Virtual Localisation Formula for $S{L}_{\hat{I}}\cdot$-Oriented Theories

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 $S{L}_{\hat{I}}\cdot$-oriented spectra $\mathrm{A}$, we obtain an additive presentation of $\mathrm{A}(BN)$, for $N$ the normaliser of the torus in $S{L}_2$. 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 $\mathrm{KW}(BN)$.

This is part of the author's PhD thesis. Comments are very welcome