The cohomology of $BO (n)$ with twisted integer coefficients

Abstract

Let $H^{*}(BO(n), \mathbf{Z}^{t})$ be the graded cohomology group of the classifying space $BO(n)$ with twisted integer coefficients. Then $H^{*}(BO(n); \mathbf{Z}) \bigoplus H^{*}(BO(n); \mathbf{Z}^{t})$ has a structure of a $\mathbf{Z} \bigoplus \mathbf{Z}_{2}$ graded ring. In the paper this ring is described in terms of generators and relations. It extends the results on the integer cohomology ring $H^{*}(BO(n); \mathbf{Z})$ derived in [B] and [F].