Timeline for The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from that of topological manifolds?
Current License: CC BY-SA 4.0
18 events
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| Oct 16, 2019 at 19:33 | comment | added | Yonatan Harpaz | Remark 3.29 of the paper arxiv.org/abs/1206.5522 claims that the answer is yes, and that this follows from Kirby-Siebenmann, but there is no precise reference. I was not able to find it in KS, but if you manage I would be quite interested to see it myself. | |
| Oct 15, 2019 at 21:09 | comment | added | Ben Wieland | Yeah, I guess the diagonal topology on $Homeo(M)$ works. I was being overcautious because a lot of people use a simplicial modification. Probably they're doing it to deal with subgroups that are not closed. But if you're only dealing with the one homotopy type, I can't see anything that could go wrong. | |
| S Oct 15, 2019 at 14:00 | history | bounty ended | CommunityBot | ||
| S Oct 15, 2019 at 14:00 | history | notice removed | CommunityBot | ||
| Oct 12, 2019 at 20:46 | comment | added | Tim Campion | If you really want an answer to this question, it's probably worth rephrasing to avoid mention of $\infty$-categories. I imagine there may well be somebody around who knows enough differential topology to answer the question conclusively or else explain why it's an open problem, but may not be 100% certain what is needed to get the $\infty$-categorical statement. This would be a worthwhile exercise in "compiling out" the $\infty$-categories for its own sake anyway. You could leave the $\infty$-categorical statement in, but not feature it so prominently. | |
| S Oct 7, 2019 at 12:26 | history | bounty started | Saal Hardali | ||
| S Oct 7, 2019 at 12:26 | history | notice added | Saal Hardali | Authoritative reference needed | |
| Oct 6, 2019 at 5:07 | history | edited | Saal Hardali | CC BY-SA 4.0 |
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| Oct 5, 2019 at 17:52 | comment | added | skupers | By Kisters theorem, BTop(n) defined as classifying n-dimensional topological microbundles is weakly equivalent to BHomeo(R^n). Anyway, it's Essay V.1 you want to look at. Another reference is Lashof's Embedding Spaxes | |
| Oct 5, 2019 at 16:53 | history | edited | Saal Hardali | CC BY-SA 4.0 |
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| Oct 5, 2019 at 16:52 | comment | added | Saal Hardali | @BenWieland I thought $BHomeo(\mathbb{R}^n)$ works. I see your point about the stability, i'll fix that. | |
| Oct 5, 2019 at 16:45 | comment | added | Ben Wieland | The product structure theorems give $n$-equivalences, but here you ask for full homotopy equivalences. It is true that $Top(n)/PL(n)=Top/PL=K(\mathbb Z/2,3)$, though, so that square is OK either way. Unstable PL bundles realize all Pontrjagin classes, while smooth ones don't. So there is a PL $S^3$ bundle on $S^{4n}$ with nontrivial $p_n$ that is not smoothable, but which your stable statement would imply smoothable. . . Also, $Top(n)$ probably isn't a topological group, either, and you should do something simplicial there, too. | |
| Oct 5, 2019 at 16:36 | history | edited | Saal Hardali | CC BY-SA 4.0 |
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| Oct 5, 2019 at 16:18 | history | edited | Saal Hardali | CC BY-SA 4.0 |
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| Oct 5, 2019 at 16:17 | comment | added | Saal Hardali | @skupers Having the stable versions is part of the statement which I think shiuld be true due to Product Structure theorems. Could you perhaps point at the exact statements from Kirby and Siebenmann you have in mind? | |
| Oct 5, 2019 at 16:09 | comment | added | skupers | I think you need to replace BO, BPL and BTop by their unstable versions. Then it sounds like a reformulation of smoothing theory as in Kirby-Siebenmann (which would even work for dimension 4 when going from PL to Diff). | |
| Oct 5, 2019 at 11:52 | history | edited | Harry Gindi | CC BY-SA 4.0 |
fixed some TeX formatting.
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| Oct 5, 2019 at 11:20 | history | asked | Saal Hardali | CC BY-SA 4.0 |