Circle actions on oriented manifolds with discrete fixed point sets and classification in dimension 4

In this paper, we study a circle action on a compact oriented manifold with a discrete fixed point set. The fixed point data consists of the weights of the $S^1$-representations at the fixed points. We prove various results and properties of the action, in terms of the fixed point data. We show that the manifold can be described by a multigraph associated to it. Specializing into the case of dimension 4, we classify the fixed point data. Moreover, we prove that there exist oriented $S^1$-manifolds with these fixed point data. Finally, we show that a certain multigraph behaves like a manifold.