Timeline for If $M$ and $N$ are closed and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true that $M$ and $N$ are diffeomorphic?
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13 events
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| Jun 25, 2020 at 16:00 | comment | added | Igor Belegradek | @MichaelAlbanese: no worries. I was too lazy to post an answer, and it is good that you did that, since the answers get more visibility. | |
| Jun 25, 2020 at 14:04 | comment | added | Michael Albanese | @IgorBelegradek: Sorry, I didn't see your comment when I posted my answer which is effectively your first sentence. | |
| Jun 25, 2020 at 13:18 | history | edited | Willie Wong |
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| Jun 25, 2020 at 11:44 | answer | added | Michael Albanese | timeline score: 23 | |
| Jun 25, 2020 at 11:26 | comment | added | Igor Belegradek | There are closed flat manifolds that are not homotopy equivalent but become diffeomorphic after product with $S^1$, see math.stackexchange.com/questions/396608/…. Also if $M$ is closed manifold of dimension $\ge 6$ (maybe $5$?) such that the Whitehead group of $\pi_1(M)$ is nonzero, then there is a $h$-cobordism $W$ from $M$ to a manifold $L$ not diffeomorphic to $M$, and $W\times S^1$ is trivial, so $M\times S^1$, $L\times S^1$ are diffeomorphic. | |
| Jun 25, 2020 at 3:17 | vote | accept | Mohammad Farajzadeh-Tehrani | ||
| Jun 25, 2020 at 3:16 | answer | added | Anubhav Mukherjee | timeline score: 29 | |
| Jun 25, 2020 at 3:06 | comment | added | Mohammad Farajzadeh-Tehrani | @ Mukherjee: Definitely means you know a result, please let me know. | |
| Jun 25, 2020 at 3:04 | comment | added | Mohammad Farajzadeh-Tehrani | @ Chris: I am sure it is. I don't know where to find it. I googled but did not find anything. I asked our topologists; no response yet. The one with S^2 instead of S^1 is (negative) pretty famous and important. | |
| Jun 25, 2020 at 3:04 | comment | added | Anubhav Mukherjee | Definitely not true if M is of dim 4. | |
| Jun 25, 2020 at 3:02 | comment | added | Chris Gerig | This has to be in the literature somewhere... Internet suggests it's true for simply-connected n-manifolds with n > 4. | |
| Jun 25, 2020 at 3:01 | history | edited | Mohammad Farajzadeh-Tehrani |
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| Jun 25, 2020 at 2:43 | history | asked | Mohammad Farajzadeh-Tehrani | CC BY-SA 4.0 |