Comparing A^1-h-cobordism and A^1-weak equivalence

Abstract

We study the problem of classifying projectivizations of rank-two vector bun- dles over P 2 up to various notions of equivalence that arise naturally in A 1 -homotopy the- ory, namely A 1 -weak equivalence and A 1 -h-cobordism. First, we classify such varieties up to A 1 -weak equivalence: over algebraically closed fields having characteristic unequal to two the classificati on can be given in terms of charac- teristic classes of the underlying vector bundle. When the base field is C, this classification result can be compared to a corresponding topological result and we find that the algebraic and topological homotopy classifications agree. Second, we study the problem of classifying such varieties up to A 1 -h-cobordism using techniques of deformation theory. To this end, we establish a deformation rigidity result for P 1 -bundles over P 2 which links A 1 -h-cobordisms to deformations of the underlying vector bundles. Using results from the deformation theory of vector bundles we show that if X is a P 1 -bundle over P 2 and Y is the projectivization of a direct sum of line bundles on P 2 , then if X is A 1 -weakly equivalent to Y , X is also A 1 -h-cobordant to Y. Finally, we discuss some subtleties inherent in the definiti on of A 1 -h-cobordism. We show, for instance, that direct A 1 -h-cobordism fails to be an equivalence relation.