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

Abstract

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