Cohomology $\mathrm{Mod}\, 2$ of the Classifying Space of $\textit{Spin}^c(n)$

Abstract

In this paper we determine the mod 2 cohomology ring and the integral cohomology ring of the classifying space of the compact., connected Lie group Spinc(ri), which is a subgroup of the group of units in the complex Clifford algebra Cn®C (see [1]). The group Spinc(ri) is very important for the orientations in the KO-theory. We also determine (the mod 2 reduction of) the Ghern classes of the complex spin representations and the Hopf algebra structure of the mod 2 cohomology ring of Spinc(n), The first section is devoted to studying an ideal of a polynomial ring over F2 which is associated to a symplectic bilinear form on a F2 vector space and whose variety of geometric points is the union of the maximal isotropic subspaces rational over F2. We show that the generators of the ideal form a regular sequence and we determine the decomposition of the ideal into prime ideals. These algebraicgeometric results are applied in the second and third sections to compute the mod 2 and integral cohomology ring of BSpinc(n} and determine the Ghern classes of the spin representation of Spinc(ri)e In the last section we compute the Steenrod operations and the coproducts of the mod 2 cohomology of Spinc(n)a Throughout the paper H*(X) denotes the mod 2 cohomology ring, 1. Let V be an ^-dimensional vector space over F2, V* its dua!3 S(V*) the symmetric algebra over F* and B a symplectic bilinear form on V. Let h' be the codimension of a 5-isotropic subspace of maximum dimension. Consider the following sequence of homogeneous