Abstract
In this paper we are going to compute the
KW
$ \mathrm {KW} $
upper K upper W
-Euler classes for rank 2 vector bundles on the classifying stack
B
N
$ \mathrm {B} \mathrm{N} $
normal upper B normal upper N
, where N is the normaliser of the standard torus in
SL
2
$\mathrm{SL}_2$
upper S upper L 2
and
KW
$\mathrm {KW}$
upper K upper W
represents Balmer’s derived Witt groups. Using these computations we will recover, through a new and different strategy, the formulas previously obtained by Levine in Witt-sheaf cohomology. In order to obtain our results, we will prove Künneth formulas for products of
GL
n
$\mathrm{GL}_n$
upper G upper L Subscript n
’s and
SL
n
$\mathrm{SL}_n$
upper S upper L Subscript n
’s classifying spaces and we will develop from scratch the basic theory of twisted symplectic bundles with their associated twisted Borel classes in
SL
$\mathrm{SL}$
upper S upper L
-oriented theories.