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

Thesis (Ph.D) - Indian Statistical Institute, 2024

Includes bibliography

Introduction -- A1-homotopy theory: An Introduction -- A1-invariance of πA1/ 0 (−) -- Birational Connected Components -- Existence of A1 and A1-Connectedness of a Surface -- A1-homotopy theory and log-uniruledness -- Kan Fibrant Property of Sing∗(X)(−) -- Characterisation of the Affine Space -- A1-homotopy type of A2 \ {(0, 0)} -- Regular Functions on S(X) -- Naive 0-th A1-homology --

Guided by Prof. Utsav Choudhury

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)}.