Timeline for Which automorphisms on $H_{1}(M^{3})$ are induced by homotopy equivalences?
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11 events
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| May 24, 2015 at 6:18 | comment | added | Dylan Wilson | I agree it should be simpler because of Mostow, but rephrasing it this way didn't seem to help me- though I didn't think about it too long. I imagine that if you run the general machine above one will find that you can rephrase the obstruction in elementary terms. | |
| May 24, 2015 at 4:06 | comment | added | Qiaochu Yuan | @Dylan: I think here all you need to know about hyperbolic manifolds $M$ is that they're aspherical. So, writing $G = \pi_1(M)$, the question is equivalent to: when does an automorphism $G/[G, G] \to G/[G, G]$ lift to an automorphism $G \to G$? I'd expect the obstruction theory to be simpler because of this, but maybe not. | |
| May 23, 2015 at 12:56 | answer | added | Paul Siegel | timeline score: 1 | |
| May 23, 2015 at 9:23 | comment | added | Dylan Wilson | (cont'd) so presumably someone who knows more about these manifolds could compute them. That gives you something up to p-completion, so you'll have to do something rationally as well and then glue. See, e.g., Goerss-Jardine VIII.4 for this obstruction theory spelled out, and also the work of Lannes etc. | |
| May 23, 2015 at 9:21 | comment | added | Dylan Wilson | I don't know anything about hyperbolic 3-manifolds, but there is a general obstruction theory for lifting maps in (co)homology to actual homotopy classes of maps. The first obstruction is that the map on cohomology be a map of modules over the Steenrod algebra, but I don't think that's much of a condition in your case since 3 is a small number. After that there is a sequence of obstructions living in algebraically defined Andre-Quillen style groups in the category of unstable modules over the Steenrod algebra. That sounds very complicated and it is, but in this case there's not much to compute | |
| May 23, 2015 at 7:46 | history | edited | Haimiao Chen | CC BY-SA 3.0 |
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| May 23, 2015 at 7:41 | history | edited | Haimiao Chen | CC BY-SA 3.0 |
added 24 characters in body
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| May 23, 2015 at 7:40 | comment | added | Haimiao Chen | @AlexDegtyarev Yes. Sorry for carelessness. | |
| May 23, 2015 at 7:38 | comment | added | Alex Degtyarev | Do you mean self-homotopy equivalences? | |
| May 23, 2015 at 6:39 | review | First posts | |||
| May 23, 2015 at 7:02 | |||||
| May 23, 2015 at 6:38 | history | asked | Haimiao Chen | CC BY-SA 3.0 |