Cyclic group actions and embedded spheres in 4-manifolds

TL;DR

This paper establishes an upper bound on the number of fixed 2-spheres in certain smooth cyclic group actions on simply-connected 4-manifolds, refining previous bounds and revealing restrictions on the existence of such actions.

Contribution

It provides a sharper upper bound for fixed 2-spheres in prime order cyclic actions and shows conditions under which these actions cannot exist or must be pseudofree.

Findings

Upper bound on fixed 2-spheres improves previous estimates

Certain cyclic actions are impossible on some 4-manifolds

Some actions must be pseudofree due to topological constraints

Abstract

In this note we derive an upper bound on the number of 2-spheres in the fixed point set of a smooth and homologically trivial cyclic group action of prime order on a simply-connected 4-manifold. This improves the a priori bound which is given by one half of the Euler characteristic of the 4-manifold. The result also shows that in some cases the 4-manifold does not admit such actions of a certain order at all or that any such action has to be pseudofree.