Relative $\mathbb{A}^1$-Contractibility of Koras-Russell Prototypes and Exotic Motivic Spheres

TL;DR

This paper extends the $A^1$-contractibility of Koras-Russell threefolds from fields to general base schemes, providing a strategy for higher dimensions and establishing the existence of exotic motivic spheres in all dimensions at least 4 over infinite perfect fields.

Contribution

It generalizes $A^1$-contractibility results for Koras-Russell varieties to broader base schemes and introduces a method to construct exotic motivic spheres in higher dimensions.

Findings

$A^1$-contractibility extends to general base schemes.

Strategy for higher-dimensional Koras-Russell prototypes.

Existence of exotic motivic spheres in all dimensions ≥ 4.

Abstract

The Koras-Russell threefolds are a certain family of smooth, affine contractible threefolds exhibiting "exotic" behavior in the algebro-geometric context. Our goal in this note is to extend its $\mathbb{A}^1$-contractibility from a field to a general base scheme. As a consequence, we also give a general strategy to extend the $\mathbb{A}^1$-contractibility of Koras-Russell prototypes in higher dimensions over a general base scheme. As a major consequence, we establish the existence of "exotic" motivic spheres in all dimensions at least 4 over infinite perfect fields.