Families of \mathbb{A}^1-contractible affine threefolds

TL;DR

This paper constructs families of affine threefolds that are contractible in the unstable -homotopy category and are pairwise non-isomorphic, including the Koras-Russell threefolds, answering a question in algebraic geometry.

Contribution

It introduces new families of -contractible affine threefolds and demonstrates the contractibility of the Koras-Russell threefolds of the first kind.

Findings

Constructed pairwise non-isomorphic -contractible threefolds.

Extended contractibility results to Koras-Russell threefolds.

Answered a question posed by Asok and Doran.

Abstract

We provide families of affine threefolds which are contractible in the unstable $\mathbb A^1$-homotopy category of Morel-Voevodsky and pairwise non-isomorphic, thus answering a question of A. Asok and B. Doran. As a particular case, we show that the Koras-Russell threefolds of the first kind are contractible, extending results of M. Hoyois, A. Krishna and P. A. Ostvaer.