  We study some properties of A^1-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 A^1-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
A^1-homotopy groups of smooth toric varieties. We give simple combinatorial
conditions (in terms of fans) guaranteeing vanishing of low degree A^1-homotopy
groups of smooth (proper) toric varieties. Finally, in certain cases, we can
actually compute the "next" non-vanishing A^1-homotopy group (beyond
\pi_1^{A^1}) of a smooth toric variety. From this point of view, A^1-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.

Algebraic Geometry;Algebraic Topology;K-Theory and Homology