{"id":28,"date":"2019-07-28T21:46:48","date_gmt":"2019-07-28T18:46:48","guid":{"rendered":"https:\/\/philokatholos.getway.org\/?p=28"},"modified":"2019-07-28T21:55:26","modified_gmt":"2019-07-28T18:55:26","slug":"classification-of-dimensions-for-the-non-existence-of-exotic-spheres","status":"publish","type":"post","link":"https:\/\/philokatholos.getway.org\/?p=28","title":{"rendered":"Classification of dimensions for the non-existence of exotic spheres"},"content":{"rendered":"\n<p>It turns out that the only dimensions $n$ for which a sphere $S^n$ admits a unique differentiable structure are exactly $n=1, 2, 3, 5, 6, 12, 56, 61$.<\/p>\n\n\n\n<p>In all other dimensions, except $n=4$, we get the existence of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Exotic_sphere\">exotic spheres<\/a>.  <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img loading=\"lazy\" width=\"960\" height=\"896\" src=\"https:\/\/philokatholos.getway.org\/wp-content\/uploads\/2019\/07\/Vilhelm-Hammershoi-The-farm-1883.jpg\" alt=\"\" class=\"wp-image-32\" srcset=\"https:\/\/philokatholos.getway.org\/wp-content\/uploads\/2019\/07\/Vilhelm-Hammershoi-The-farm-1883.jpg 960w, https:\/\/philokatholos.getway.org\/wp-content\/uploads\/2019\/07\/Vilhelm-Hammershoi-The-farm-1883-300x280.jpg 300w, https:\/\/philokatholos.getway.org\/wp-content\/uploads\/2019\/07\/Vilhelm-Hammershoi-The-farm-1883-768x717.jpg 768w\" sizes=\"(max-width: 767px) 89vw, (max-width: 1000px) 54vw, (max-width: 1071px) 543px, 580px\" \/><figcaption>Vilhelm Hammershoi The farm, 1883<\/figcaption><\/figure><\/div>\n\n\n\n<p>What happens at dimension $4$ is still largely <em>unknown<\/em> and it goes by the name of <a href=\"https:\/\/arxiv.org\/abs\/0906.5177\">smooth Poincare conjecture<\/a>, the only open case of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Generalized_Poincar%C3%A9_conjecture\">generalized Poincare conjecture<\/a>. <\/p>\n","protected":false},"excerpt":{"rendered":"<p>It turns out that the only dimensions $n$ for which a sphere $S^n$ admits a unique differentiable structure are exactly $n=1, 2, 3, 5, 6, 12, 56, 61$. In all other dimensions, except $n=4$, we get the existence of exotic spheres. What happens at dimension $4$ is still largely unknown and it goes by the &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/philokatholos.getway.org\/?p=28\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Classification of dimensions for the non-existence of exotic spheres&#8221;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/posts\/28"}],"collection":[{"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=28"}],"version-history":[{"count":4,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/posts\/28\/revisions"}],"predecessor-version":[{"id":34,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=\/wp\/v2\/posts\/28\/revisions\/34"}],"wp:attachment":[{"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=28"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=28"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/philokatholos.getway.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=28"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}