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, and the map $B\varphi $ sends a bundles $\xi $ to $\xi \otimes det(\xi )$ , we conclude that for a $4m$-dimensional vector bundle $\xi $ (Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$ thanks @Matthias Wendt), $\xi $ has $Pin _+$ structure if and only if $\xi \otimes det(\xi )$ has $Pin_{-}$ structure.
Now, my questions are
- Is this documented somewhere?
- Can this be seen directly using the definition of $Pin$ structures by Clifford algebras without cohomological computation?
The computation of $B\varphi ^*(w_2)$ goes as follows. Since $H^*(BO(2n);Z/2)$ injects to $H^*(B(Z/2)^{2n};Z/2)$ it suffices to compute the map induced by $B(Z/2)^{2n}\rightarrow BO(2n)\stackrel{\varphi}{\rightarrow}BO(2n)$. This composition factors through an automorphism of $Z/2)^n$ given by $$(a_1,\ldots ,a_{2n} )\mapsto (t+a_1, \cdots ,t +a_{2n})\mbox{ where }t=\Sigma _i a_i$$ Write $H^*(B(Z/2)^{2n};Z/2)\cong Z/2[x_1,\ldots x_{2n}]$, and identify $$H^*(BO(2n);Z/2)\cong Z/2[w_1,\ldots w_{2n}]$$ with its image in $H^*(B(Z/2)^{2n};Z/2)$ so that we have $$w_1=\Sigma _i{x_i},w_2=\Sigma _{i<j}(x_ix_j) \mbox{ etc.}$$ The above factorization of $\varphi$ implies that $$B\varphi ^*(x_i)=w_1+x_i$$ which leads to $$B\varphi ^*(w_2)=\Sigma _{i<j}(w_1+x_1)(w_1+x_j)=\Sigma _{i<j}(w_1^2)+w_1\Sigma _{i<j}(x_i+x_j)+w_2$$ Noting that that there are $n(2n-1)$ couples $(i,j)$ and since we are working modulo $2$, the latter equals to $$n(2n-1) w_1^2 + (2n-1) w_1^2+w_2 =(n+1)w_1^2+w_2,$$ as claimed.
