Timeline for Must the number of smooth structures be countable or continuum?
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15 events
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| Jul 2 at 6:24 | comment | added | Hanul Jeon | Hjorth and Kechris wrote a paper about the classification of Riemann surfaces, which might be relevant. I once heard from Solecki that descriptive-set-theoretic classification for the class of manifolds has not been well-studied. | |
| Jul 2 at 5:00 | comment | added | Elliot Glazer | Actually, from the equivalence of 4D smooth and PL manifolds, can isomorphism between two 4D smooth manifolds be expressed combinatorially via triangulations $T_i$ of $M_i$? Something like “there are simplices $s \in T_1, t \in T_2$ such that for all $n,$ there is a combinatorial isomorphism from the set of simplices within $n$ adjacencies of $s$ to those of $t$”? I’m out of my depth here geometrically, but that could confirm the equivalence relation to be Borel. | |
| Jul 2 at 4:10 | comment | added | Elliot Glazer | Naively this equivalence relation is analytic, because it posits the existence of a diffeomorphism. I don’t think there is a simple trick to proving this conjecture. In the 4D case, it will probably requiring actually overcoming the longstanding problem of finding nontrivial invariants for smooth structures. | |
| Jul 2 at 1:30 | comment | added | 183orbco3 | @NoahSchweber Could you indicate how to show (or why we should expect) that this equivalence relation is coanalytic? | |
| Jul 1 at 20:31 | history | edited | 183orbco3 | CC BY-SA 4.0 |
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| Jul 1 at 19:26 | history | edited | LSpice | CC BY-SA 4.0 |
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| Jul 1 at 16:44 | comment | added | 183orbco3 | @MoisheKohan I guess his question was whether there is a (connected) non-compact manifold with exactly countably infinite many smooth structures. | |
| Jul 1 at 15:40 | comment | added | Moishe Kohan | math.stackexchange.com/questions/2129843/… and math.stackexchange.com/questions/2122240/… @AlessandroCodenotti | |
| Jul 1 at 14:53 | history | edited | 183orbco3 | CC BY-SA 4.0 |
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| Jul 1 at 14:48 | history | edited | 183orbco3 | CC BY-SA 4.0 |
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| Jul 1 at 11:38 | comment | added | Tyrone | The long line supports $2^{\aleph_1}$ many pairwise non-diffeomorphic differential structures. So at least its consistent that the statement fails for nonmetrisable manifolds. | |
| Jul 1 at 8:27 | comment | added | Alessandro Codenotti | Out of curiosity, what is an example of such an $M$ admitting countably many smooth structures? | |
| Jul 1 at 4:40 | comment | added | David Roberts♦ | If one were to get some possibly intermediate uncountable cardinality, then it would almost surely have to be something that appears on Cichoń's diagram, a bit like how Brooke-Taylor's theorem about products of CW-complexes with tight cardinality bounds, using the bounding number $\mathfrak{b}$. | |
| Jul 1 at 4:06 | comment | added | Noah Schweber | The general strategy for such questions is to show that there is a natural topology on the set of relevant objects - in this case, smooth structures - which yields a (Borel subset of a) Polish space and with respect to which the relevant equivalence relation (in this case, diffeomorphism) is coanalytic or better. If you can do that, then Silver's dichotomy tells you that you do indeed have either $\le\aleph_0$ or exactly $2^{\aleph_0}$ many objects of the relevant type. I believe this works in this case, but my differential geometry isn't good enough to be certain. | |
| Jul 1 at 3:59 | history | asked | 183orbco3 | CC BY-SA 4.0 |