3
$\begingroup$

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 $RP^4$?

$\endgroup$
4
$\begingroup$

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 $S^4\to \mathbb{RP}^4$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.