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

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.