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

## TL;DR

This paper investigates circle actions on oriented manifolds with discrete fixed points, providing a classification in four dimensions by analyzing fixed point data and associated multigraphs, and establishing existence results.

## Contribution

It introduces a classification of fixed point data for 4-dimensional manifolds with circle actions and shows how to realize these data through actual manifolds.

## Key findings

- Manifolds can be described by associated multigraphs.
- Classification of fixed point data in dimension 4.
- Existence of manifolds with prescribed fixed point data.

## 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 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.