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KW-Euler Classes via Twisted Symplectic Bundles

Alessandro D'Angelo

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 Künneth 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.

KW-Euler Classes via Twisted Symplectic Bundles

Abstract

In this paper we are going to compute the -Euler classes for rank 2 vector bundles on the classifying stack , where is the normaliser of the standard torus in and 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 's and 's classifying spaces and we will develop from scratch the basic theory of twisted symplectic bundles with their associated twisted Borel classes in -oriented theories.

Paper Structure

This paper contains 3 sections, 3 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notations and Conventions

Key Result

Proposition 1

Let $S \in \mathbf{Sm}_{ {\raisebox{0em}{$$}\!\!\!\;\!\!\;\left/\!\!\;\!\!\;\raisebox{-0em}{$\mathbbm{k}$}\right.} }$. Let $\mathrm{A}$ be an $SL_{\eta}$-oriented ring spectrum. Let $\mathcal{X}=\prod_{i=1}^{s} \mathcal{B}GL_{n_i,S} \times \prod_{j=s+1}^{s+r} \mathcal{B}SL_{n_j,S}$. Then we have: with $L:=L_1 \boxtimes \ldots \boxtimes L_{s+r}$ and $L_i\in \mathrm{Pic}(\mathcal{B}GL_{n_i})$ for

Theorems & Definitions (3)

  • Proposition 1: Proposition \ref{['ch3:_Kunneth_Formula']}
  • Theorem 2: Theorem \ref{['ch3:_Twisted_Proj_Bun_Thm']}
  • Theorem 3: \ref{['ch3:_recursive_formulas_proposition']}