic group and \(\phi\) is a free action, \textit{I. Hambleton} and \textit{M. Kreck} [Math. Ann. 280, 85-104 (1988; Zbl 0616.57009)] have classified such actions up to G-homeomorphism. The main result of this paper is a non-trivial extension of the Hambleton-Kreck classification theorem for certain pseudo-free actions: Theorem. Let G be a cyclic group of odd order. A semifree, locally linear G-action with isolated fixed points on a closed, oriented, simply- connected 4-manifold M is determined up to oriented homeomorphism by its quadratic 2-type and the Kirby-Siebenmann invariant of M. For actions on \({\mathbb{C}}{\mathbb{P}}^ 2\), the author extends his earlier result [Math. Ann. 281, 333-339 (1988; Zbl 0627.57025)]. Theorem. Every pseudofree, locally linear action of a finite group on \({\mathbb{C}}{\mathbb{P}}^ 2\) is topologically conjugate to a linear action.
topological locally linear finite group-action
actions on \({\mathbb{C}}{\mathbb{P}}^ 2\)
topologically conjugate to a linear action
scientific article; zbMATH DE number 4178143