Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$

Question

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 $[X,BO(2n)]\cong Vect_{2n}(X)$ where $Vect_{2n}(X)$ is the set of isomorphic classes of $2n$-dimensional real vector bundles over the topological space $X$. Or in terms of charts and transition functions, given a bundle $E$ with trivializing neighborhoods $U_i$ and transition functions $g_{i,j}$, one can construct a bundle $B\varphi ^*(E)$ with same trivializing neighborhoods $U_i$ and transition functions $\varphi \circ g_{i,j}$.

Question: is there any geometric interpretation of this involution? For example, can we describe geometrically $B\varphi ^*(T(RP^4))$ where $T(RP^4)$ denotes the tangent bundle over $RP^4$?

Answer 1

The involution described could also be called "twisting by the determinant line bundle". I think that the following is a geometric description: let $\mathcal{E}$ be an orthogonal bundle on $X$, let $\mathcal{L}=\det \mathcal{E}$ be the determinant line bundle (whose transition functions are given by the determinant as in the question). Take $f:Y\to X$ to be the associated double covering. Then the projection formula implies $$ f_\ast f^\ast \mathcal{E}\cong \mathcal{E}\oplus(\mathcal{E}\otimes\mathcal{L}^{-1}). $$ But for the real bundle, the line bundles $\mathcal{L}$ and $\mathcal{L}^{-1}$ are isomorphic. So the result of the involution can be described as $\operatorname{coker}(\mathcal{E}\to f_\ast f^\ast\mathcal{E})$. For the tangent bundle on $\mathbb{RP}^4$, the covering associated to the determinant line bundle is the orientation double cover $S^4\to \mathbb{RP}^4$.