Cohomology mod 2 of the classifying space of Spin\(^ c(n)\)
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 \(Spin^ c(n)\).
cohomology ring mod 2 of \(BSpin^ c(n)\)
scientific article; zbMATH DE number 4019997