A1-homotopy groups, excision, and solvable quotients

We study some properties of A1-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of A1-homotopy theory. These concepts and results are well suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate A1-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree A1-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the “next” non-vanishing A1-homotopy group (beyond ) of a smooth toric variety. From this point of view, A1-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost “as tractable” (in low degrees) as ordinary homotopy for large classes of interesting varieties.