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
