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A smooth 4-manifold homeomorphic yet not diffeomorphic to euclidean space
In mathematics, an exotic is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.[1][2] There is a continuum of non-diffeomorphic differentiable structures of as was shown first by Clifford Taubes.[3]
Prior to this construction, non-diffeomorphic smooth structures on spheres – exotic spheres – were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and still remains open as of 2024). For any positive integer n other than 4, there are no exotic smooth structures on in other words, if n ≠ 4 then any smooth manifold homeomorphic to is diffeomorphic to [4]
In mathematics, an exotic R 4 {\displaystyle \mathbb {R} ^{4}} is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic...
reactor built in Sweden ExoticR4, in mathematics, a differentiable manifold R4, abbreviation for rear-engine, four-wheel drive layout R4, a sub-class of Group...
Look up exotic in Wiktionary, the free dictionary. Exotic may refer to: ExoticR4, a differentiable 4-manifold, homeomorphic but not diffeomorphic to...
construction ExoticR4 Cerf theory Seven-dimensional space Behrens, M.; Hill, M.; Hopkins, M. J.; Mahowald, M. (2020). "Detecting exotic spheres in low...
n = 4, there are uncountably many such types. One refers to these by exoticR4. The following table lists the number of smooth types of the topological...
generalized Poincaré conjecture. Freedman and Robion Kirby showed that an exoticR4 manifold exists. Freedman was born in Los Angeles, California, in the...
4-manifold. Important consequences of this theorem include the existence of an ExoticR4 and the failure of the smooth h-cobordism theorem in 4 dimensions. The...
only case where Rn admits a non-standard differential structure: see exoticR4. One could define many norms on the vector space Rn. Some common examples...
phenomena, most strikingly the uncountably infinitely many exotic differentiable structures on R4. Notably, differentiable 4-manifolds is the only remaining...
in R4 with x12 + x22 + x32 + x42 = 1. The Hopf fibration p: S3 → S2 of the 3-sphere over the 2-sphere can be defined in several ways. Identify R4 with...
overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in...
group R3Si−O−CR=CR2, composed of an enolate (R3C−O−R) bonded to a silane (SiR4) through its oxygen end and an ethene group (R2C=CR2) as its carbon end. They...
VFPv3-D16 As above, but with only 16 64-bit FPU registers. Implemented on Cortex-R4 and R5 processors and the Tegra 2 (Cortex-A9). VFPv3-F16 Uncommon; it supports...
wavelength. The electrical power in the fields, respectively, decrease as 1/r4 and 1/r2. See near and far field for a more detailed discussion. See near-field...
September 2020. "P&O PRINCESS CRUISES TO ACQUIRE FORMER RENAISSANCE SHIPS R3 AND R4". "Carnival cruises towards P&O deal". 25 October 2002. Retrieved 30 October...
boiling water reactor, by contrast, has only one coolant loop, while more exotic designs such as breeder reactors use substances other than water for coolant...
Nekrasov, Nikita; Schwarz, Albert (1998). "Instantons on noncommutative R4 and (2,0) superconformal six dimensional theory". Communications in Mathematical...
basal wing cell and basal wing cell), but are without a discoidal wing cell. R4+5 is simple or branched; at most, only three branches of R developed. The...
(mainly German speaking countries). This car was intended to replace the R4 F6 panel van, which had ceased production in 1986. In 1989, the Belgian company...
c , (R3) sin b = sin B sin c , (R8) cos A = sin B cos a , (R4) tan a = tan A sin b , (R9) cos B = sin A cos b , (R5) tan b...
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