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 S1-manifold.
Jang, Donghoon, and Oleg R. Musin. "Circle actions on oriented 4-manifolds." arXiv preprint arXiv:2307.04576 (2023).