BooksStandalone TitlesCollected Volumepp. 355–371

# An invitation to circle actions

• ### Leonor Godinho

Instituto Superior Técnico, Lisboa, Portugal
• ### Silvia Sabatini

Universität Köln, Germany
The problem of determining whether a manifold admits symmetries has been widely studied in mathematics and physics. It is in general hard to determine whether, given a Lie group $G$ and a manifold $M$, there exists a nontrivial action of $G$ on $M$ that preserves a prescribed structure. When $M$ is symplectic, for instance when $M$ is the phase space of a particle, having one conserved quantity whose associated (Hamiltonian) flow on the manifold is periodic, is equivalent to having a Hamiltonian circle action. The following questions are therefore natural: Which symplectic manifolds admit symplectic circle actions? What are their topological properties? Here we discuss these problems when the fixed point set is discrete. In particular we review some of the consequences of the fact that the Chern number $c_1c_{n-1}[M]$ is completely determined by the fixed point data of the action.