Timeline for a question about the isotropy subgroup of circle action on manifolds with isolated fixed point
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 31, 2012 at 6:01 | comment | added | Richard Montgomery | Thanks, Tom. Yes, that is what I was trying to say. -Richard M. | |
| Jan 23, 2012 at 1:56 | comment | added | Tom Goodwillie | I believe that that Richard Montgomery was answering your second question with the slice theorem. In general, for any smooth action of compact $G$ on $M$, for every point $p\in M$ with trivial isotropy group, let $D\subset M$ be a disk transverse to the orbit $Gp$ at $p$, of dimension $dim(M)-dim(G)$. The map $G\times D\to M$, $(g,x)\mapsto gx$, is a submersion at every point in $G\times p$, therefore in a neighborhood of $G\times p$, and it's not hard to see that for a small enough disk nbhd $\Delta \subset D$ of $p$ in $D$ it gives a diffeomorphism of $G\times \Delta$ onto its image. | |
| Jan 23, 2012 at 1:22 | comment | added | Ping | some literatures take "isolated” as "nonempty and finitely many fixed points" while some are not. So it depends on concrete situation. | |
| Jan 22, 2012 at 12:57 | comment | added | Tom Goodwillie | Irrelevant in the sense of being unneeded hypotheses for the conclusion you are asking about. And does "the fixed points are isolated" imply that there is at least one fixed point? (Sorry.) | |
| Jan 22, 2012 at 7:59 | comment | added | Ping | By the way, the condition "even-dimensional" is of course can be deduced from the fact that the fixed points are isolated. But the conditions "closed, orientable" are relevant:-) | |
| Jan 22, 2012 at 7:55 | vote | accept | Ping | ||
| Jan 22, 2012 at 7:56 | |||||
| Jan 22, 2012 at 7:52 | comment | added | Ping | Thank you. But what about my second question?:-)(Moreover, if the answer is yes, can we extend...) | |
| Jan 22, 2012 at 7:46 | vote | accept | Ping | ||
| Jan 22, 2012 at 7:46 | |||||
| Jan 22, 2012 at 4:07 | history | edited | Tom Goodwillie | CC BY-SA 3.0 |
added 15 characters in body
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| Jan 22, 2012 at 3:57 | history | answered | Tom Goodwillie | CC BY-SA 3.0 |