See fundamental group in All languages combined, or Wiktionary
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Massey, A Basic Course in Algebraic Topology, Springer, page 35:", "text": "From the definition it will be clear the group is a topological invariant of X; i.e., if two spaces are homeomorphic, their fundamental groups are isomorphic.", "type": "quote" }, { "ref": "2011, John Coates, Minhyong Kim, Florian Pop, Mohamed Saïdi, Peter Schneider, editors, Non-abelian Fundamental Groups and Iwasawa Theory, Cambridge University Press, page vii:", "text": "Therein came into focus the far-reaching vision that the perspective of non-abelian fundamental groups could lead to a fundamentally new understanding of deep arithmetic phenomena, including the arithmetic theory of moduli and Diophantine finiteness on hyperbolic curves.", "type": "quote" }, { "ref": "2012, Jakob Stix, editor, The Arithmetic of Fundamental Groups: PIA 2010, Springer, page ix:", "text": "During the more than 100 years of its existence, the notion of the fundamental group has undergone a considerable evolution.", "type": "quote" } ], "glosses": [ "For a specified topological space, the group whose elements are homotopy classes of loops (images of some arbitrary closed interval whose endpoints are both mapped to a designated point) and whose group operation is concatenation." ], "id": "en-fundamental_group-en-noun-V6LksscT", "links": [ [ "topology", "topology" ], [ "topological space", "topological space" ], [ "group", "group" ], [ "element", "element" ], [ "homotopy", "homotopy" ], [ "class", "class" ], [ "image", "image" ], [ "closed", "closed" ], [ "interval", "interval" ], [ "endpoint", "endpoint" ], [ "map", "map" ], [ "point", "point" ], [ "operation", "operation" ], [ "concatenation", "concatenation" ] ], "raw_glosses": [ "(topology) For a specified topological space, the group whose elements are homotopy classes of loops (images of some arbitrary closed interval whose endpoints are both mapped to a designated point) and whose group operation is concatenation." ], "related": [ { "word": "fundamental groupoid" } ], "topics": [ "mathematics", "sciences", "topology" ], "translations": [ { "code": "ca", "lang": "Catalan", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "masculine" ], "word": "grup fonamental" }, { "code": "cs", "lang": "Czech", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "feminine" ], "word": "fundamentální grupa" }, { "code": "nl", "lang": "Dutch", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "feminine" ], "word": "fundamentaalgroep" }, { "code": "fi", "lang": "Finnish", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "word": "perusryhmä" }, { "code": "fr", "lang": "French", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "masculine" ], "word": "groupe fondamental" }, { "code": "de", "lang": "German", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "feminine" ], "word": "Fundamentalgruppe" }, { "code": "hu", "lang": "Hungarian", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "word": "fundamentális csoport" }, { "code": "is", "lang": "Icelandic", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "feminine" ], "word": "undirstöðugrúpa" }, { "code": "it", "lang": "Italian", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "masculine" ], "word": "gruppo fondamentale" }, { "code": "pl", "lang": "Polish", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "feminine" ], "word": "grupa podstawowa" }, { "code": "ru", "lang": "Russian", "roman": "fundamentalʹnaja gruppa", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "feminine" ], "word": "фундаментальная группа" }, { "code": "sv", "lang": "Swedish", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "common-gender" ], "word": "fundamentalgrupp" }, { "code": "tr", "lang": "Turkish", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "word": "temel grup" }, { "code": "uk", "lang": "Ukrainian", "roman": "fundamentalʹna hrupa", "sense": "group of equivalence classes of the homotopies of loops in a given topological space", "tags": [ "feminine" ], "word": "фундаментальна група" } ], "wikipedia": [ "fundamental group" ] } ], "word": "fundamental group" }
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