Circle actions on oriented 4-manifolds

TL;DR

This paper studies circle group actions on compact oriented 4-manifolds, deriving formulas and associating graphs to fixed points, and characterizes manifolds via these graphs.

Contribution

It derives the Atiyah-Hirzebruch formula for such manifolds and establishes a correspondence between fixed point graphs and the existence of manifolds.

Findings

Derived the Atiyah-Hirzebruch formula for 4-manifolds with circle actions.

Associated a graph structure to fixed points and characterized manifolds via these graphs.

Proved existence of manifolds corresponding to certain fixed point graphs.

Abstract

In the present paper, we consider an action of the circle group on a compact oriented 4-manifold. We derive the Atiyah-Hirzebruch formula for the manifold, and associate a graph in terms of data on the fixed point set. We show in the case of isolated fixed points that if an abstract graph satisfies the Atiyah-Hirzebruch formula, then there exists a corresponding 4-dimensional oriented $S^1$-manifold.