Talk:Circle actions on 3-manifolds (Ex)
From Manifold Atlas
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Insight: Try to quotient the manifold by the action. Note that you get something that looks kind of reasonable but isn't a manifold. | 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 | + | 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 take a quotient and use the lemma on Slide 14 of http://www.him.uni-bonn.de/lueck/data/Melbourne_I_Introduction_to_3-manifolds190107.pdf. |
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Latest revision as of 22:31, 9 January 2019
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 
. Then turn the 
action into one with trivial isotropy groups. Now take a quotient and use the lemma on Slide 14 of http://www.him.uni-bonn.de/lueck/data/Melbourne_I_Introduction_to_3-manifolds190107.pdf.
