On stably trivial spin torsors over low-dimensional schemes

The paper discusses stably trivial torsors for spin and orthogonal groups over smooth affine schemes over infinite perfect fields of characteristic unequal to 2. We give a complete description of all the invariants relevant for the classification of such objects over schemes of dimension at most $3$, along with many examples. The results are based on the $\mathbb{A}^1$-representability theorem for torsors and transfer of known computations of $\mathbb{A}^1$-homotopy sheaves along the sporadic isomorphisms to spin groups.