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.