Atiyah-Bott localization in equivariant Witt cohomology

arXiv2203.13882MaRDI QIDQ6394737

Marc Levine

Publication date: 25 March 2022

Abstract: Let $N$ be a normalizer of the diagonal torus $T_{1}congmathbb{G}_m$ in $ext{SL}_2$. We prove localization theorems for $ext{SL}_2^n$ and $N^n$ for equivariant cohomology with coefficients in the (twisted) Witt sheaf, along the lines of the classical Atiyah-Bott localization theorems for equivariant cohomology for a torus action. We also have an analog of the Bott residue formula for $ext{SL}_2^n$ and $N$. In the case of an $ext{SL}_2^n$-action, there is a rather serious restriction on the orbit type. For an $N$-action, there is no restriction for the localization result, but for the Bott residue theorem, one requires a certain type of decomposition of the fixed points for the $T_1$-action.

Mathematics Subject Classification ID

Motivic cohomology; motivic homotopy theory (14F42)

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