On nearly semifree circle actions

Abstract

Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if $(M, \om)$ is a coadjoint orbit of a compact Lie group $G$ then every element of $\pi_1(G)$ may be represented by a semifree $S^1$-action. A theorem of McDuff--Slimowitz then implies that $\pi_1(G)$ injects into $\pi_1(\Ham(M, \om))$, which answers a question raised by Weinstein. We also show that a circle action on a manifold $M$ which is semifree near a fixed point $x$ cannot contract in a compact Lie subgroup $G$ of the diffeomorphism group unless the action is reversed by an element of $G$ that fixes the point $x$. Similarly, if a circle acts in a Hamiltonian fashion on a manifold $(M,\omega)$ and the stabilizer of every point has at most two components, then the circle cannot contract in a compact Lie subgroup of the group of Hamiltonian symplectomorphism unless the circle is reversed by an element of $G$