Timeline for Topological version of Bogomolov’s question
Current License: CC BY-SA 4.0
14 events
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| S May 3, 2022 at 0:48 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
fixed broken link to springerlink.com, linked to relevant answer on MO, mentioned link to MO question inline so that the title auto-expands, changed HTTP to HTTPS in another URL, used blockquote formatting for quoted portion of Gromov's paper
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| May 2, 2022 at 13:09 | review | Suggested edits | |||
| S May 3, 2022 at 0:48 | |||||
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Dec 17, 2009 at 19:38 | history | edited | Ian Agol | CC BY-SA 2.5 |
added refined question
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| Dec 17, 2009 at 5:15 | answer | added | Greg Kuperberg | timeline score: 6 | |
| Dec 14, 2009 at 17:53 | comment | added | HJRW | Igor, of course, you're quite right. For some reason, I was only thinking about H_1. | |
| Dec 14, 2009 at 13:41 | comment | added | Igor Belegradek | Henry, this does not work because homology of SxF vanish in degrees >3, while G could be the fundamental group of a closed aspherical n-manifold with n>3. I wish to add that the existence of a degree one map is a subtle condition to express group-theoretically, and my attempt above (to substitute "degree one" assumption with surjectivity in homology in all degrees) may be not a perfect one. But I figured a counterexample to the group-theoretic version may shed some light to the topological case. | |
| Dec 13, 2009 at 16:22 | comment | added | Tim Perutz | Two who have thought about the topological version (one at least being a 4-manifold topologist) are Kotschick and Loeh, arXiv:0806.4540. Kotschick gave a pretty talk about it at Columbia, with Gromov in the audience. DK: "For the next five minutes I'm going to explain something standard. Misha, I expect you know this as Preismann's theorem?" MG: "It's just obvious." | |
| Dec 13, 2009 at 16:09 | comment | added | HJRW | Igor, here's a silly answer to your question as stated. Let F be a free group that surjects G, let S be a surface group and let H=SxF. Now the obvious map H->F->G is a surjection in homology! | |
| Dec 13, 2009 at 15:05 | comment | added | Igor Belegradek | It would be good to have group-theoretic version of your question. Here is one. Given a finitely presented group $G$, does there exists a finite index subgroup $G_0$ and and a surjective homomorphism $H\to G_0$ inducing a surjection in homology and such that $H$ is finitely presented and contains a surface subgroup that is normal. Note that degree one maps are surjective in homology (in all degrees). | |
| Dec 13, 2009 at 2:05 | comment | added | HJRW | JSE - some of the most profound recent work on this is by, er, Agol (see eg arxiv.org/abs/0707.4522). | |
| Dec 13, 2009 at 1:26 | comment | added | JSE | I don't know the answer to the question about Bogomolov, but certainly for hyperbolic 3-manifolds M it's a much-thought-about question whether M necessarily has a finite cover smoothly fibered over the circle. | |
| Dec 13, 2009 at 1:08 | history | edited | Ryan Budney |
edited tags
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| Dec 13, 2009 at 1:01 | history | asked | Ian Agol | CC BY-SA 2.5 |