KW KWupper K upper W-EULER CLASSES VIA TWISTED SYMPLECTIC BUNDLES

In this paper we are going to compute the KW KW upper K upper W -Euler classes for rank 2 vector bundles on the classifying stack B N B N normal upper B normal upper N , where N is the normaliser of the standard torus in SL 2 SL₂ upper S upper L 2 and KW 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 GLₙ upper G upper L Subscript n ’s and SL n SLₙ 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 SL upper S upper L -oriented theories.