Daniel Asimov, Florian Frick, Michael Harrison, Wesley Pegden
{"title":"Unit sphere fibrations in Euclidean space","authors":"Daniel Asimov, Florian Frick, Michael Harrison, Wesley Pegden","doi":"10.1017/s0013091524000038","DOIUrl":null,"url":null,"abstract":"<p>We show that if an open set in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306133538207-0692:S0013091524000038:S0013091524000038_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb{R}^d$</span></span></img></span></span> can be fibered by unit <span>n</span>-spheres, then <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306133538207-0692:S0013091524000038:S0013091524000038_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$d \\geq 2n+1$</span></span></img></span></span>, and if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306133538207-0692:S0013091524000038:S0013091524000038_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$d = 2n+1$</span></span></img></span></span>, then the spheres must be pairwise linked, and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306133538207-0692:S0013091524000038:S0013091524000038_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$n \\in \\left\\{0, 1, 3, 7 \\right\\}$</span></span></img></span></span>. For these values of <span>n</span>, we construct unit <span>n</span>-sphere fibrations in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306133538207-0692:S0013091524000038:S0013091524000038_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb{R}^{2n+1}$</span></span></img></span></span>.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0013091524000038","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We show that if an open set in $\mathbb{R}^d$ can be fibered by unit n-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{0, 1, 3, 7 \right\}$. For these values of n, we construct unit n-sphere fibrations in $\mathbb{R}^{2n+1}$.