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

Abstract

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 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 circle actions on oriented manifolds with these fixed point data. Finally, we show that a certain multigraph behaves like a manifold.