KW-Euler Classes via Twisted Symplectic Bundles

TL;DR

This paper computes KW-Euler classes for rank 2 bundles on a classifying stack, introduces twisted symplectic bundles, and offers a new approach to formulas previously obtained by Levine, including K"unneth formulas.

Contribution

It develops the theory of twisted symplectic bundles and provides a novel method to compute KW-Euler classes, recovering known formulas through a different strategy.

Findings

Computed KW-Euler classes for rank 2 bundles on classifying stacks.

Established K"unneth formulas for products of $GL_n$ and $SL_n$ classifying spaces.

Developed foundational theory of twisted symplectic bundles in $SL$-oriented theories.

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 $SL_2$ 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 K\"unneth formulas for products of $GL_n$'s and $SL_n$'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.