Timeline for Where was it first shown that every homotopy self-equivalence of $S^1\times S^2$ is homotopic to a homeomorphism?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 16, 2021 at 17:22 | vote | accept | Igor Belegradek | ||
| Feb 16, 2021 at 15:56 | comment | added | Andy Putman | @AllenHatcher: Oh, that’s a good point! I had just assumed that Sawashita was correct without thinking very hard about it. | |
| Feb 16, 2021 at 15:36 | comment | added | Allen Hatcher | @IgorBelegradek: The theorem of Sawashita appears to be incorrect in the case of $S^1\times S^2$. The kernel is ${\mathbb Z}_2$ rather than ${\mathbb Z}$ (and the extension is a product). The ${\mathbb Z}$ presumably comes from $\pi_1SO(2)$ but it should be $\pi_1SO(3)$, corresponding to a homeomorphism of $S^1\times S^2$ rotating $\{t\}\times S^2$ by the angle $t$. As in my comment to Andy Putman there is no difference between basepoint-preserving homotopy equivalences and those that do not preserve basepoint in this situation. | |
| Feb 16, 2021 at 14:55 | comment | added | Allen Hatcher | @AndyPutman: For $S^1\times S^2$ the mapping class group fixing a basepoint is the same as the one with the basepoint not fixed. This is because in the Birman exact sequence the map $\pi_1 {\rm Diff}(S^1\times S^2)\to\pi_1(S^1\times S^2)$ induced by evaluating diffeomorphisms at the basepoint is surjective. (For others: the Birman exact sequence is the exact sequence of homotopy groups for the fibration ${\rm Diff}(S^1\times S^2)\to S^1\times S^2$ given by evaluating at a basepoint, with fiber the basepoint-preserving diffeomorphisms.) | |
| Feb 16, 2021 at 5:12 | answer | added | Allen Hatcher | timeline score: 13 | |
| Feb 16, 2021 at 4:06 | comment | added | Andy Putman | Yes, I think that does it. You could also easily prove from Gluck's theorem that the mapping class group of basepoint-preserving orientation-preserving diffeomorphisms is an extension of $(\mathbb{Z}/2)^2$ by $\mathbb{Z}$ (by the way, I don't know if this split or not). I recently wrote a paper on connect sums of $n$ copies of $S^1 \times S^2$ where we improve a theorem of Laudenbach to show that the mapping class group is a split extension of $Out(F_n)$ by $(\mathbb{Z}/2)^n$, which reduces to Gluck's theorem for $n=1$. See here: arxiv.org/abs/2012.01529 | |
| Feb 16, 2021 at 3:35 | comment | added | Igor Belegradek | @AndyPutman: thanks! Sawashita in theorem 8.8 of [On the Group of Self-Equivalences of the Product of Spheres", Hiroshima Math J, 1975] projecteuclid.org/download/pdf_1/euclid.hmj/1206136785 computed the group of based homotopy self-equivalences of $S^1\times S^2$; it maps onto $(\mathbb Z_2)^2$ with kernel $\mathbb Z$, and presumably, one can compare the two results and extract what I want. I was just hoping for a clean reference. | |
| Feb 16, 2021 at 3:17 | comment | added | Andy Putman | I don't know an earlier proof, but you might be interested in the following paper which shows that the mapping class group of $S^1 \times S^2$ is $(\mathbb{Z}/2)^2$: H. Gluck, The embedding of two-spheres in the four-sphere. Trans. Amer. Math. Soc. 104 (1962), 308–333. | |
| Feb 16, 2021 at 1:19 | history | asked | Igor Belegradek | CC BY-SA 4.0 |