[過去ﾛｸﾞ] 現代数学の系譜11 ガロア理論を読む8 (779ﾚｽ)
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386(3): 現代数学の系譜11 ガロア理論を読む 2014/03/09(日)09:50 AAS
>>384
４次元球は難しいみたい
http://en.wikipedia.org/wiki/Exotic_sphere
Exotic sphere
From Wikipedia, the free encyclopedia
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 Poincare 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 for exotic 4-spheres are given by Gluck twists (Gluck 1962). These 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. But in most cases it is unknown whether or not the result is diffeomorphic to S4. (If the 2-sphere is unknotted,
or given by spinning a knot in the 3-sphere, then the Gluck twist is known to be diffeomorphic to S4, but there are plenty of other ways to knot a 2-sphere in S4.)
Akbulut (2009) showed that a certain family of candidates for 4-dimensional exotic spheres constructed by Cappell and Shaneson are in fact standard.
404(2): 現代数学の系譜11 ガロア理論を読む 2014/03/21(金)07:00 AAS
>>386
しばらく、Exotic sphere 4次元微分ポアンカレ予想にはまっていた
>The statement that they do not exist is known as the "smooth Poincare conjecture", and is discussed by Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.
http://arxiv.org/abs/0906.5177
Freedman, Michael; Gompf, Robert; Morrison, Scott; Walker, Kevin (2010), "Man and machine thinking about the smooth 4-dimensional Poincare conjecture", Quantum Topology 1 (2): 171?208, arXiv:0906.5177
面白かった
Freedmanは、マイクロソフトに移っていたんだ
http://ja.wikipedia.org/wiki/%E3%83%9E%E3%82%A4%E3%82%B1%E3%83%AB%E3%83%BB%E3%83%95%E3%83%AA%E3%83%BC%E3%83%89%E3%83%9E%E3%83%B3
で、コンピュータパワーで、結び目理論で計算したらしい
5.3 Results
Computing the two-variable polynomial for K2 took approximately 4 weeks on a
dual core AMD Opteron 285 with 32 gb of RAM. At this point, we haven’t been
able to do the calculation for K3 . With the current version of the program, after
about two weeks the program runs out of memory and aborts.
と書いてあって、計算は完了しなかったと
488: 現代数学の系譜11 ガロア理論を読む 2014/04/27(日)07:21 AAS
>>487
補足
”
球面に許される微分構造の数を表にしてみると，
球面の次元 微分構造
１ 1
２ 1
３ 1
４ 1 ”
この４次元球面の微分構造１は、未解決。
これは、>>386 http://en.wikipedia.org/wiki/Exotic_sphere に詳しい解説がある
多分野口の本に間違いがあるので、そこからだろう
http://books.google.co.jp/books/about/%E3%82%A8%E3%82%AD%E3%82%BE%E3%83%81%E3%83%83%E3%82%AF%E3%81%AA%E7%90%83%E9%9D%A2.html?id=mRsScAAACAAJ&redir_esc=y
エキゾチックな球面 野口廣 筑摩書房, 2010 - 295 ページ
（引用おわり）
この本のP195に表があるが、これの４次元が間違いなんだ
494(2): 現代数学の系譜11 ガロア理論を読む 2014/04/27(日)21:46 AAS
>>386
補足
http://en.wikipedia.org/wiki/Exotic_sphere
Exotic sphere
この中に
http://www.nilesjohnson.net/seven-manifolds.html
An animation of exotic 7-spheres Video from a presenation by Niles Johnson at the Second Abel conference in honor of John Milnor.
この７次元Exotic球面のアニメーションがめちゃ面白い
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