Cohomology mod 2 of the classifying space of Spin\(^ c(n)\)
type: Item
instance of: scholarly article
title: Cohomology mod 2 of the classifying space of Spin\(^ c(n)\)
zbMATH Open document ID: 0627.55011
author: Masana Harada
author: Akira Kono
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1986-01-01T00:00:00Z
review text: The paper deals with the calculation of the cohomology ring mod 2 of \(BSpin^ c(n)\), that is the classifying space of Spin\(^ c(n)=S^ 1\times_{{\mathbb{Z}}_ 2}Spin(n)\). There exist natural extensions \[ 0 \to S^ 1 \to Spin^ c(n)\to^{\pi}SO(n) \to 0 \text{ and } 0 \to S^ 1 \to \tilde V^ c \to V \to 0, \] where V denotes the group of diagonal matrices in SO(n) and \(\tilde V^ c=\pi^{-1}(V)\). Using methods and results of \textit{D. Quillen} [Math. Ann. 194, 197-212 (1971; Zbl 0225.55015)] the authors calculate the mod 2 cohomology rings of \(B\tilde V^ c\) and of \(BSpin^ c(n)\). In addition they determine the integral cohomology of \(BSpin^ c(n)\), the mod 2 Chern classes of the Spin representation of \(Spin^ c(n)\), and the Hopf algebra structure of the mod 2 cohomology ring of \(Spin^ c(n)\).
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 4019997
zbMATH Keywords: cohomology ring mod 2 of \(BSpin^ c(n)\)
zbMATH Keywords: classifying space
zbMATH Keywords: mod 2 Chern classes
zbMATH Keywords: Spin representation
zbMATH Keywords: Hopf algebra structure
reviewed by: K. H. Mayer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195177851
OpenAlex ID: W2076879186
cites work: Clifford modules
cites work: Sur L'Homologie et la Cohomologie des Groupes de Lie Compacts Connexes
cites work: Hopf algebra structure of mod 2 cohomology of simple Lie groups
cites work: On the integral cohomology of BSpin(n)
cites work: The mod 2 cohomology rings of extra-special 2-groups and the spinor groups
cites work: The cohomology of classifying spaces of 𝐻-spaces
DOI: 10.2977/PRIMS/1195177851
description: scientific article; zbMATH DE number 4019997