. "\u0420\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0435 \u0425\u043E\u043F\u0444\u0430"@ru . . . . . "En la rama de las matem\u00E1ticas denominada topolog\u00EDa, la fibraci\u00F3n de Hopf (tambi\u00E9n denominada el haz de Hopf o mapa de Hopf) describe una 3-esfera (una hiperesfera en el espacio de cuatro dimensiones) mediante circunferencias y una esfera ordinaria. Descubierta en 1931 por Heinz Hopf, es un ejemplo inicial importante de un haz de fibras. T\u00E9cnicamente, Hopf descubri\u00F3 una funci\u00F3n continua (o \"mapa\") de varios a uno de la 3-esfera en la 2-esfera tal que cada punto en particular de la 2-esfera proviene de una circunferencia espec\u00EDfica de la 3-esfera. Por lo tanto la 3-esfera se compone de fibras, donde cada fibra es una circunferencia \u2014 uno para cada punto de la 2-esfera."@es . . "\u970D\u666E\u592B\u7EA4\u7EF4\u5316"@zh . "\uD638\uD504 \uC62C\uBB49\uCE58"@ko . . "\u5728\u62D3\u6251\u5B66\u4E2D\uFF0C\u970D\u666E\u592B\u7E96\u7DAD\u5316\uFF08Hopf fibration\uFF0C\u4EA6\u79F0\u970D\u666E\u592B\u7E96\u7DAD\u4E1B\uFF09\u662F\u6700\u65E9\u63D0\u51FA\u7684\u7EA4\u7EF4\u5316\uFF0C\u5176\u4E2D\u7684\u7EA4\u7EF4\u662F\u5706\u5708\uFF081-\u7403\u9762\uFF0CS1\uFF09\uFF0C\u57FA\u7A7A\u95F4\u662F\u4E09\u7EF4\u7A7A\u95F4\u4E2D\u7684\u7403\u9762\uFF082-\u7403\u9762\uFF0CS2\uFF09\uFF0C\u800C\u5168\u7A7A\u95F4\u662F\u56DB\u7EF4\u7A7A\u95F4\u4E2D\u7684\u8D85\u7403\u9762\uFF083-\u7403\u9762\uFF0CS3\uFF09\u3002\u5BB9\u6613\u9A8C\u8BC1\uFF0C\u5B83\u662F\u975E\u5E73\u51E1\u7684\u3002\u5373\u5168\u7A7A\u95F4S3\u4E0E\u79EF\u7A7A\u95F4S1\u00D7S2\u4E0D\u662F\u62D3\u6251\u540C\u6784\u7684\u3002"@zh . . . "In geometria, la fibrazione di Hopf \u00E8 una particolare mappa dalla sfera tridimensionale a quella bidimensionale, tale che la controimmagine di ogni punto \u00E8 una circonferenza."@it . "\u0420\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0413\u043E\u043F\u0444\u0430"@uk . "Die Hopf-Faserung (nach Heinz Hopf) ist eine bestimmte Abbildung im mathematischen Teilgebiet der Topologie. Es handelt sich um eine Abbildung der 3-Sph\u00E4re, die man sich als den dreidimensionalen Raum zusammen mit einem unendlich fernen Punkt vorstellen kann, in die 2-Sph\u00E4re, also eine Kugeloberfl\u00E4che:"@de . . . . "Fibration de Hopf"@fr . . . . . "580384"^^ . . "p/h047980"@en . . . . "\u0420\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0435 \u0425\u043E\u043F\u0444\u0430 \u2014 \u043F\u0440\u0438\u043C\u0435\u0440 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u043E \u0442\u0440\u0438\u0432\u0438\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u044F \u0442\u0440\u0451\u0445\u043C\u0435\u0440\u043D\u043E\u0439 \u0441\u0444\u0435\u0440\u044B \u043D\u0430\u0434 \u0434\u0432\u0443\u043C\u0435\u0440\u043D\u043E\u0439 \u0441\u043E \u0441\u043B\u043E\u0435\u043C-\u043E\u043A\u0440\u0443\u0436\u043D\u043E\u0441\u0442\u044C\u044E: . \u0420\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0435 \u0425\u043E\u043F\u0444\u0430 \u043D\u0435 \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u0442\u0440\u0438\u0432\u0438\u0430\u043B\u044C\u043D\u044B\u043C. \u042F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u0442\u0430\u043A\u0436\u0435 \u0432\u0430\u0436\u043D\u044B\u043C \u043F\u0440\u0438\u043C\u0435\u0440\u043E\u043C \u0433\u043B\u0430\u0432\u043D\u043E\u0433\u043E \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u044F. \u041E\u0434\u043D\u0438\u043C \u0438\u0437 \u0441\u0430\u043C\u044B\u0445 \u043F\u0440\u043E\u0441\u0442\u044B\u0445 \u0441\u043F\u043E\u0441\u043E\u0431\u043E\u0432 \u0437\u0430\u0434\u0430\u043D\u0438\u044F \u044D\u0442\u043E\u0433\u043E \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u044F \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u043F\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043B\u0435\u043D\u0438\u0435 \u0442\u0440\u0451\u0445\u043C\u0435\u0440\u043D\u043E\u0439 \u0441\u0444\u0435\u0440\u044B \u043A\u0430\u043A \u0435\u0434\u0438\u043D\u0438\u0447\u043D\u043E\u0439 \u0441\u0444\u0435\u0440\u044B \u0432 , \u0430 \u0434\u0432\u0443\u043C\u0435\u0440\u043D\u043E\u0439 \u0441\u0444\u0435\u0440\u044B \u043A\u0430\u043A \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u043E\u0439 \u043F\u0440\u043E\u0435\u043A\u0442\u0438\u0432\u043D\u043E\u0439 \u043F\u0440\u044F\u043C\u043E\u0439 . \u0422\u043E\u0433\u0434\u0430 \u043E\u0442\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u0435: \u0438 \u0437\u0430\u0434\u0430\u0451\u0442 \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0435 \u0425\u043E\u043F\u0444\u0430. \u041F\u0440\u0438 \u044D\u0442\u043E\u043C \u0441\u043B\u043E\u044F\u043C\u0438 \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u044F \u0431\u0443\u0434\u0443\u0442 \u043E\u0440\u0431\u0438\u0442\u044B \u0441\u0432\u043E\u0431\u043E\u0434\u043D\u043E\u0433\u043E \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044F \u0433\u0440\u0443\u043F\u043F\u044B : , \u0433\u0434\u0435 \u043E\u043A\u0440\u0443\u0436\u043D\u043E\u0441\u0442\u044C \u043F\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043B\u0435\u043D\u0430 \u043A\u0430\u043A \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u043E \u0435\u0434\u0438\u043D\u0438\u0447\u043D\u044B\u0445 \u043F\u043E \u043C\u043E\u0434\u0443\u043B\u044E \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u044B\u0445 \u0447\u0438\u0441\u0435\u043B: ."@ru . . . . . "Hopf fibration"@en . . . . . . . . . . . . . . . . . "\u0420\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0425\u043E\u043F\u0444\u0430 \u2014 \u043F\u0440\u0438\u043A\u043B\u0430\u0434 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u043E \u0442\u0440\u0438\u0432\u0456\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0442\u0440\u0438\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0441\u0444\u0435\u0440\u0438 \u043D\u0430\u0434 \u0434\u0432\u043E\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u044E \u0437 \u0448\u0430\u0440\u043E\u043C-\u043A\u043E\u043B\u043E\u043C: . \u0420\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0425\u043E\u043F\u0444\u0430 \u043D\u0435 \u0454 \u0442\u0440\u0438\u0432\u0456\u0430\u043B\u044C\u043D\u0438\u043C. \u0404 \u0442\u0430\u043A\u043E\u0436 \u0432\u0430\u0436\u043B\u0438\u0432\u0438\u043C \u043F\u0440\u0438\u043A\u043B\u0430\u0434\u043E\u043C \u0433\u043E\u043B\u043E\u0432\u043D\u043E\u0433\u043E \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F. \u041E\u0434\u043D\u0438\u043C \u0437 \u043D\u0430\u0439\u043F\u0440\u043E\u0441\u0442\u0456\u0448\u0438\u0445 \u0441\u043F\u043E\u0441\u043E\u0431\u0456\u0432 \u0437\u0430\u0432\u0434\u0430\u043D\u043D\u044F \u0446\u044C\u043E\u0433\u043E \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0454 \u043F\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043B\u0435\u043D\u043D\u044F \u0442\u0440\u0438\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0441\u0444\u0435\u0440\u0438 \u044F\u043A \u043E\u0434\u0438\u043D\u0438\u0447\u043D\u043E\u0457 \u0441\u0444\u0435\u0440\u0438 \u0432 , \u0430 \u0434\u0432\u043E\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0441\u0444\u0435\u0440\u0438 \u044F\u043A \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u043E\u0457 \u043F\u0440\u043E\u0454\u043A\u0442\u0438\u0432\u043D\u043E\u0457 \u043F\u0440\u044F\u043C\u043E\u0457 . \u0422\u043E\u0434\u0456 \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F: \u0456 \u0437\u0430\u0434\u0430\u0454 \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0425\u043E\u043F\u0444\u0430. \u041F\u0440\u0438 \u0446\u044C\u043E\u043C\u0443 \u0448\u0430\u0440\u0430\u043C\u0438 \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0431\u0443\u0434\u0443\u0442\u044C \u043E\u0440\u0431\u0456\u0442\u0438 \u0432\u0456\u043B\u044C\u043D\u043E\u0457 \u0434\u0456\u0457 \u0433\u0440\u0443\u043F\u0438 : , \u0434\u0435 \u043A\u043E\u043B\u043E \u043F\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043B\u0435\u043D\u0430 \u044F\u043A \u043C\u043D\u043E\u0436\u0438\u043D\u0430 \u043E\u0434\u0438\u043D\u0438\u0447\u043D\u0438\u0445 \u0437\u0430 \u043C\u043E\u0434\u0443\u043B\u0435\u043C \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u0438\u0445 \u0447\u0438\u0441\u0435\u043B: ."@uk . . . . . . "Fibra\u00E7\u00E3o de Hopf"@pt . . . . "\uC704\uC0C1\uC218\uD559\uC5D0\uC11C \uD638\uD504 \uC62C\uBB49\uCE58(\uC601\uC5B4: Hopf fibration)\uB294 \uAD6C\uAC00 \uB2E4\uB978 \uCC28\uC6D0\uC758 \uAD6C \uC704\uC758 \uC62C\uB2E4\uBC1C\uC744 \uC774\uB8E8\uB294 \uD604\uC0C1\uC774\uB2E4. \uAC00\uC7A5 \uB300\uD45C\uC801\uC778 \uACBD\uC6B0\uB294 3\uCC28\uC6D0 \uAD6C\uAC00 2\uCC28\uC6D0 \uAD6C \uC704\uC5D0 \uB2E4\uBC1C\uC744 \uC774\uB8E8\uB294 \uACBD\uC6B0\uBA70, \uC720\uC0AC\uD558\uAC8C 7\uCC28\uC6D0 \uAD6C\uAC00 4\uCC28\uC6D0 \uAD6C \uC704\uC5D0, 15\uCC28\uC6D0 \uAD6C\uAC00 8\uCC28\uC6D0 \uAD6C \uC704\uC5D0 \uC62C\uB2E4\uBC1C\uC744 \uC774\uB8EC\uB2E4."@ko . . "\u0420\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0425\u043E\u043F\u0444\u0430 \u2014 \u043F\u0440\u0438\u043A\u043B\u0430\u0434 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u043E \u0442\u0440\u0438\u0432\u0456\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0442\u0440\u0438\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0441\u0444\u0435\u0440\u0438 \u043D\u0430\u0434 \u0434\u0432\u043E\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u044E \u0437 \u0448\u0430\u0440\u043E\u043C-\u043A\u043E\u043B\u043E\u043C: . \u0420\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0425\u043E\u043F\u0444\u0430 \u043D\u0435 \u0454 \u0442\u0440\u0438\u0432\u0456\u0430\u043B\u044C\u043D\u0438\u043C. \u0404 \u0442\u0430\u043A\u043E\u0436 \u0432\u0430\u0436\u043B\u0438\u0432\u0438\u043C \u043F\u0440\u0438\u043A\u043B\u0430\u0434\u043E\u043C \u0433\u043E\u043B\u043E\u0432\u043D\u043E\u0433\u043E \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F. \u041E\u0434\u043D\u0438\u043C \u0437 \u043D\u0430\u0439\u043F\u0440\u043E\u0441\u0442\u0456\u0448\u0438\u0445 \u0441\u043F\u043E\u0441\u043E\u0431\u0456\u0432 \u0437\u0430\u0432\u0434\u0430\u043D\u043D\u044F \u0446\u044C\u043E\u0433\u043E \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0454 \u043F\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043B\u0435\u043D\u043D\u044F \u0442\u0440\u0438\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0441\u0444\u0435\u0440\u0438 \u044F\u043A \u043E\u0434\u0438\u043D\u0438\u0447\u043D\u043E\u0457 \u0441\u0444\u0435\u0440\u0438 \u0432 , \u0430 \u0434\u0432\u043E\u0432\u0438\u043C\u0456\u0440\u043D\u043E\u0457 \u0441\u0444\u0435\u0440\u0438 \u044F\u043A \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u043E\u0457 \u043F\u0440\u043E\u0454\u043A\u0442\u0438\u0432\u043D\u043E\u0457 \u043F\u0440\u044F\u043C\u043E\u0457 . \u0422\u043E\u0434\u0456 \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F: \u0456 \u0437\u0430\u0434\u0430\u0454 \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0425\u043E\u043F\u0444\u0430. \u041F\u0440\u0438 \u0446\u044C\u043E\u043C\u0443 \u0448\u0430\u0440\u0430\u043C\u0438 \u0440\u043E\u0437\u0448\u0430\u0440\u0443\u0432\u0430\u043D\u043D\u044F \u0431\u0443\u0434\u0443\u0442\u044C \u043E\u0440\u0431\u0456\u0442\u0438 \u0432\u0456\u043B\u044C\u043D\u043E\u0457 \u0434\u0456\u0457 \u0433\u0440\u0443\u043F\u0438 : , \u0434\u0435 \u043A\u043E\u043B\u043E \u043F\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043B\u0435\u043D\u0430 \u044F\u043A \u043C\u043D\u043E\u0436\u0438\u043D\u0430 \u043E\u0434\u0438\u043D\u0438\u0447\u043D\u0438\u0445 \u0437\u0430 \u043C\u043E\u0434\u0443\u043B\u0435\u043C \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u0438\u0445 \u0447\u0438\u0441\u0435\u043B: ."@uk . . . "Hopf fibration"@en . . . . . . . . . . . "1113345027"^^ . "\u0627\u0647\u062A\u0632\u0627\u0632 \u0647\u0648\u0628\u0641 \u0641\u064A \u0627\u0644\u0645\u062C\u0627\u0644 \u0627\u0644\u0631\u064A\u0627\u0636\u064A \u0644\u0644\u0637\u0648\u0628\u0648\u0644\u0648\u062C\u064A\u0627 \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629\u060C \u064A\u0635\u0641 \u0627\u0647\u062A\u0632\u0627\u0632 \u0647\u0648\u0628\u0641 (\u0627\u0644\u0645\u0639\u0631\u0648\u0641 \u0623\u064A\u0636\u064B\u0627 \u0628\u0639\u062F\u0629 \u0623\u0633\u0645\u0627\u0621 \u0648\u0647\u064A \u062D\u0632\u0645\u0629 \u0647\u0648\u0628\u0641 \u0623\u0648 \u062E\u0631\u064A\u0637\u0629 \u0647\u0648\u0628\u0641) \u0643\u0631\u0629 \u062B\u0644\u0627\u062B\u064A\u0629 (\u0643\u0631\u0629 \u0632\u0627\u0626\u062F\u0629 \u0641\u064A \u0641\u0636\u0627\u0621 \u0631\u0628\u0627\u0639\u064A \u0627\u0644\u0623\u0628\u0639\u0627\u062F) \u0645\u0646 \u062D\u064A\u062B \u0627\u0644\u062F\u0648\u0627\u0626\u0631 \u0648\u0627\u0644\u0645\u062C\u0627\u0644 \u0627\u0644\u0639\u0627\u062F\u064A."@ar . . . . . . "In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function (or \"map\") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped from a distinct great circle of the 3-sphere. Thus the 3-sphere is composed of fibers, where each fiber is a circle \u2014 one for each point of the 2-sphere. This fiber bundle structure is denoted meaning that the fiber space S1 (a circle) is embedded in the total space S3 (the 3-sphere), and p : S3 \u2192 S2 (Hopf's map) projects S3 onto the base space S2 (the ordinary 2-sphere). The Hopf fibration, like any fiber bundle, has the important property that it is locally a product space. However it is not a trivial fiber bundle, i.e., S3 is not globally a product of S2 and S1 although locally it is indistinguishable from it. This has many implications: for example the existence of this bundle shows that the higher homotopy groups of spheres are not trivial in general. It also provides a basic example of a principal bundle, by identifying the fiber with the circle group. Stereographic projection of the Hopf fibration induces a remarkable structure on R3, in which all of 3-dimensional space, except for the z-axis, is filled with nested tori made of linking Villarceau circles. Here each fiber projects to a circle in space (one of which is a line, thought of as a \"circle through infinity\"). Each torus is the stereographic projection of the inverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product of two circles.) These tori are illustrated in the images at right. When R3 is compressed to the boundary of a ball, some geometric structure is lost although the topological structure is retained (see Topology and geometry). The loops are homeomorphic to circles, although they are not geometric circles. There are numerous generalizations of the Hopf fibration. The unit sphere in complex coordinate space Cn+1 fibers naturally over the complex projective space CPn with circles as fibers, and there are also real, quaternionic, and octonionic versions of these fibrations. In particular, the Hopf fibration belongs to a family of four fiber bundles in which the total space, base space, and fiber space are all spheres: By Adams's theorem such fibrations can occur only in these dimensions. The Hopf fibration is important in twistor theory."@en . . . . "En g\u00E9om\u00E9trie la fibration de Hopf donne une partition de la sph\u00E8re \u00E0 3-dimensions S3 par des grands cercles. Plus pr\u00E9cis\u00E9ment, elle d\u00E9finit une structure fibr\u00E9e sur S3. L'espace de base est la sph\u00E8re \u00E0 2-dimensions S2, la fibre mod\u00E8le est un cercle S1. Ceci signifie notamment qu'il existe une application p de projection de S3 sur S2, telle que les images r\u00E9ciproques de chaque point de S2 soient des cercles. Cette structure a \u00E9t\u00E9 d\u00E9couverte par Heinz Hopf en 1931. Cette fibration peut aussi \u00EAtre interpr\u00E9t\u00E9e comme un fibr\u00E9 principal, dont le groupe structural est le groupe S1 des complexes de module 1."@fr . . . . . . . "In geometria, la fibrazione di Hopf \u00E8 una particolare mappa dalla sfera tridimensionale a quella bidimensionale, tale che la controimmagine di ogni punto \u00E8 una circonferenza."@it . "No campo matem\u00E1tico da topologia, a fibra\u00E7\u00E3o de Hopf (tamb\u00E9m conhecida como fibrado de Hopf ou mapa de Hopf) descreve uma 3-esfera (uma hiperesfera no espa\u00E7o quadri-dimensional) em termos de c\u00EDrculos e uma esfera ordin\u00E1ria. Descoberto por Heinz Hopf em 1931, \u00E9 um exemplo primordial influente de fibrado de linhas. Tecnicamente, Hopf encontrou uma fun\u00E7\u00E3o cont\u00EDnua (ou \"mapa\") de \"muitos para um\" da 3-esfera para a 2-esfera tal que cada ponto distinto da 2-esfera torna-se um c\u00EDrculo distinto da 3-esfera. Assim a 3-esfera \u00E9 composta de \"fibras\", onde cada \"fibra\" \u00E9 um c\u00EDrculo \u2014 um para cada ponto da 2-esfera. Esta estrutura de fibras \u00E9 denotada o que quer dizer que a fibra S\u00B9 (um c\u00EDrculo) est\u00E1 imersa no espa\u00E7o total S\u00B3 (a 3-esfera), e p: S\u00B3\u2192S\u00B2 (mapa de Hopf) projeta S\u00B3 na base S\u00B2 (a 2-esfera ordin\u00E1ria). A fibra\u00E7\u00E3o de Hopf, como qualquer fibrado, tem a propriedade de ser um espa\u00E7o produto. Todavia este n\u00E3o \u00E9 um fibrado trivial, i.e., S\u00B3 n\u00E3o \u00E9 (globalmente) o produto de S\u00B2 e S\u00B9. Isto apresenta algumas implica\u00E7\u00F5es: por exemplo, a exist\u00EAncia deste fibrado mostra que os mais altos n\u00E3o s\u00E3o triviais em geral. Tamb\u00E9m prov\u00EA um exemplo b\u00E1sico de fibrado principal pela identifica\u00E7\u00E3o da fibra com o . A proje\u00E7\u00E3o estereogr\u00E1fica do fibrado de Hopf induz a uma estrutura marcante em R\u00B3, na qual o espa\u00E7o \u00E9 preenchido com toros aninhados feitos de interligados. Aqui cada fibra \u00E9 projetada num c\u00EDrculo no espa\u00E7o (um dos quais \u00E9 um \"c\u00EDrculo passando pelo infinito\" \u2014 uma reta). Cada toro \u00E9 a proje\u00E7\u00E3o estereogr\u00E1fica da imagem inversa de um c\u00EDrculo de latitude da 2-esfera. (Topologicamente, um toro \u00E9 o produto de dois c\u00EDrculos.) Estes toros s\u00E3o ilustrados pelas imagens \u00E0 direita. Quando o R3 \u00E9 comprimido numa bola, sua estrutura geom\u00E9trica \u00E9 perdida mas a estrutura topol\u00F3gica se mant\u00E9m (ver Topologia e Geometria). Os la\u00E7os s\u00E3o a c\u00EDrculos, apesar de n\u00E3o serem c\u00EDrculos geom\u00E9tricos. H\u00E1 in\u00FAmeras generaliza\u00E7\u00F5es da fibra\u00E7\u00E3o de Hopf. A esfera unit\u00E1ria em Cn+1 projeta-se naturalmente em CPn tendo c\u00EDrculos como fibras, e h\u00E1 tamb\u00E9m vers\u00F5es reais, em e em destas fibra\u00E7\u00F5es. Em particular, a fibra\u00E7\u00E3o de Hopf pertence a uma fam\u00EDlia de quatro fibrados cujo espa\u00E7o total, a base e a fibra s\u00E3o todos esferas: Pelo , tais fibra\u00E7\u00F5es podem ocorrer apenas nestas dimens\u00F5es. A fibra\u00E7\u00E3o de Hopf \u00E9 importante na Teoria dos twistores."@pt . . . . . . . . . . . . . . . . . "No campo matem\u00E1tico da topologia, a fibra\u00E7\u00E3o de Hopf (tamb\u00E9m conhecida como fibrado de Hopf ou mapa de Hopf) descreve uma 3-esfera (uma hiperesfera no espa\u00E7o quadri-dimensional) em termos de c\u00EDrculos e uma esfera ordin\u00E1ria. Descoberto por Heinz Hopf em 1931, \u00E9 um exemplo primordial influente de fibrado de linhas. Tecnicamente, Hopf encontrou uma fun\u00E7\u00E3o cont\u00EDnua (ou \"mapa\") de \"muitos para um\" da 3-esfera para a 2-esfera tal que cada ponto distinto da 2-esfera torna-se um c\u00EDrculo distinto da 3-esfera. Assim a 3-esfera \u00E9 composta de \"fibras\", onde cada \"fibra\" \u00E9 um c\u00EDrculo \u2014 um para cada ponto da 2-esfera."@pt . . . . . "Fibrazione di Hopf"@it . . . . . . . . . . . "In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function (or \"map\") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped from a distinct great circle of the 3-sphere. Thus the 3-sphere is composed of fibers, where each fiber is a circle \u2014 one for each point of the 2-sphere."@en . "En la rama de las matem\u00E1ticas denominada topolog\u00EDa, la fibraci\u00F3n de Hopf (tambi\u00E9n denominada el haz de Hopf o mapa de Hopf) describe una 3-esfera (una hiperesfera en el espacio de cuatro dimensiones) mediante circunferencias y una esfera ordinaria. Descubierta en 1931 por Heinz Hopf, es un ejemplo inicial importante de un haz de fibras. T\u00E9cnicamente, Hopf descubri\u00F3 una funci\u00F3n continua (o \"mapa\") de varios a uno de la 3-esfera en la 2-esfera tal que cada punto en particular de la 2-esfera proviene de una circunferencia espec\u00EDfica de la 3-esfera. Por lo tanto la 3-esfera se compone de fibras, donde cada fibra es una circunferencia \u2014 uno para cada punto de la 2-esfera. Esta estructura de haz de fibras queda expresada mediante la expresi\u00F3n que significa que el espacio de fibra S1 (un c\u00EDrculo) se encuentra encajado en el espacio total S3 (la 3-esfera), y p: S3\u2192S2 (Mapa de Hopf) proyecta S3 en el espacio base S2 (la 2-esfera ordinaria). La fibraci\u00F3n de Hopf, al igual que todo haz de fibras, posee la propiedad que es un producto espacial local. Sin embargo es un haz de fibras no trivial, o sea S3 no es en sentido global un producto de S2 y S1 aunque a nivel local es indistinguible de este.Existen numerosas generalizaciones de la fibraci\u00F3n de Hopf. La esfera unidad en Cn+1 se fibra naturalmente en CPn con circunferencias como fibras, existen tambi\u00E9n versiones de estas fibraciones reales, cuaterni\u00F3nicas, y octoni\u00F3nicas. En particular, la fibraci\u00F3nn de Hopf corresponde a una familia de cuatro haces de fibras en los cuales el espacio total, el espacio base, y el espacio fibra son todos esferas: Seg\u00FAn establece el estas fibraciones solo pueden presentarse en estas dimensiones. La fibraci\u00F3n de Hopf es importante en el \u00E1mbito de la teor\u00EDa de twistores."@es . . . . . "Fibraci\u00F3n de Hopf"@es . "35431"^^ . . . . . "\u0627\u0647\u062A\u0632\u0627\u0632 \u0647\u0648\u0628\u0641"@ar . . . . . . . . . "\u0627\u0647\u062A\u0632\u0627\u0632 \u0647\u0648\u0628\u0641 \u0641\u064A \u0627\u0644\u0645\u062C\u0627\u0644 \u0627\u0644\u0631\u064A\u0627\u0636\u064A \u0644\u0644\u0637\u0648\u0628\u0648\u0644\u0648\u062C\u064A\u0627 \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629\u060C \u064A\u0635\u0641 \u0627\u0647\u062A\u0632\u0627\u0632 \u0647\u0648\u0628\u0641 (\u0627\u0644\u0645\u0639\u0631\u0648\u0641 \u0623\u064A\u0636\u064B\u0627 \u0628\u0639\u062F\u0629 \u0623\u0633\u0645\u0627\u0621 \u0648\u0647\u064A \u062D\u0632\u0645\u0629 \u0647\u0648\u0628\u0641 \u0623\u0648 \u062E\u0631\u064A\u0637\u0629 \u0647\u0648\u0628\u0641) \u0643\u0631\u0629 \u062B\u0644\u0627\u062B\u064A\u0629 (\u0643\u0631\u0629 \u0632\u0627\u0626\u062F\u0629 \u0641\u064A \u0641\u0636\u0627\u0621 \u0631\u0628\u0627\u0639\u064A \u0627\u0644\u0623\u0628\u0639\u0627\u062F) \u0645\u0646 \u062D\u064A\u062B \u0627\u0644\u062F\u0648\u0627\u0626\u0631 \u0648\u0627\u0644\u0645\u062C\u0627\u0644 \u0627\u0644\u0639\u0627\u062F\u064A."@ar . . . . "\u5728\u62D3\u6251\u5B66\u4E2D\uFF0C\u970D\u666E\u592B\u7E96\u7DAD\u5316\uFF08Hopf fibration\uFF0C\u4EA6\u79F0\u970D\u666E\u592B\u7E96\u7DAD\u4E1B\uFF09\u662F\u6700\u65E9\u63D0\u51FA\u7684\u7EA4\u7EF4\u5316\uFF0C\u5176\u4E2D\u7684\u7EA4\u7EF4\u662F\u5706\u5708\uFF081-\u7403\u9762\uFF0CS1\uFF09\uFF0C\u57FA\u7A7A\u95F4\u662F\u4E09\u7EF4\u7A7A\u95F4\u4E2D\u7684\u7403\u9762\uFF082-\u7403\u9762\uFF0CS2\uFF09\uFF0C\u800C\u5168\u7A7A\u95F4\u662F\u56DB\u7EF4\u7A7A\u95F4\u4E2D\u7684\u8D85\u7403\u9762\uFF083-\u7403\u9762\uFF0CS3\uFF09\u3002\u5BB9\u6613\u9A8C\u8BC1\uFF0C\u5B83\u662F\u975E\u5E73\u51E1\u7684\u3002\u5373\u5168\u7A7A\u95F4S3\u4E0E\u79EF\u7A7A\u95F4S1\u00D7S2\u4E0D\u662F\u62D3\u6251\u540C\u6784\u7684\u3002"@zh . . . . . . . . . "\uC704\uC0C1\uC218\uD559\uC5D0\uC11C \uD638\uD504 \uC62C\uBB49\uCE58(\uC601\uC5B4: Hopf fibration)\uB294 \uAD6C\uAC00 \uB2E4\uB978 \uCC28\uC6D0\uC758 \uAD6C \uC704\uC758 \uC62C\uB2E4\uBC1C\uC744 \uC774\uB8E8\uB294 \uD604\uC0C1\uC774\uB2E4. \uAC00\uC7A5 \uB300\uD45C\uC801\uC778 \uACBD\uC6B0\uB294 3\uCC28\uC6D0 \uAD6C\uAC00 2\uCC28\uC6D0 \uAD6C \uC704\uC5D0 \uB2E4\uBC1C\uC744 \uC774\uB8E8\uB294 \uACBD\uC6B0\uBA70, \uC720\uC0AC\uD558\uAC8C 7\uCC28\uC6D0 \uAD6C\uAC00 4\uCC28\uC6D0 \uAD6C \uC704\uC5D0, 15\uCC28\uC6D0 \uAD6C\uAC00 8\uCC28\uC6D0 \uAD6C \uC704\uC5D0 \uC62C\uB2E4\uBC1C\uC744 \uC774\uB8EC\uB2E4."@ko . . . . 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"Hopf-Faserung"@de . "Die Hopf-Faserung (nach Heinz Hopf) ist eine bestimmte Abbildung im mathematischen Teilgebiet der Topologie. Es handelt sich um eine Abbildung der 3-Sph\u00E4re, die man sich als den dreidimensionalen Raum zusammen mit einem unendlich fernen Punkt vorstellen kann, in die 2-Sph\u00E4re, also eine Kugeloberfl\u00E4che:"@de . . . . . . . . . . . . "En g\u00E9om\u00E9trie la fibration de Hopf donne une partition de la sph\u00E8re \u00E0 3-dimensions S3 par des grands cercles. Plus pr\u00E9cis\u00E9ment, elle d\u00E9finit une structure fibr\u00E9e sur S3. L'espace de base est la sph\u00E8re \u00E0 2-dimensions S2, la fibre mod\u00E8le est un cercle S1. Ceci signifie notamment qu'il existe une application p de projection de S3 sur S2, telle que les images r\u00E9ciproques de chaque point de S2 soient des cercles."@fr . . . . . . . .