Splitting vector bundles and A^1-fundamental groups of higher dimensional varieties

TL;DR

This paper investigates the A^1-homotopy groups of higher-dimensional smooth varieties, constructing examples with isomorphic A^1-homotopy groups that are not A^1-weakly equivalent, revealing nuanced differences in their homotopy structures.

Contribution

It provides explicit computations of A^1-homotopy groups for certain varieties and demonstrates how vector bundle splitting affects the A^1-fundamental group and higher homotopy groups.

Findings

Constructed pairs of varieties with isomorphic A^1-homotopy groups but not A^1-weakly equivalent.

Analyzed the impact of vector bundle splitting on the A^1-fundamental group.

Identified the role of obstruction classes in the structure of A^1-homotopy groups.

Abstract

We study aspects of the A^1-homotopy classification problem in dimensions >= 3 and, to this end, we investigate the problem of computing A^1-homotopy groups of some A^1-connected smooth varieties of dimension >=. Using these computations, we construct pairs of A^1-connected smooth proper varieties all of whose A^1-homotopy groups are abstractly isomorphic, yet which are not A^1-weakly equivalent. The examples come from pairs of Zariski locally trivial projective space bundles over projective spaces and are of the smallest possible dimension. Projectivizations of vector bundles give rise to A^1-fiber sequences, and when the base of the fibration is an A^1-connected smooth variety, the associated long exact sequence of A^1-homotopy groups can be analyzed in detail. In the case of the projectivization of a rank 2 vector bundle, the structure of the A^1-fundamental group depends on the splitting behavior of the vector bundle via a certain obstruction class. For projective bundles of vector bundles of rank >=, the A^1-fundamental group is insensitive to the splitting behavior of the vector bundle, but the structure of higher A^1-homotopy groups is influenced by an appropriately defined higher obstruction class.