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

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.