拓扑群中的高同伦规范性

Pub Date : 2023-02-17 DOI:10.1112/topo.12282

Mitsunobu Tsutaya

{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-02-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/topo.12282\",\"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://onlinelibrary.wiley.com/doi/10.1112/topo.12282","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

本文的目的是介绍Nk(ℓ)$N_ k（\ell）$-maps（1⩽k，ℓ⩽∞$1\leqslant k，\ell\leqslant\infty$），并研究了它们的基本性质和例子。An Nk(ℓ)$N_k（\ell）$-map是用更高的同位条件定义的。证明了同态是Nk(ℓ)$N_k（\ell）$-map当且仅当在具有某些性质的纤维状投影空间之间存在纤维状映射。此外，如果Nk（k）$N_k（k）$映射的LS范畴不大于k$k$，则其同伦商被证明是H$H$空间。作为一个应用，我们研究了当夹杂物SU（m）→SU（n）$\运算符名称｛SU｝（m）\rightarrow\运算符名称{SU}（n）$和SO（2m+1）→SO（2n+1）$\运算符名称｛SO｝（2m+1）\rightarrow\运算符名称{SO｝(ℓ)$N_k（\ell）$映射。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

Higher homotopy normalities in topological groups

The purpose of this paper is to introduce $N_{k} (ℓ)$ -maps ( $1 ⩽ k, ℓ ⩽ \infty$ ), which describe higher homotopy normalities, and to study their basic properties and examples. An $N_{k} (ℓ)$ -map is defined with higher homotopical conditions. It is shown that a homomorphism is an $N_{k} (ℓ)$ -map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an $N_{k} (k)$ -map is shown to be an $H$ -space if its LS category is not greater than $k$ . As an application, we investigate when the inclusions $SU (m) \to SU (n)$ and $SO (2 m + 1) \to SO (2 n + 1)$ are $p$ -locally $N_{k} (ℓ)$ -maps.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助