Universal birational invariants and $\mathbb{A}^1$-homology

Abstract: Let $k$ be a field admitting a resolution of singularities. In this paper, we prove that the functor of zeroth $\mathbb{A}^1$-homology $\mathbf{H}^{\mathbb{A}^1}_0$ is universal as a functorial birational invariant of smooth proper $k$-varieties taking values in a category enriched by abelian groups. For a smooth proper $k$-variety $X$, we also prove that the dimension of $\mathbf{H}^{\mathbb{A}^1}_0(X;\mathbb{Q})(\mathrm{Spec} k)$ coincides with the number of $R$-equivalence classes of $X(k)$. We deduce these results as consequences of the structure theorem that for a smooth proper $k$-variety $X$, the sheaf $\mathbf{H}^{\mathbb{A}^1}_0(X)$ is the free abelian presheaf generated by the birational $\mathbb{A}^1$-connected components $\pi_0^{b\mathbb{A}^1}(X)$ of Asok-Morel.