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Geometry-Topology Seminar

Thursday, April 28, 2022 - 2:00pm

Guchuan Li

University of Michigan

Location

University of Pennsylvania

4C8

PLEASE NOTE THE SPECIAL TIME FOR THIS SEMINAR.

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 immediate application, we establish a bound for the orientation order $\Theta(h, G)$, the smallest number such that the $\Theta(h, G)$-fold direct sum of any real vector bundle is $E_h^{hG}$-orientable. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.