Birational Algebraic Topology

Abstract

Over a qcqs scheme $S$, we analyze the birational localization $L_{\mathrm{bir}}{\mathcal{H}}^{{\mathbb{A}}^1}(S)$ of the motivic $\infty$-category ${\mathcal{H}}^{{\mathbb{A}}^1}(S)$. We establish that the associated localization functor $L_{bir}$ commutes with the bar construction, thereby preserving connectivity over fields. When the field is perfect, we show that a sheaf of groups is birational exactly when it is strongly ${\mathbb{A}}^1$-invariant and has trivial ${\mathbb{G}}_m$-contraction. For connected motivic spaces over such fields, this yields a canonical equivalence between $L_{bir}$ and the $S^{2,1}$-nullification functor $L^{2,1}$. For a general field $k$, we identify ${\pi }_0^b(X)$ with ${\pi }_0^{b{\mathbb{A}}^1}(X)$ for proper $k$-schemes and show that ${\mathbb{A}}^1$-connectedness is equivalent to birational connectedness for (ind-)proper $k$-schemes. This also confirms that ${\pi }_0^{b{\mathbb{A}}^1}(-)$ and ${\mathbb{H}}_0^{{\mathbb{A}}^1}(-)$ are stable birational invariants of smooth proper $k$-schemes, satisfying the expected universal property for invariants valued in birational sheaves (of sets and abelian groups respectively). Finally, we show that etale birational equivalences to $S$ are precisely the dense open immersions into $S$.