{"title":"论$S^2$上$SO(n)$几何群分类空间的同调性","authors":"Yuki Minowa","doi":"10.4310/hha.2024.v26.n2.a6","DOIUrl":null,"url":null,"abstract":"Let $\\mathcal{G}_\\alpha (X,G)$ be the $G$-gauge group over a space $X$ corresponding to a map $\\alpha : X \\to B\\mathcal{G}_1$. We compute the integral cohomology of $B\\mathcal{G}_1 (S^2, SO(n))$ for $n = 3, 4$. We also show that the homology of $B\\mathcal{G}_1 (S^2, SO(n))$ is torsion free if and only if $n \\leqslant 4$. As an application, we classify the homotopy types of $SO(n)$-gauge groups over a Riemann surface for $n \\leqslant 4$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"18 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the cohomology of the classifying spaces of $SO(n)$-gauge groups over $S^2$\",\"authors\":\"Yuki Minowa\",\"doi\":\"10.4310/hha.2024.v26.n2.a6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $\\\\mathcal{G}_\\\\alpha (X,G)$ be the $G$-gauge group over a space $X$ corresponding to a map $\\\\alpha : X \\\\to B\\\\mathcal{G}_1$. We compute the integral cohomology of $B\\\\mathcal{G}_1 (S^2, SO(n))$ for $n = 3, 4$. We also show that the homology of $B\\\\mathcal{G}_1 (S^2, SO(n))$ is torsion free if and only if $n \\\\leqslant 4$. As an application, we classify the homotopy types of $SO(n)$-gauge groups over a Riemann surface for $n \\\\leqslant 4$.\",\"PeriodicalId\":55050,\"journal\":{\"name\":\"Homology Homotopy and Applications\",\"volume\":\"18 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Homology Homotopy and Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/hha.2024.v26.n2.a6\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Homology Homotopy and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/hha.2024.v26.n2.a6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the cohomology of the classifying spaces of $SO(n)$-gauge groups over $S^2$
Let $\mathcal{G}_\alpha (X,G)$ be the $G$-gauge group over a space $X$ corresponding to a map $\alpha : X \to B\mathcal{G}_1$. We compute the integral cohomology of $B\mathcal{G}_1 (S^2, SO(n))$ for $n = 3, 4$. We also show that the homology of $B\mathcal{G}_1 (S^2, SO(n))$ is torsion free if and only if $n \leqslant 4$. As an application, we classify the homotopy types of $SO(n)$-gauge groups over a Riemann surface for $n \leqslant 4$.
期刊介绍：
Homology, Homotopy and Applications is a refereed journal which publishes high-quality papers in the general area of homotopy theory and algebraic topology, as well as applications of the ideas and results in this area. This means applications in the broadest possible sense, i.e. applications to other parts of mathematics such as number theory and algebraic geometry, as well as to areas outside of mathematics, such as computer science, physics, and statistics. Homotopy theory is also intended to be interpreted broadly, including algebraic K-theory, model categories, homotopy theory of varieties, etc. We particularly encourage innovative papers which point the way toward new applications of the subject.
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