Motivic Homotopy Groups of Spheres and Free Summands of Stably Free Modules
Sebastian Gant, Ben Williams
TL;DR
This work shows that over an algebraically closed field of characteristic zero, the motivic stable homotopy groups of the sphere can be recovered from the $p$-completed motivic spheres and motivic cohomology, except in the $0$- and $-1$-stem. Consequently, the complex realization map to classical stable homotopy groups is an isomorphism in broad bidegree ranges, and this extends to unstable homotopy groups of Stiefel varieties, enabling a full determination of when projection maps admit right inverses. The authors reduce global motivic questions to prime-by-prime data via Ext-completions and apply recent advances in motivic homotopy theory (notably Ananyevskiy) to bridge motivic and classical computations. A central application is a complete solution to when universal stably free modules of type $(n,n-1)$ have free summands, characterized by James numbers $b_r$ dividing $n$, which resolves a long-standing question of Raynaud in this arithmetic setting.
Abstract
Working over an algebraically closed field of characteristic $0$, we show that the motivic stable homotopy groups of the sphere spectrum can be determined entirely from the motivic homotopy groups of the $p$-completed sphere spectra and the motivic cohomology of the ground field, except possibly for the $0$ and $-1$-stems. Using this, we show that the complex realization map from the motivic homotopy group to the classical stable homotopy group is an isomorphism in a range of bidegrees. We apply this to deduce that complex realization also induces isomorphisms on unstable homotopy groups for Stiefel varieties $V_r(\mathbb{A}^n_k)$ in a range of bidegrees. This allows a complete solution of the question of when the projection map $V_r(\mathbb{A}^n_k) \to V_1(\mathbb{A}^n_k)$ admits a right inverse. Equivalently, this settles the question of when the universal stably-free module of type $(n,n-1)$ admits a free summand of given rank.