On Nearly Semifree Circle Actions

Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M,ω) is a coadjoint orbit of a compact Lie group G then every element of π1(G) may be represented by a semifree S-action. A theorem of McDuff– Slimowitz then implies that π1(G) injects into π1(Ham(M,ω)), 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,ω) 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