高连接流形上的球体纤维化

IF 1 2区 数学 Q1 MATHEMATICS
Samik Basu, Aloke Kr Ghosh
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More precisely, for <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> even, we construct fibrations <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>S</mi>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>→</mo>\n <msup>\n <mo>#</mo>\n <mrow>\n <mi>k</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>S</mi>\n <mi>n</mi>\n </msup>\n <mo>×</mo>\n <msup>\n <mi>S</mi>\n <mrow>\n <mn>2</mn>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>)</mo>\n </mrow>\n <mo>→</mo>\n <msub>\n <mi>M</mi>\n <mi>k</mi>\n </msub>\n </mrow>\n <annotation>$S^{n-1} \\rightarrow \\#^{k-1}(S^n \\times S^{2n-1}) \\rightarrow M_k$</annotation>\n </semantics></math>, where <span></span><math>\n <semantics>\n <msub>\n <mi>M</mi>\n <mi>k</mi>\n </msub>\n <annotation>$M_k$</annotation>\n </semantics></math> is a <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$(n-1)$</annotation>\n </semantics></math>-connected <span></span><math>\n <semantics>\n <mrow>\n <mn>2</mn>\n <mi>n</mi>\n </mrow>\n <annotation>$2n$</annotation>\n </semantics></math>-dimensional Poincaré duality complex that satisfies <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>H</mi>\n <mi>n</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>M</mi>\n <mi>k</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <mo>≅</mo>\n <msup>\n <mi>Z</mi>\n <mi>k</mi>\n </msup>\n </mrow>\n <annotation>$H_n(M_k)\\cong {\\mathbb {Z}}^k$</annotation>\n </semantics></math>, in a localised category of spaces. The construction of the fibration is proved for <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n <mo>⩾</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$k\\geqslant 2$</annotation>\n </semantics></math>, where the prime 2, and the primes that occur as torsion in <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>π</mi>\n <mrow>\n <mn>2</mn>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msub>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>S</mi>\n <mi>n</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\pi _{2n-1}(S^n)$</annotation>\n </semantics></math> are inverted. In specific cases, by either assuming <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> is small, or assuming <span></span><math>\n <semantics>\n <mi>k</mi>\n <annotation>$k$</annotation>\n </semantics></math> is large we can reduce the number of primes that need to be inverted. Integral results are obtained for <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>=</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$n=2$</annotation>\n </semantics></math> or 4, and if <span></span><math>\n <semantics>\n <mi>k</mi>\n <annotation>$k$</annotation>\n </semantics></math> is bigger than the number of cyclic summands in the stable stem <span></span><math>\n <semantics>\n <msubsup>\n <mi>π</mi>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <mi>s</mi>\n </msubsup>\n <annotation>$\\pi _{n-1}^s$</annotation>\n </semantics></math>, we obtain results after inverting 2. Finally, we prove some applications for fibrations over <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n <mo>#</mo>\n <msub>\n <mi>M</mi>\n <mi>k</mi>\n </msub>\n </mrow>\n <annotation>$N\\# M_k$</annotation>\n </semantics></math>, and for looped configuration spaces.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Sphere fibrations over highly connected manifolds\",\"authors\":\"Samik Basu,&nbsp;Aloke Kr Ghosh\",\"doi\":\"10.1112/jlms.70002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We construct sphere fibrations over <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(n-1)$</annotation>\\n </semantics></math>-connected <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>2</mn>\\n <mi>n</mi>\\n </mrow>\\n <annotation>$2n$</annotation>\\n </semantics></math>-manifolds such that the total space is a connected sum of sphere products. 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引用次数: 0

摘要

我们在 ( n - 1 ) $(n-1)$ 连通的 2 n $2n$ -manifold 上构建球体纤维,使得总空间是球体乘积的连通和。更确切地说,对于 n $n$ 偶数,我们构建了纤维 S n - 1 → # k - 1 ( S n × S 2 n - 1 ) → M k $S^{n-1} \rightarrow \#^{k-1}(S^n \times S^{2n-1}) \rightarrow M_k$ ,其中 M k $M_k$ 是一个 ( n - 1 ) $(n-1)$ 连接的 2 n $2n$ -dimensional Poincaré duality complex,满足 H n ( M k ) ≅ Z k $H_n(M_k)\cong {\mathbb {Z}}^k$ , 在一个局部化的空间类别中。在 k ⩾ 2 $k\geqslant 2$ 的情况下,证明了纤维的构造,其中素数 2 以及作为扭转出现在 π 2 n - 1 ( S n ) $\pi _{2n-1}(S^n)$ 中的素数都是反转的。在特定情况下,通过假设 n $n$ 较小或假设 k $k$ 较大,我们可以减少需要倒置的素数。对于 n = 2 $n=2$ 或 4,如果 k $k$ 大于稳定干π n - 1 s $p\i _{n-1}^s$中循环和的个数,我们就能得到反转 2 后的积分结果。最后,我们证明了在 N # M k $N\# M_k$ 上的纤化以及循环配置空间的一些应用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Sphere fibrations over highly connected manifolds

We construct sphere fibrations over ( n 1 ) $(n-1)$ -connected 2 n $2n$ -manifolds such that the total space is a connected sum of sphere products. More precisely, for n $n$ even, we construct fibrations S n 1 # k 1 ( S n × S 2 n 1 ) M k $S^{n-1} \rightarrow \#^{k-1}(S^n \times S^{2n-1}) \rightarrow M_k$ , where M k $M_k$ is a ( n 1 ) $(n-1)$ -connected 2 n $2n$ -dimensional Poincaré duality complex that satisfies H n ( M k ) Z k $H_n(M_k)\cong {\mathbb {Z}}^k$ , in a localised category of spaces. The construction of the fibration is proved for k 2 $k\geqslant 2$ , where the prime 2, and the primes that occur as torsion in π 2 n 1 ( S n ) $\pi _{2n-1}(S^n)$ are inverted. In specific cases, by either assuming n $n$ is small, or assuming k $k$ is large we can reduce the number of primes that need to be inverted. Integral results are obtained for n = 2 $n=2$ or 4, and if k $k$ is bigger than the number of cyclic summands in the stable stem π n 1 s $\pi _{n-1}^s$ , we obtain results after inverting 2. Finally, we prove some applications for fibrations over N # M k $N\# M_k$ , and for looped configuration spaces.

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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
186
审稿时长
6-12 weeks
期刊介绍: The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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