Cellular ${\mathbb{A}}^1$-homology and the motivic version of Matsumoto's theorem

We define a new version of ${\mathbb{A}}^1$-homology, called cellular $\mathbb A^1$-homology, for smooth schemes over a field that admit an increasing filtration by open subschemes with cohomologically trivial closed strata. We provide several explicit computations of cellular ${\mathbb{A}}^1$-homology and use them to determine the ${\mathbb{A}}^1$-fundamental group of a split reductive group over an arbitrary field, thereby obtaining the motivic version of Matsumoto’s theorem on universal central extensions of split, semisimple, simply connected algebraic groups. As applications, we uniformly explain and generalize results due to Brylinski-Deligne and Esnault-Kahn-Levine-Viehweg, determine the isomorphism classes of central extensions of such an algebraic group by an arbitrary strictly ${\mathbb{A}}^1$-invariant sheaf and also reprove classical results of E. Cartan on homotopy groups of complex Lie groups.