##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology seminar

## Guchuan Li

#### University of Michigan

## Vanishing results in Chromatic homotopy theory at prime 2

##### Abstract:

Chromatic homotopy theory is a powerful tool to study periodic phenomena in the stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the fixed points of Lubin--Tate theories $E_h$. These fixed points are computed via homotopy fixed points spectral sequences. In this talk, we prove that at the prime 2, for all heights $h$ and all finite subgroups $G$ of the Morava stabilizer group, the $G$-homotopy fixed point spectral sequence of $E_h$ collapses after the $N(h,G)$-page and admits a horizontal vanishing line of filtration $N(h,G)$.

This vanishing result has proven to be computationally powerful, as demonstrated by Hill--Shi--Wang--Xu’s recent computation of $E_4^{hC_4}$. Our proof uses new equivariant techniques developed by Hill--Hopkins--Ravenel in their solution of the Kervaire invariant one problem. As an application, we extend Kitchloo--Wilson’s $E_n^{hC_2}$-orientation results to all $E_n^{hG}$-orientations at the prime 2. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.

Host: Zhouli Xu

### February 22, 2022

### 1:00 PM

https://ucsd.zoom.us/j/99777474063

Password: topology

Research Areas

Geometry and Topology****************************