## Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]

Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the post). In particular, if $n$ is even, it switches $w_2$ and $w_2+w_1^2$. As these classes are the obstructions for existence of $Pin_{+}$ and $Pin_{-}$ structures, … Read more