Integral Weyl Invariants in Chow Characteristic Images of Spin and Special Clifford Groups

Abstract

Let $G=\Spin(n)$ be the split spin group over an arbitrary field, with $n\ge 7$. Extending a Steenrod-theoretic obstruction of Karpenko, we classify the recursively defined integral Weyl invariants $q_i$ in the Benson--Wood generating set that lie in the Chow characteristic image: the only such invariant is $q_3$ for $\Spin(10)$. We obtain the analogous classification for the recursive invariants $f_i$ of the special Clifford group ${\Gamma }^{+}(n)$: in their finite generating range, the only such invariant is $f_2$ for ${\Gamma }^{+}(7)$. Over $\mathbb{C}$, the class corresponding to $q_i$ in the torsion-free quotient of the integral cohomology of the classifying space $BG$ is algebraic precisely when $(n,i)=(10,3)$. For each $n$, a single smooth projective approximation simultaneously realizes all the corresponding classes in the finite range. Every nonexceptional class remains nonalgebraic after the addition of any torsion class, as detected by a Bockstein--Steenrod operation.