Timeline for Fundamental group of the space of immersions of the 2-sphere in 3-space modulo diffeomorphisms of the first
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Sep 21, 2015 at 9:38 | comment | added | Arnaud Chéritat | I just read in the Morin-Apery paper Eversion of the sphere (page 145 Lemma 7) that Morin claims pi_1(Im(S^2,R^3)) is generated by one of his eversions (the one in the 1970's Nelson Max movie) followed by a mirror image of the time-reversed eversion. I do not know if he quotients by Diff^+, or just forgot Z/2. He also wrote that this claim is from [T. BANCHOFF AND N. MAX, Every sphere eversion has a quadruple point, in Contributions to Analysis and Geometry (Baltimore, Md., 1980), Johns Hopkins Univ. Press, Baltimore, Md., 1981, pp. 191-209.]. I'll try to find and read this article. | |
| Sep 12, 2015 at 9:58 | comment | added | Arnaud Chéritat | Thanks. In the light of your last paragraph, it makes it very likely that Morin's eversion is a generator of the fundamental group when we quotient by the full diffeomorphism group of $S^2$. The intuition being that this is a loop that is non-trivial (because it links two different points in the $2$-cover you mention) and minimizes the number of topological transitions (14 generic transitions, according to Bernard Morin). As another consequence, a generator of the $\pi_1$ for the quotient by $Diff^+$ would be just following Morin's eversion twice. | |
| Sep 12, 2015 at 6:23 | vote | accept | Arnaud Chéritat | ||
| Sep 12, 2015 at 5:46 | history | edited | mme | CC BY-SA 3.0 |
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| Sep 11, 2015 at 22:22 | history | edited | mme | CC BY-SA 3.0 |
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| Sep 11, 2015 at 22:15 | history | answered | mme | CC BY-SA 3.0 |