{"title":"关于𝜋₁(𝐷𝑖𝑓𝑓_{∂}𝐷^{4𝑘})的简短说明，适用于 𝑘≥3","authors":"Wei Wang","doi":"10.1090/proc/16908","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D i f f Subscript partial-differential Baseline left-parenthesis upper D Superscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Diff</mml:mi> <mml:mrow> <mml:mi mathvariant=\"normal\">∂</mml:mi> </mml:mrow> </mml:msub> <mml:mo></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\operatorname {Diff}_{\\partial }(D^{n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the topological group of diffeomorphisms of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D Superscript n\"> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">D^{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which agree with the identity near the boundary. In this short note, we compute the fundamental group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi 1 upper D i f f Subscript partial-differential Baseline left-parenthesis upper D Superscript 4 k Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>Diff</mml:mi> <mml:mrow> <mml:mi mathvariant=\"normal\">∂</mml:mi> </mml:mrow> </mml:msub> <mml:mo></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\pi _1 \\operatorname {Diff}_{\\partial }(D^{4k})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k greater-than-or-equal-to 3\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">k\\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A short note on 𝜋₁(𝐷𝑖𝑓𝑓_{∂}𝐷^{4𝑘}) for 𝑘≥3\",\"authors\":\"Wei Wang\",\"doi\":\"10.1090/proc/16908\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D i f f Subscript partial-differential Baseline left-parenthesis upper D Superscript n Baseline right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Diff</mml:mi> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">∂</mml:mi> </mml:mrow> </mml:msub> <mml:mo></mml:mo> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\operatorname {Diff}_{\\\\partial }(D^{n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the topological group of diffeomorphisms of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D Superscript n\\\"> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">D^{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which agree with the identity near the boundary. In this short note, we compute the fundamental group <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"pi 1 upper D i f f Subscript partial-differential Baseline left-parenthesis upper D Superscript 4 k Baseline right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>Diff</mml:mi> <mml:mrow> <mml:mi mathvariant=\\\"normal\\\">∂</mml:mi> </mml:mrow> </mml:msub> <mml:mo></mml:mo> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\pi _1 \\\\operatorname {Diff}_{\\\\partial }(D^{4k})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"k greater-than-or-equal-to 3\\\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">k\\\\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-05-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16908\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16908","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让 Diff ∂ ( D n ) （operatorname {Diff}_{\partial }(D^{n}）是 D n D^{n} 的差分变形的拓扑群，它与边界附近的同一性一致。在这篇短文中，我们计算了 k ≥ 3 k\geq 3 时的基群 π 1 Diff ∂ ( D 4 k ) \pi _1 \operatorname {Diff}_{\partial }(D^{4k}) 。
Let Diff∂(Dn)\operatorname {Diff}_{\partial }(D^{n}) be the topological group of diffeomorphisms of DnD^{n} which agree with the identity near the boundary. In this short note, we compute the fundamental group π1Diff∂(D4k)\pi _1 \operatorname {Diff}_{\partial }(D^{4k}) for k≥3k\geq 3.
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