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We define a new version of $\mathbb A^1$-homology, called cellular $\mathbbA^1$-homology, for smooth schemes over a field that admit an increasingfiltration by open subschemes with cohomologically trivial closed strata. Weprovide several explicit computations of cellular $\mathbb A^1$-homology anduse them to determine the $\mathbb A^1$-fundamental group of a split reductivegroup over an arbitrary field, thereby obtaining the motivic version ofMatsumoto's theorem on universal central extensions of split, semisimple,simply connected algebraic groups. As applications, we uniformly explain andgeneralize results due to Brylinski-Deligne and Esnault-Kahn-Levine-Viehweg,determine the isomorphism classes of central extensions of such an algebraicgroup by an arbitrary strictly $\mathbb A^1$-invariant sheaf and also reproveclassical results of E. Cartan on homotopy groups of complex Lie groups.

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