Vanishing lines in chromatic homotopy theory

Duan, Zhipeng; Li, Guchuan; Shi, XiaoLin Danny

doi:10.2140/gt.2025.29.903

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Vanishing lines in chromatic homotopy theory

Zhipeng Duan, Guchuan Li and XiaoLin Danny Shi

Geometry & Topology 29 (2025) 903–930

DOI: 10.2140/gt.2025.29.903

Abstract

We show that at the prime 2, for any height $h$ and any finite subgroup $G \subset 𝔾_{h}$ of the Morava stabilizer group, the $RO (G)$ -graded homotopy fixed point spectral sequence for the Lubin–Tate spectrum $E_{h}$ has a strong horizontal vanishing line of filtration $N_{h, G}$ , a specific number depending on $h$ and $G$ . It is a consequence of the nilpotence theorem that such homotopy fixed point spectral sequences all admit strong horizontal vanishing lines at some finite filtration. Here, we establish specific bounds for them. Our bounds are sharp for all the known computations of $E_{h}^{h G}$ .

Our approach involves investigating the effect of the Hill–Hopkins–Ravenel norm functor on the slice differentials. As a result, we also show that the $RO (G)$ -graded slice spectral sequence for ${(N_{C_{2}}^{G} {\bar{v}}_{h})}^{- 1} {BP}^{((G))}$ shares the same horizontal vanishing line at filtration $N_{h, G}$ . As an application, we utilize this vanishing line to establish a bound on the orientation order $Θ (h, G)$ , the smallest number such that the $Θ (h, G)$ -fold direct sum of any real vector bundle is $E_{h}^{h G}$ -orientable.

Keywords

chromatic homotopy theory, equivariant homotopy theory, Lubin–Tate theories, vanishing lines, slice spectral sequence

Mathematical Subject Classification

Primary: 55P91, 55R25, 55T25

References

Publication

Received: 19 May 2022

Revised: 23 January 2024

Accepted: 4 March 2024

Published: 21 April 2025

Proposed: Stefan Schwede

Seconded: Haynes R Miller, Mark Behrens

Authors

Zhipeng Duan
School of Mathematical Sciences Nanjing Normal University Nanjing China

Guchuan Li
School of Mathematical Sciences Peking University Beijing China

XiaoLin Danny Shi
Department of Mathematics University of Washington Seattle, WA United States

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Volume 29, issue 2 (2025)