Virtual Localisation Formula for $ SL_{\eta} $-Oriented Theories

TL;DR

This paper generalizes the Virtual Localization Formula to motivic ring spectra, providing new localization results for virtual classes in Witt theory and an explicit description of the ring structure of KW(BN).

Contribution

It extends Levine's localization formula to a broad class of motivic spectra, including a localization formula for Witt theory and an explicit ring structure for KW(BN).

Findings

Established an equivariant Atiyah-Bott localization theorem.

Derived an N-equivariant virtual localization formula.

Provided an additive presentation of A(BN) for SL_2 normalizer.

Abstract

In this paper, we extend the Virtual Localization Formula of Levine to a wide class of motivic ring spectra, obtaining in particular a localization formula for virtual fundamental classes in Witt theory $ \mathrm{KW} $. Applying standard tools of $\mathbb A^1$-intersection theory to any $ SL_{\eta} $-oriented spectra $ \mathrm A $, we obtain an additive presentation of $ \mathrm A(BN) $, for $ N $ the normaliser of the torus in $ SL_2 $. Then we establish an equivariant Atiyah-Bott localization theorem for $ \mathrm A_N^{\mathrm{BM}}(X) $ and we conclude with the $N$-equivariant virtual localisation formula. Of independent interest, we also describe the ring structure of $ \mathrm{KW}(BN) $.