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. As introduced in [\cite{bachmann2019voevodsky}], this is obtained by localizing $\mathcal{H}^{\mathbb{A}^1}(S)$ at all dense open immersions in $Sm_S$. We establish that the associated localization functor $L_{bir}$ commutes with the bar construction, and thus preserves connectivity. Over a perfect field $k$, we demonstrate 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 $L^{2,1}$ of [\cite{asok2023p}]. Finally, identifying $π_0^b(X)$ of a proper scheme $X/k$ with $π_0^{b\mathbb{A}^1}(X)$ [\cite{asok2011smooth}], we prove that: 1. the canonical morphism from $π_0^{\mathbb{A}^1}(X) $ to $π_0^{b\mathbb{A}^1}(X)$ is the universal birationalization, 2. $π_0^{b\mathbb{A}^1}(-)$ is a birational invariant of proper schemes, 3. (ind-) proper schemes are $\mathbb{A}^1$-connected if and only if they are birationally connected.