2
$\begingroup$

I am reading Marc Levine's paper 'Motivic Euler characteristics and Witt-valued characteristic classes'. In that paper he considers the $BN_T(SL_n)$, namely the classifying space of the group of matrices generated by permutation matrices and torus (det=1).

Every oriented vector bundle of rank n on $X$ corresponds to a map $X\to BSL_n$. I wonder for which vector bundles do this map factor through $BN_T(SL_n)$?

$\endgroup$
1
  • 1
    $\begingroup$ It factors if and only if there exists an etale $n$-sheeted cover of $X$ such that the vector bundle is the pushforward of an invertible sheaf from the cover. $\endgroup$ Mar 30, 2023 at 13:38

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.