KW-Euler Classes via Twisted Symplectic Bundles

Abstract

In this paper we are going to compute the $\mathrm{KW}$-Euler classes for rank 2 vector bundles on the classifying stack $\mathcal{B}N$, where $N$ is the normaliser of the standard torus in $S{L}_2$ and $\mathrm{KW}$ 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 $G{L}_n$'s and $S{L}_n$'s classifying spaces and we will develop from scratch the basic theory of twisted symplectic bundles with their associated twisted Borel classes in $SL$-oriented theories.