Birational Algebraic Topology

Over a qcqs scheme $S$, we analyze the birational localization $L_{\mathrm{b}\mathrm{i}\mathrm{r}}{\mathcal{H}}^{{\mathbb{A}}^1}(S)$ of the motivic $\mathrm{\infty }$-category. As introduced in [\cite{bachmann2019voevodsky}], this is obtained by localizing ${\mathcal{H}}^{{\mathbb{A}}^1}(S)$ at all dense open immersions in $S{m}_S$. 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 ${\pi }_0^b(X)$ of a proper scheme $X\mathrm{/}k$ with ${\pi }_0^{b{\mathbb{A}}^1}(X)$ [\cite{asok2011smooth}], we prove that: 1. the canonical morphism from ${\pi }_0^{{\mathbb{A}}^1}(X)$ to ${\pi }_0^{b{\mathbb{A}}^1}(X)$ is the universal birationalization, 2. ${\pi }_0^{b{\mathbb{A}}^1}(-)$ is a birational invariant of proper schemes, 3. (ind-) proper schemes are ${\mathbb{A}}^1$-connected if and only if they are birationally connected.