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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 ...
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Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$
Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution)
$B\varphi ^*$ on $[...
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