Research Seminars @ Illinois

Ningchuan Zhang (Indiana): Profinite transfers in $K(n)$-local homotopy theory

Event Type

Seminar/Symposium

Sponsor

UIUC Math Department

Location

TBD

Date

Oct 21, 2025   11:00 am  

Views

37

Originating Calendar

Topology Seminar

 After $K(1)$-localization, the classical $J$-homomorphism can be interpreted as a profinite transfer map. More precisely, it is a transfer map $\Sigma^{-1}KO^\wedge_2 \to \mathbb{S}_{K(1)}$ from the $C_2$-homotopy fixed points (with a twist) to the $\mathbb{Z}_2^\times$-homotopy fixed points of the $2$-complete complex topological $K$-theory. In joint work in progress with Guchuan Li, we extend this idea to define and study profinite transfers between homotopy fixed points of the Morava $E$-theory by closed subgroups of the Morava stabilizer group.

 

We introduce two definitions of the profinite transfer maps. The first defines them as duals to the profinite restriction maps in the appropriate category. At large primes, we show that the image of the transfer map $\Sigma^{-n^2}E_n \to \mathbb{S}_{K(n)}$ on homotopy groups is the $n^2$-th filtration in the homotopy fixed point spectral sequence. A second definition is based on the 6-functor formalism for smooth representations of $p$-adic Lie groups after Heyer--Mann. We prove that the two definitions of profinite transfers are equivalent for homotopy fixed points of the Morava $E$-theory.