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Exotic R4
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<h2 id="small-exotic-r4s">Small exotic R4s</h2> <p>An exotic R 4 {\displaystyle \mathbb {R} ^{4}} is called small if it can be smoothly embedded as an open subset of the standard R 4 . {\displaystyle \mathbb {R} ^{4}.} </p><p>Small exotic R 4 {\displaystyle \mathbb {R} ^{4}} can be constructed by starting with a non-trivial smooth 5-dimensional <i>h</i>-<a href="/facts/Cobordism/ctN2FSTB">cobordism</a> (which exists by Donaldson's proof that the <a href="/facts/H-cobordism/gh4z1VVW"><i>h</i>-cobordism theorem</a> fails in this dimension) and using Freedman's theorem that the topological <i>h</i>-cobordism theorem holds in this dimension. </p> <h2 id="large-exotic-r4s">Large exotic R4s</h2> <p>An exotic R 4 {\displaystyle \mathbb {R} ^{4}} is called large if it cannot be smoothly embedded as an open subset of the standard R 4 . {\displaystyle \mathbb {R} ^{4}.} </p><p>Examples of large exotic R 4 {\displaystyle \mathbb {R} ^{4}} 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). </p><p>Michael Hartley Freedman and Laurence R. Taylor (1986) showed that there is a maximal exotic R 4 , {\displaystyle \mathbb {R} ^{4},} into which all other R 4 {\displaystyle \mathbb {R} ^{4}} can be smoothly embedded as open subsets. </p> <h2 id="related-exotic-structures">Related exotic structures</h2> <p><a href="/facts/Casson_handle/pJVm0gKu">Casson handles</a> are homeomorphic to D 2 × R 2 {\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}} by Freedman's theorem (where D 2 {\displaystyle \mathbb {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 . {\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}.} In other words, some Casson handles are exotic D 2 × R 2 . {\displaystyle \mathbb {D} ^{2}\times \mathbb {R} ^{2}.} </p><p>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 <a href="/facts/Generalized_Poincar%C3%A9_conjecture/9wvLDqTq">generalized Poincaré conjecture</a> in dimension 4. Some plausible candidates are given by <a href="/facts/Exotic_sphere/3YKNwX1I">Gluck twists</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Akbulut_cork/77u9y6Tw">Akbulut cork</a> - tool used to construct exotic R 4 {\displaystyle \mathbb {R} ^{4}} 's from classes in H 3 ( S 3 , R ) {\displaystyle H^{3}(S^{3},\mathbb {R} )} <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li><a href="/facts/Atlas_(topology)/pjoya29t">Atlas (topology)</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Michael_Freedman/FxESCuA7">Freedman, Michael H.</a>; <a href="/facts/Frank_Quinn_(mathematician)/Q6aLxgF2">Quinn, Frank</a> (1990). <a href="https://archive.org/details/topologyof4manif0000free"><i>Topology of 4-manifolds</i></a>. Princeton Mathematical Series. Vol. 39. Princeton, NJ: Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-08577-3.</li> <li><a href="/facts/Michael_Freedman/FxESCuA7">Freedman, Michael H.</a>; Taylor, Laurence R. (1986). <a href="https://projecteuclid.org/euclid.jdg/1214440258">"A universal smoothing of four-space"</a>. <i>Journal of Differential Geometry</i>. 24 (1): 69–78. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4310%2Fjdg%2F1214440258">10.4310/jdg/1214440258</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0022-040X">0022-040X</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0857376">0857376</a>.</li> <li><a href="/facts/Robion_Kirby/QMxyDhtv">Kirby, Robion C.</a> (1989). <i>The topology of 4-manifolds</i>. Lecture Notes in Mathematics. Vol. 1374. Berlin: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-51148-2.</li> <li>Scorpan, Alexandru (2005). <i>The wild world of 4-manifolds</i>. Providence, RI: American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3749-8.</li> <li><a href="/facts/John_R._Stallings/GRIr0QcQ">Stallings, John</a> (1962). <a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2053736&fulltextType=RA&fileId=S0305004100036756">"The piecewise-linear structure of Euclidean space"</a>. <i><a href="/facts/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society/KUUzcS7S">Mathematical Proceedings of the Cambridge Philosophical Society</a></i>. 58 (3): 481–488. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1962PCPS...58..481S">1962PCPS...58..481S</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2Fs0305004100036756">10.1017/s0305004100036756</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120418488">120418488</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0149457">0149457</a></li> <li><a href="/facts/Robert_Gompf/Io8S08RT">Gompf, Robert E.</a>; Stipsicz, András I. (1999). <i>4-manifolds and Kirby calculus</i>. <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>. Vol. 20. Providence, RI: American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0994-6.</li> <li><a href="/facts/Clifford_Henry_Taubes/fB11uJb7">Taubes, Clifford Henry</a> (1987). <a href="http://projecteuclid.org/euclid.jdg/1214440981">"Gauge theory on asymptotically periodic 4-manifolds"</a>. <i>Journal of Differential Geometry</i>. 25 (3): 363–430. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4310%2Fjdg%2F1214440981">10.4310/jdg/1214440981</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0882829">0882829</a>. <a href="/facts/Project_Euclid/UXdRpPcW">PE</a> <a href="http://projecteuclid.org/euclid.jdg/1214440981">1214440981</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Kirby (1989), p. 95 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Freedman and Quinn (1990), p. 122 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Taubes (1987), Theorem 1.1 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Stallings (1962), in particular Corollary 5.2 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Asselmeyer-Maluga, Torsten; Król, Jerzy (2014-08-28). "Abelian gerbes, generalized geometries and foliations of small exotic R^4". arXiv:0904.1276 [hep-th]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>