ガロア第一論文と乗数イデアル他関連資料スレ11

[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ11 (1002ﾚｽ)
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843: １３２人目の素数さん [] 2024/12/29(日) 22:05:32.84 ID:aRTKq65A

>>841-842
>Q.　Exotic R^4には、通常のC^2とは異なる複素構造が入る？
>それは未解決だと思う

それは、ずいぶん面白い問いだと思う
まず、Exotic R4とは？
SmallとLargeがあるらしい

そのまえに、通常のC^2には、通常のR^4と微分同相か？ という問いがあるだろう。多分Yesかな
とすると、C^2にも Exoticな（通常と非微分同相な）微分可能構造が入るか？ という問題設定かな？ 多分Yesかな

Cをリーマン球に丸めて、C'と書く。C'^2 はどうか？ 頭が働かない・・ ；ｐ）
ところで、exotic 4-sphereについて
”a counterexample to the smooth generalized Poincaré conjecture in dimension 4. Some plausible candidates are given by Gluck twists.”
とあるね

(参考)
en.wikipedia.org/wiki/Exotic_R4
Exotic R4

Small exotic R4s

Large exotic R4s
Michael Hartley Freedman and Laurence R. Taylor (1986) showed that there is a maximal exotic
R4, into which all other
R4 can be smoothly embedded as open subsets.

Related exotic structures
Casson handles are homeomorphic to
D2×R2 by Freedman's theorem (where
D2 is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to
D2×R2.
In other words, some Casson handles are exotic
D2×R2.

It is not known (as of 2024) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincaré conjecture in dimension 4. Some plausible candidates are given by Gluck twists.

en.wikipedia.org/wiki/Exotic_sphere#4-dimensional_exotic_spheres_and_Gluck_twists
4-dimensional exotic spheres and Gluck twists
In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. The statement that they do not exist is known as the "smooth Poincaré conjecture", and is discussed by Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.

Some candidates proposed for exotic 4-spheres are the Cappell–Shaneson spheres (Sylvain Cappell and Julius Shaneson (1976)) and those derived by Gluck twists (Gluck 1962). Gluck twist spheres are constructed by cutting out a tubular neighborhood of a 2-sphere S in S4 and gluing it back in using a diffeomorphism of its boundary S2×S1. The result is always homeomorphic to S4. Many cases over the years were ruled out as possible counterexamples to the smooth 4 dimensional Poincaré conjecture. For example, Cameron Gordon (1976), José Montesinos (1983), Steven P. Plotnick (1984), Gompf (1991), Habiro, Marumoto & Yamada (2000), Selman Akbulut (2010), Gompf (2010), Kim & Yamada (2017).

ja.wikipedia.org/wiki/%E3%82%A8%E3%82%AD%E3%82%BE%E3%83%81%E3%83%83%E3%82%AF_R4
エキゾチック R4

http://rio2016.5ch.net/test/read.cgi/math/1724969804/843

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