Timeline for Is there a topological Chevalley-Shephard-Todd Theorem?
Current License: CC BY-SA 3.0
29 events
| when toggle format | what | by | license | comment | |
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| Jul 25, 2015 at 21:16 | answer | added | Nico Bellic | timeline score: 4 | |
| Jul 25, 2015 at 21:10 | vote | accept | Nico Bellic | ||
| Jul 22, 2015 at 19:23 | comment | added | Jason Starr | The topological Chevalley-Shephard-Todd theorem is false in dimensions $n\geq 3$. I just ran across counterexamples in an old MO answer by Greg Kuperberg. | |
| Jul 20, 2015 at 14:14 | answer | added | Jason Starr | timeline score: 9 | |
| S Jul 19, 2015 at 0:03 | history | bounty ended | CommunityBot | ||
| S Jul 19, 2015 at 0:03 | history | notice removed | CommunityBot | ||
| Jul 15, 2015 at 8:32 | comment | added | Geordie Williamson | So now we arrive at the converse, and Jason's argument. I agree there is something left to show... | |
| Jul 15, 2015 at 7:45 | comment | added | Geordie Williamson | @Nico: so now I think I understand better: the quotient $\mathbb{C}^n \to \mathbb{C}^n/\Gamma$ is a quotient both in the category of topological spaces, and algebraic varieties. In particular, if $\Gamma$ is generated by pseudoreflections then it will be a topological manifold after all (by classical Chevalley-Shephard-Todd). | |
| Jul 15, 2015 at 7:44 | comment | added | Geordie Williamson | @Nico: Sorry, I wrote that comment after a glass of wine. | |
| Jul 15, 2015 at 2:46 | comment | added | Nico Bellic | Geordie: How many seems, believes and shoulds were in your last math paper? | |
| Jul 14, 2015 at 20:37 | comment | added | Geordie Williamson | I don't know why there is a bounty ... it seems Jason has already answered the question! | |
| Jul 11, 2015 at 12:50 | history | edited | Nico Bellic | CC BY-SA 3.0 |
added 5 characters in body
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| S Jul 10, 2015 at 22:49 | history | bounty started | Nico Bellic | ||
| S Jul 10, 2015 at 22:49 | history | notice added | Nico Bellic | Authoritative reference needed | |
| Jul 10, 2015 at 22:30 | history | edited | Nico Bellic | CC BY-SA 3.0 |
streamlined the question
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| Jul 10, 2015 at 18:35 | comment | added | Nico Bellic | Let us continue this discussion in chat. | |
| Jul 9, 2015 at 22:08 | comment | added | Jason Starr | "Isn't it surprising if there is no reference for it?" In the 2-dimensional case, the local fundamental group of the origin in $Q$ is nontrivial, so $Q$ is not locally Euclidean near the origin. I suppose that in the case of higher dimensions, there is the possibility that the homeomorphism of $Q$ with $\mathbb{R}^{2n}$ might restrict to a wild embedding on the strata of $B$. Presumably if the homeomorphism is piecewise linear, this is easy to rule out. | |
| Jul 9, 2015 at 21:37 | comment | added | Nico Bellic | To me it seems to be a very important statement. Isn't it surprising if there is no reference for it? | |
| Jul 9, 2015 at 19:44 | comment | added | Jason Starr | It "should" follow from van Kampen's theorem, and the existence of a stratification of $B$ by manifolds. If I remove a submanifold of real codimension $\geq 3$ from a manifold, then the fundamental group of the punctured tubular neighborhood of the submanifold projects isomorphically to the fundamental group of the submanifold. So, by van Kampen's theorem, the fundamental group of the open complement of the submanifold equals the fundamental group of the original manifold. Now iterate this with each stratum of $B$. | |
| Jul 9, 2015 at 19:36 | comment | added | Nico Bellic | Jason Starr: can you please shed any light on why should it so? | |
| Jul 9, 2015 at 19:35 | history | edited | Nico Bellic |
edited tags
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| Jul 9, 2015 at 19:11 | comment | added | Jason Starr | I believe the answer is "no". First, use Chevalley-Shephard-Todd to reduce to the case that $G$ contains no pseudo-reflections. Thus, for the quotient $q:\mathbb{C}^n\to Q$, the branch locus $B$ in $Q$ has complex codimension $\geq 2$, i.e., real codimension $\geq 4$. But now if you take a small sphere about the origin (assuming locally Euclidean), and remove this real analytic subset of real codimension $\geq 4$, that should still leave a simply connected topological space, contradicting the existence of the cover $q:(\mathbb{C}^n\setminus q^{-1}(B)) \to (Q\setminus B)$. | |
| Jul 9, 2015 at 18:41 | comment | added | Nico Bellic | Jason Starr: yes, you are right. I made the corrections | |
| Jul 9, 2015 at 18:40 | history | edited | Nico Bellic | CC BY-SA 3.0 |
I made the corrections as per Ali Taghavi's comments
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| S Jul 8, 2015 at 20:43 | history | suggested | Ali Taghavi |
I add two tags
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| Jul 8, 2015 at 20:20 | review | Suggested edits | |||
| S Jul 8, 2015 at 20:43 | |||||
| Jul 8, 2015 at 17:51 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
Reformatted the question for better readability.
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| Jul 8, 2015 at 17:49 | comment | added | Jason Starr | For (1), did you mean to write that $G$ is not generated by pseudo-reflections? If $G$ is generated by pseudo-reflections, then the Chevalley-Shephard-Todd theorem says that the quotient space is a manifold. | |
| Jul 8, 2015 at 17:44 | history | asked | Nico Bellic | CC BY-SA 3.0 |