Sebastian Gant | Search arXiv

arXiv2026

The nonexistence of sections of Stiefel varieties and stably free modules

Let $V_r(\mathbb{A}^n)$ denote the Stiefel variety ${\rm GL}_n/{\rm GL}_{n-r}$ over a field. There is a natural projection $p: V_{r+\ell}(\mathbb{A}^n) \to V_r(\mathbb{A}^n)$. The question of whether this projection admits a section was asked by M. Raynaud in 1968. We focus on the case of $r \ge 2$ and provide examples of triples $(r,n,\ell)$ for which a section does not exist. Our results produce examples of stably free modules that do not have free summands of a given rank. To this end, we also construct a splitting of $V_2(\mathbb{A}^n)$ in the motivic stable homotopy category over a field, analogous to the classical stable splitting of the Stiefel manifolds due to I. M. James.

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arXiv2026

Motivic Homotopy Groups of Spheres and Free Summands of Stably Free Modules

Working over an algebraically closed field $k$ 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 maps from the motivic homotopy groups to the classical stable homotopy groups are isomorphisms 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. We use this to determine when the projection map $V_r(\mathbb{A}^n_k) \to V_1(\mathbb{A}^n_k)$ admits a right inverse, settling the question of when the universal stably-free module of type $(n,n-1)$ admits a free summand of given rank.

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