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

TL;DR

This paper classifies projectivizations of rank-two vector bundles over ${f P}^2$ up to ${f A}^1$-weak equivalence and ${f A}^1$-$h$-cobordism, linking algebraic and topological classifications and exploring deformation theory implications.

Contribution

It provides the first classification of such varieties up to ${f A}^1$-weak equivalence over various fields and establishes deformation rigidity results connecting ${f A}^1$-$h$-cobordism to vector bundle deformations.

Findings

Algebraic and topological classifications agree over ${f C}$.

${f A}^1$-$h$-cobordism is linked to vector bundle deformations.

Direct ${f A}^1$-$h$-cobordism is not an equivalence relation.

Abstract

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