同调与双模子的扭曲作用

IF 0.5 4区 数学 Q3 MATHEMATICS
VLADIMIR SHCHIGOLEV
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Considering this action as a right action, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000065_inline4.png\"/> <jats:tex-math> $H_L^{\\bullet }(X,\\Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> becomes a bimodule together with the canonical left action of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788724000065_inline5.png\"/> <jats:tex-math> $H_L^{\\bullet }(\\mathrm {pt},\\Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott–Samelson varieties and to a geometric construction of the bimodule morphisms between them.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"TWISTED ACTIONS ON COHOMOLOGIES AND BIMODULES\",\"authors\":\"VLADIMIR SHCHIGOLEV\",\"doi\":\"10.1017/s1446788724000065\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For closed subgroups <jats:italic>L</jats:italic> and <jats:italic>R</jats:italic> of a compact Lie group <jats:italic>G</jats:italic>, a left <jats:italic>L</jats:italic>-space <jats:italic>X</jats:italic>, and an <jats:italic>L</jats:italic>-equivariant continuous map <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1446788724000065_inline1.png\\\"/> <jats:tex-math> $A:X\\\\to G/R$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we introduce the twisted action of the equivariant cohomology <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1446788724000065_inline2.png\\\"/> <jats:tex-math> $H_R^{\\\\bullet }(\\\\mathrm {pt},\\\\Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> on the equivariant cohomology <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1446788724000065_inline3.png\\\"/> <jats:tex-math> $H_L^{\\\\bullet }(X,\\\\Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Considering this action as a right action, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1446788724000065_inline4.png\\\"/> <jats:tex-math> $H_L^{\\\\bullet }(X,\\\\Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> becomes a bimodule together with the canonical left action of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1446788724000065_inline5.png\\\"/> <jats:tex-math> $H_L^{\\\\bullet }(\\\\mathrm {pt},\\\\Bbbk )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. 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引用次数: 0

摘要

对于紧凑李群 G 的封闭子群 L 和 R、左 L 空间 X 以及 L-变量连续映射 $A:X\to G/R$,我们引入了等变同调 $H_R^{\bullet }(\mathrm {pt},\Bbbk )$ 对等变同调 $H_L^{\bullet }(X,\Bbbk )$ 的扭曲作用。把这个作用看作右作用,$H_L^{\bullet }(X,\Bbbk )$ 就变成了一个双模,同时还有$H_L^{\bullet }(\mathrm {pt},\Bbbk )$ 的典型左作用。利用这个双模块结构,我们证明了库奈特同构的等变版本。我们将这一结果应用于计算博特-萨缪尔森(Bott-Samelson)变体的等变同构,以及它们之间的双模态的几何构造。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
TWISTED ACTIONS ON COHOMOLOGIES AND BIMODULES
For closed subgroups L and R of a compact Lie group G, a left L-space X, and an L-equivariant continuous map $A:X\to G/R$ , we introduce the twisted action of the equivariant cohomology $H_R^{\bullet }(\mathrm {pt},\Bbbk )$ on the equivariant cohomology $H_L^{\bullet }(X,\Bbbk )$ . Considering this action as a right action, $H_L^{\bullet }(X,\Bbbk )$ becomes a bimodule together with the canonical left action of $H_L^{\bullet }(\mathrm {pt},\Bbbk )$ . Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott–Samelson varieties and to a geometric construction of the bimodule morphisms between them.
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来源期刊
CiteScore
1.70
自引率
0.00%
发文量
36
审稿时长
6 months
期刊介绍: The Journal of the Australian Mathematical Society is the oldest journal of the Society, and is well established in its coverage of all areas of pure mathematics and mathematical statistics. It seeks to publish original high-quality articles of moderate length that will attract wide interest. Papers are carefully reviewed, and those with good introductions explaining the meaning and value of the results are preferred. Published Bi-monthly Published for the Australian Mathematical Society
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