{"title":"拓扑群中的高同伦规范性","authors":"Mitsunobu Tsutaya","doi":"10.1112/topo.12282","DOIUrl":null,"url":null,"abstract":"<p>The purpose of this paper is to introduce <math>\n <semantics>\n <mrow>\n <msub>\n <mi>N</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>ℓ</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$N_k(\\ell )$</annotation>\n </semantics></math>-maps (<math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>⩽</mo>\n <mi>k</mi>\n <mo>,</mo>\n <mi>ℓ</mi>\n <mo>⩽</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$1\\leqslant k,\\ell \\leqslant \\infty$</annotation>\n </semantics></math>), which describe higher homotopy normalities, and to study their basic properties and examples. An <math>\n <semantics>\n <mrow>\n <msub>\n <mi>N</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>ℓ</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$N_k(\\ell )$</annotation>\n </semantics></math>-map is defined with higher homotopical conditions. It is shown that a homomorphism is an <math>\n <semantics>\n <mrow>\n <msub>\n <mi>N</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>ℓ</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$N_k(\\ell )$</annotation>\n </semantics></math>-map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an <math>\n <semantics>\n <mrow>\n <msub>\n <mi>N</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>k</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$N_k(k)$</annotation>\n </semantics></math>-map is shown to be an <math>\n <semantics>\n <mi>H</mi>\n <annotation>$H$</annotation>\n </semantics></math>-space if its LS category is not greater than <math>\n <semantics>\n <mi>k</mi>\n <annotation>$k$</annotation>\n </semantics></math>. As an application, we investigate when the inclusions <math>\n <semantics>\n <mrow>\n <mo>SU</mo>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n <mo>→</mo>\n <mo>SU</mo>\n <mo>(</mo>\n <mi>n</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{SU}(m)\\rightarrow \\operatorname{SU}(n)$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mo>SO</mo>\n <mo>(</mo>\n <mn>2</mn>\n <mi>m</mi>\n <mo>+</mo>\n <mn>1</mn>\n <mo>)</mo>\n <mo>→</mo>\n <mo>SO</mo>\n <mo>(</mo>\n <mn>2</mn>\n <mi>n</mi>\n <mo>+</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{SO}(2m+1)\\rightarrow \\operatorname{SO}(2n+1)$</annotation>\n </semantics></math> are <math>\n <semantics>\n <mi>p</mi>\n <annotation>$p$</annotation>\n </semantics></math>-locally <math>\n <semantics>\n <mrow>\n <msub>\n <mi>N</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>ℓ</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$N_k(\\ell )$</annotation>\n </semantics></math>-maps.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Higher homotopy normalities in topological groups\",\"authors\":\"Mitsunobu Tsutaya\",\"doi\":\"10.1112/topo.12282\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The purpose of this paper is to introduce <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>N</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>ℓ</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$N_k(\\\\ell )$</annotation>\\n </semantics></math>-maps (<math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>⩽</mo>\\n <mi>k</mi>\\n <mo>,</mo>\\n <mi>ℓ</mi>\\n <mo>⩽</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$1\\\\leqslant k,\\\\ell \\\\leqslant \\\\infty$</annotation>\\n </semantics></math>), which describe higher homotopy normalities, and to study their basic properties and examples. An <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>N</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>ℓ</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$N_k(\\\\ell )$</annotation>\\n </semantics></math>-map is defined with higher homotopical conditions. It is shown that a homomorphism is an <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>N</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>ℓ</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$N_k(\\\\ell )$</annotation>\\n </semantics></math>-map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>N</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>k</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$N_k(k)$</annotation>\\n </semantics></math>-map is shown to be an <math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$H$</annotation>\\n </semantics></math>-space if its LS category is not greater than <math>\\n <semantics>\\n <mi>k</mi>\\n <annotation>$k$</annotation>\\n </semantics></math>. As an application, we investigate when the inclusions <math>\\n <semantics>\\n <mrow>\\n <mo>SU</mo>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n <mo>→</mo>\\n <mo>SU</mo>\\n <mo>(</mo>\\n <mi>n</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{SU}(m)\\\\rightarrow \\\\operatorname{SU}(n)$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mo>SO</mo>\\n <mo>(</mo>\\n <mn>2</mn>\\n <mi>m</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n <mo>→</mo>\\n <mo>SO</mo>\\n <mo>(</mo>\\n <mn>2</mn>\\n <mi>n</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{SO}(2m+1)\\\\rightarrow \\\\operatorname{SO}(2n+1)$</annotation>\\n </semantics></math> are <math>\\n <semantics>\\n <mi>p</mi>\\n <annotation>$p$</annotation>\\n </semantics></math>-locally <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>N</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>ℓ</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$N_k(\\\\ell )$</annotation>\\n </semantics></math>-maps.</p>\",\"PeriodicalId\":56114,\"journal\":{\"name\":\"Journal of Topology\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-02-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Topology\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/topo.12282\",\"RegionNum\":2,\"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":"Journal of Topology","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/topo.12282","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The purpose of this paper is to introduce -maps (), which describe higher homotopy normalities, and to study their basic properties and examples. An -map is defined with higher homotopical conditions. It is shown that a homomorphism is an -map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an -map is shown to be an -space if its LS category is not greater than . As an application, we investigate when the inclusions and are -locally -maps.
期刊介绍：
The Journal of Topology publishes papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal.
The Journal of Topology was founded in 2008. It is published quarterly with articles published individually online prior to appearing in a printed issue.
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