Norms and Hermitian $\mathrm{K}$-Theory

Abstract

Over the past century, cohomology operations have played a crucial role in homotopy theory and its applications. A powerful framework for constructing such operations is the theory of commutative algebras in spectra. In this article, we discuss an algebro-geometric analogue of this framework, called the theory of normed algebras in motivic spectra. Specifically, we show that the motivic spectrum $\mathrm{ko}$ representing very effective hermitian $\mathrm{K}$-theory can be equipped with a normed algebra structure, and that the orientation map $\mathrm{MSL}\to \mathrm{ko}$ respects this structure. The main step will be showing that the motivic infinite loop space machine is compatible with norms.

An expository account of arXiv:2305.12684; Comments welcome!