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