- Exotic R4
In

mathematics , an**exotic****R**^{4}is adifferentiable manifold that ishomeomorphic to theEuclidean space **R**^{4}, but not diffeomorphic. The first examples were found byRobion Kirby andMichael Freedman , by using the contrast between Freedman's theorems about topological 4-manifolds, andSimon Donaldson 's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphicdifferentiable structure s of**R**^{4}, as was shown first byClifford Taubes .Prior to this construction, non-diffeomorphic smooth structures on spheres —

exotic sphere s — were already known to exist, although the question of the existence of such structures for the particular case of the4-sphere remains open. For any positive integer "n" other than 4, there are no exotic smooth structures on**R**^{"n"}; in other words, if "n" ≠ 4 then any smooth manifold homeomorphic to**R**^{"n"}is diffeomorphic to**R**^{"n"}.**mall exotic****R**^{4}sAn exotic

**R**^{4}is called**small**if it can be smoothly embedded as an open subset of the standard**R**^{4}.Small exotic

**R**^{4}s can be constructed by starting with a non-trivial smooth 5-dimensional "h"-cobordism (which exists by Donaldson's proof that the "h"-cobordism theorem fails in this dimension) and using Freedman's theorem that the topological h-cobordism theorem holds in this dimension.**Large exotic****R**^{4}sAn exotic

**R**^{4}is called**large**if it cannot be smoothly embedded as an open subset of the standard**R**^{4}.Examples of large exotic

**R**^{4}s can be constructed using the fact that compact 4 manifolds can often be split as a topological sum (by Freedman's work), but cannot be split as a smooth sum (by Donaldson's work).There is at least one maximal exotic

**R**^{4}, into which all other**R**^{4}s can be smoothly embedded as open subsets.**Related exotic structures**Casson handle s are homeomorphic to**D**^{2}×**R**^{2}by Freedman's theorem (where**D**^{2}is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to**D**^{2}×**R**^{2}. In other words, some Casson handles are exotic**D**^{2}×**R**^{2}s.It is not known (as of 2006) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counter example to the smooth

Poincaré conjecture in dimension 4. Some plausible candidates are given byGluck twist s.**References***Alexandru Scorpan, "The wild world of 4-manifolds", ISBN 978-0-8218-3749-8

*Robert Gompf and Andras Stipsicz, "4-Manifolds and Kirby Calculus",Graduate Studies in Mathematics , vol. 20,American Mathematical Society , Providence, RI, 1999. ISBN 0-8218-0994-6

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**Exotic**— can mean:*In mathematics: **Exotic R4 differentiable manifold homeomorphic but not diffeomorphic to the Euclidean space R4 **Exotic sphere differentiable manifold homeomorphic but not diffeomorphic to the ordinary sphere*In physics: **Exotic atom … Wikipedia**Exotic**— Ex*ot ic, a. [L. exoticus, Gr. ? fr. e xw outside: cf. F. exotique. See {Exoteric}.] Introduced from a foreign country; not native; extraneous; foreign; as, an exotic plant; an exotic term or word. [1913 Webster] Nothing was so splendid and… … The Collaborative International Dictionary of English**exotic**— EXÓTIC, Ă, exotici, ce, adj. Care se află într o regiune foarte îndepărtată şi care impresionează prin aspecte neobişnuite, ciudate; care provine dintr o asemenea regiune, care aparţine, care este propriu acestor ţări. [pr.: eg zo ] – Din fr.… … Dicționar Român**exotic**— 1590s, belonging to another country, from M.Fr. exotique (16c.) and directly from L. exoticus, from Gk. exotikos foreign, lit. from the outside, from exo outside (see EXO (Cf. exo )). Sense of unusual, strange first recorded in English 1620s,… … Etymology dictionary**exotic**— ► ADJECTIVE 1) originating in or characteristic of a distant foreign country. 2) strikingly colourful or unusual. ► NOUN ▪ an exotic plant or animal. DERIVATIVES exotically adverb exoticism noun. ORIGIN … English terms dictionary**exotic**— [eg zät′ik, igzät′ik] adj. [L exoticus < Gr exōtikos < exō, outside: see EX 1] 1. foreign; not native 2. strange or different in a way that is striking or fascinating; strangely beautiful, enticing, etc. n. 1. a foreign or imported thing 2 … English World dictionary**Exotic**— Ex*ot ic, n. Anything of foreign origin; something not of native growth, as a plant, a word, a custom. [1913 Webster] Plants that are unknown to Italy, and such as the gardeners call exotics. Addison. [1913 Webster] … The Collaborative International Dictionary of English**exotic**— index nonconforming, unaccustomed, uncommon Burton s Legal Thesaurus. William C. Burton. 2006 … Law dictionary**exotic**— [adj] not native or usual; mysterious alien, alluring, avant garde, bizarre, colorful, curious, different, enticing, external, extraneous, extraordinary, extrinsic, far out*, fascinating, foreign, glamorous, imported, introduced, kinky*,… … New thesaurus**exotic**— 01. The pattern on the wallpaper in the kids room is of a tropical jungle with [exotic] birds, plants and animals. 02. Our green tea has an [exotic], refreshing, fragrant flavor. 03. Native plants are disappearing as a result of the introduction… … Grammatical examples in English