{"title":"Chern classes in equivariant bordism","authors":"Stefan Schwede","doi":"10.1017/fms.2023.124","DOIUrl":null,"url":null,"abstract":"<p>We introduce Chern classes in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105024122240-0548:S205050942300124X:S205050942300124X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$U(m)$</span></span></img></span></span>-equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105024122240-0548:S205050942300124X:S205050942300124X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf {MU}$</span></span></img></span></span>-cohomology of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105024122240-0548:S205050942300124X:S205050942300124X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$B U(m)$</span></span></img></span></span>. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240105024122240-0548:S205050942300124X:S205050942300124X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf {MU}$</span></span></img></span></span>-completion theorem of Greenlees–May and La Vecchia.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum of Mathematics Sigma","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/fms.2023.124","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce Chern classes in $U(m)$-equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the $\mathbf {MU}$-cohomology of $B U(m)$. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the $\mathbf {MU}$-completion theorem of Greenlees–May and La Vecchia.
期刊介绍：
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$U(m)$-equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the 
$\mathbf {MU}$-cohomology of 
$B U(m)$. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the 
$\mathbf {MU}$-completion theorem of Greenlees–May and La Vecchia.
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