Talk:Circle actions on 3-manifolds (Ex)

Insight: Try to quotient the manifold by the action. Note that you get something that looks kind of reasonable but isn't a manifold.

Guess: The isotropy groups must all be the same. Consider a point, then the isotropy groups of its neighborhood are the same as its own. Same thing for the boundary of an (open) region of fixed isotropy groups, so the set with identical isotropy groups is both open and closed. Then, the isotropy groups must in fact be $Z_n$$<wikitex>; Insight: Try to quotient the manifold by the action. Note that you get something that looks kind of reasonable but isn't a manifold. Guess: The isotropy groups must all be the same. Consider a point, then the isotropy groups of its neighborhood are the same as its own. Same thing for the boundary of an (open) region of fixed isotropy groups, so the set with identical isotropy groups is both open and closed. Then, the isotropy groups must in fact be $Z_n$. Then turn the $S^1$ action into one with trivial isotropy groups. Now try to take a cross section. </wikitex>Z_n$. Then turn the $S^1$$S^1$ action into one with trivial isotropy groups. Now try to take a cross section.