Timeline for Is there a PL, or topological, bordism hypothesis?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 15, 2018 at 15:52 | comment | added | Chris Schommer-Pries | And the identity functor (thought of as an ETQFT with an unusual target) gives an example which DOES distinguish these exotic spheres. | |
| Aug 15, 2018 at 15:51 | comment | added | Chris Schommer-Pries | @ManuelBärenz A PL-framing of S^7 includes (up to contractible choices) the data of a smooth structure and a smooth framing of the resulting (exotic) smooth S^7. You can think of it this way: a PL-framing is a lift of the tangent microbundle map from BPL(7) to a contractible space (e.g. EPL(7)). A smooth structure is a lift to from BPL(7) to BO(7), and a smooth framing is a further lift to a contractible space. So they are the same! PL-framed $S^7$s corresponding to distinct framed exotic $S^7$s are already distinct in the PL-framed bordism category (since the cats are equivalent). | |
| Jul 17, 2018 at 14:55 | comment | added | Manuel Bärenz | @ArunDebray, according to this answer and Lurie's article, they are. (If I'm not misunderstanding anything) | |
| Jul 17, 2018 at 14:47 | comment | added | Arun Debray | @ManuelBärenz I'm not actually sure. Are the bordism categories equivalent? I don't know enough PL topology to answer that. | |
| Jul 17, 2018 at 14:39 | comment | added | Manuel Bärenz | @ArunDebray, wow, so then ETQFTs don't distinguish exotic $S^7$s?! | |
| Jul 17, 2018 at 14:34 | comment | added | Arun Debray | @ManuelBärenz Yes, exotic $S^7$s admit framings; see this MO post for a proof. | |
| Jul 17, 2018 at 14:21 | vote | accept | Manuel Bärenz | ||
| Jul 17, 2018 at 14:21 | comment | added | Manuel Bärenz | That's a good point, thanks. Still, one conclusion is then that framed ETQFTs are the same for PL and smooth. So those won't distinguish e.g. PL-exotic smooth structures. (Not that I know of any examples. Are exotic $S^7$s framed?) And the work and magic lie in homotopy fixed points again. | |
| Jul 17, 2018 at 14:08 | comment | added | Noah Snyder | Perhaps? I mean one can't just exclude dimension 4 (because fully extended means you'll see lower dimensional boundaries), but yes it seems like a reasonable question to ask whether there's a good description of some nice subcategory of the category of topological manifolds which excludes the 4-dimensional pathological behavior. | |
| Jul 17, 2018 at 13:53 | comment | added | Manuel Bärenz | Handle decompositions do only not exist in 4 dimensions (although in some other low dimensions you have to prove hard theorems). If we look at nonsmoothable 4-manifolds like $E_8$ as some kind of pathological phenomenon, can't we ask about the topological bordism hypothesis in high dimensions? | |
| Jul 17, 2018 at 13:45 | history | answered | Noah Snyder | CC BY-SA 4.0 |