We show that if G is an anisotropic, semisimple, absolutely almost simple, simply connected group over a field k, then two elements of G over any field extension of k are R-equivalent if and only if they are A 1equivalent. As a consequence, we see that Sing * (G) cannot be A 1 -local for such groups. This implies that the A 1 -connected components of a semisimple, absolutely almost simple, simply connected group over a field k form a sheaf of abelian groups.

A conjecture of Morel asserts that the sheaf π A 1 0 (X ) of A 1 -connected components of a simplicial sheaf X is A 1 -invariant. A conjecture of Asok and Morel asserts that the sheaves of A 1 -connected components of smooth schemes over a field coincide with the sheaves of their A 1 -chain-connected components. Another conjecture of Asok and Morel states that the sheaf of A 1 -connected components is a birational invariant of smooth proper schemes. In this article, we exhibit examples of schemes for which conjectures of Asok and Morel fail to hold and whose Sing * is not A 1 -local. We also give equivalent conditions for Morel's conjecture to hold and obtain an explicit conjectural description of π A 1 0 (X ). A method suggested by these results is then used to prove Morel's conjecture for non-uniruled surfaces over a field.

Using sheaves of A 1 -connected components, we prove that the Morel-Voevodsky singular construction on a reductive algebraic group fails to be A 1 -local if the group does not satisfy suitable isotropy hypotheses. As a consequence, we show the failure of A 1 -invariance of torsors for such groups on smooth affine schemes over infinite perfect fields. We also characterize A 1connected reductive algebraic groups over a field of characteristic 0.

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