关于𝜋₁(𝐷𝑖𝑓𝑓_{∂}𝐷^{4𝑘})的简短说明，适用于 𝑘≥3

Pub Date : 2024-05-11 DOI:10.1090/proc/16908

Wei Wang

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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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16908\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16908","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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}) 。

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A short note on 𝜋₁(𝐷𝑖𝑓𝑓_{∂}𝐷^{4𝑘}) for 𝑘≥3

Let Diff ∂ ⁡ ( D n ) \operatorname {Diff}_{\partial }(D^{n}) be the topological group of diffeomorphisms of D n D^{n} which agree with the identity near the boundary. In this short note, we compute the fundamental group π 1 Diff ∂ ⁡ ( D 4 k ) \pi _1 \operatorname {Diff}_{\partial }(D^{4k}) for k ≥ 3 k\geq 3 .

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