We prove that the group of normalized cohomological invariants of degree $3$ modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group $G$ is isomorphic to the torsion part of the Chow group of codimension-$2$ cycles of the respective versal $G$-flag. In particular, if $G$ is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of $G$. As an application, we construct nontrivial cohomological invariants for indecomposable central simple algebras.

We introduce the notion of twisted jets. For a Deligne–Mumford stack $\mathscr{X}$ of finite type over $\mathbb{C}$, a twisted $\infty$-jet on $\mathscr{X}$ is a representable morphism $\mathscr{D} \to \mathscr{X}$ such that $\mathscr{D}$ is a smooth Deligne–Mumford stack with the coarse moduli space Spec$\mathbb{C}[[t]]$. We study a motivic measure on the space of the twisted $\infty$-jets on a smooth Deligne–Mumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology as a Hodge structure.