A1-homotopy types of A2 and A2 \ {(0, 0)}

Abstract

Morel-Voevodsky developed A^1-homotopy theory which is a bridge between algebraic geometry and algebraic topology. In this thesis we study the A^1-connected component of a smooth variety in great detail. We have shown that the A^1-connected component of a smooth variety contains the information about the existence of affine lines in the variety. Using this and Miyanishi-Sugie's algebraic characterisation, we determine that the affine plane is the only A^1-contractible smooth affine surface over the field of characteristic zero. In the other part of the thesis, we studied the A^1-homotopy type of A^2-{(0,0)}. We showed that over the field of characteristic zero, if an open subvariety of a smooth affine surface is A^1-weakly equivalent to A^2-{(0,0)}, then it is isomorphic to A^2-{(0,0)}.