Topology Seminar

Sue TolmanUniversity of Illinois Urbana-Champaign
Non-Hamiltonian circle actions with fewer isolated fixed points

Tuesday, October 3, 2023 - 2:55pm
Malott 206

Let the circle act on a closed manifold $M$, preserving a symplectic form $\omega$.
We say that the action is Hamiltonian if there exists a moment map, that is,
a map $\Psi \colon M \to R$ such that $\iota_\xi \omega = - d \Psi$, where $\xi$ is
the vector field that generates the action. In this case, a great deal of information about the manifold is determined by the fixed set. Therefore, it is very important to determine when symplectic actions are Hamiltonian. There has been a great deal of research on this question. It's easy to see that every Hamiltonian action has fixed points. McDuff proved that the converse wasn't true by constructing a non-Hamiltonian action with fixed tori. She then raised the following question, usually called the ``McDuff conjecture": Does there exists a non-Hamiltonian symplectic circle action with isolated fixed points on a closed, connected symplectic manifold? ? I was able to construct such an example with 32 fixed points, but this raised another question. What is the minimal number of possible fixed points? I will discuss my work with D. Jang on reducing the number of fixed points. We have already constructed an example with as few as 10 fixed points, and are now working on constructing an example with only two fixed points, which is the smallest possible number.