Timeline for Torus bundles and compact solvmanifolds
Current License: CC BY-SA 4.0
15 events
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| Feb 22, 2022 at 22:29 | comment | added | Ian Gershon Teixeira | Thanks for the new reference! I've edited my post to better reflect the truth. | |
| Feb 22, 2022 at 13:31 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Feb 22, 2022 at 12:49 | comment | added | Igor Belegradek | What you claim (about existence of exotic tori bundle over tori in each dimension $\ge 7$) might still be true but the reference I gave to Hajduk-Tralle's paper involved Hitchin homotopy spheres and those do not exist in dimension $7$. In any case you referred to me for something I did not say, and I wanted to set things straight. What I am reasonably sure of at this moment is exotic tori bundle over tori in some dimensions that start at $8$. However, section 2 of arxiv.org/pdf/1307.3223.pdf may give what you claimed but I haven't looked at it closely. | |
| Feb 22, 2022 at 12:35 | comment | added | Ian Gershon Teixeira | I don't know very much about this topic. I know that there are homotopy 7 spheres which have an exotic smooth structure. Are you saying that only certain types of exotic homotopy spheres are appropriate for this construction and that exotic spheres of that particular type do not arise until dimension 8? | |
| Feb 21, 2022 at 21:19 | comment | added | Igor Belegradek | I am worried about your claim "in every dimension $d≥7$ there are smooth torus bundles over a torus not diffeomorphic to solvmanifolds". I think Hitchin uses homotopy spheres that first show up in dimensions $8$ and $9$, and then I guess one can take products with tori. I haven't thought this through but I don't see how $d$ could be $7$ if one uses Hitchin spheres. I also think one should be able to see what happens if $d=4$, and I expect that all 4-dimensional torus bundles over tori should be diffeomorphic to solvmanifolds. | |
| Feb 21, 2022 at 19:46 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Feb 21, 2022 at 19:08 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Feb 21, 2022 at 18:28 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Feb 21, 2022 at 15:24 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Feb 21, 2022 at 15:18 | history | edited | Ian Gershon Teixeira | CC BY-SA 4.0 |
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| Feb 20, 2022 at 21:59 | history | became hot network question | |||
| Feb 20, 2022 at 17:49 | vote | accept | Ian Gershon Teixeira | ||
| Feb 20, 2022 at 16:45 | comment | added | Igor Belegradek | Smooth $T^n$ bundles over $T^m$ and classified by homotopy classed of maps from $T^m$ into $BDiff(T^n)$, and $Diff(T^n)$ is fairly complicated. In some dimensions its identity component is not homotopy equivalent to a finite CW complex (theorem B in "The non-finite homotopy type of some diffeomorphism groups" by Antonelli- Burghelea-Kahn, sciencedirect.com/science/article/pii/…) and $\pi_0(Diff(T^n))$ is often not $GL(n,\mathbb Z)$, see theorem 2.5 in "Parametrized surgery and isotopy" by Hsiang-Sharpe, msp.org/pjm/1976/67-2/pjm-v67-n2-p09-s.pdf. | |
| Feb 20, 2022 at 15:21 | answer | added | Igor Belegradek | timeline score: 7 | |
| Feb 20, 2022 at 13:52 | history | asked | Ian Gershon Teixeira | CC BY-SA 4.0 |