2026/08/03 by Haoyang Liu, Fan Yang
Mathematics · #math.AG #math.AT
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 Q2n-1 for n≥5. We then apply these calculations to give exact corank-two splitting and efficient generation criteria for projective modules.