Adjacent non-stable layers in the $\mathbb{A}^1$-homotopy of the special linear tower

Abstract

We determine the two cross-stage morphisms arising from adjacent Stiefel fiber sequences in the special linear tower. Over characteristic-zero fields, the first is the Wood morphism at even stages and vanishes at odd stages; the second is induced stably by the motivic Hopf element at even stages and is zero at odd stages. Complex realization consequently classifies oriented rank-$(n-2)$ vector bundles on the split quadric $Q_{2n-1}$ for $n\geq5$. We then apply these calculations to give exact corank-two splitting and efficient generation criteria for projective modules.