Vanishing lines in chromatic homotopy theory

Duan, Zhipeng¹ ; Li, Guchuan² ; Shi, XiaoLin Danny³

¹School of Mathematical Sciences, Nanjing Normal University, Nanjing, China
²School of Mathematical Sciences, Peking University, Beijing, China
³Department of Mathematics, University of Washington, Seattle, WA, United States

Geometry & topology, Tome 29 (2025) no. 2, pp. 903-930

Résumé

We show that at the prime 2, for any height h and any finite subgroup G ⊂ 𝔾h of the Morava stabilizer group, the RO ⁡ (G)-graded homotopy fixed point spectral sequence for the Lubin–Tate spectrum Eh has a strong horizontal vanishing line of filtration Nh,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 EhhG.

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 (NC2Gv¯h)−1 BP ⁡ ((G)) shares the same horizontal vanishing line at filtration Nh,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 EhhG-orientable.

Détail

DOI : 10.2140/gt.2025.29.903

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

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