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arXiv · 2510.21594

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

Krishna Kumar Madhavan Vijayalakshmi
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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.

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Krishna Kumar Madhavan Vijayalakshmi. 2025-10-24. Relative $\mathbb{A}^1$-Contractibility of Koras-Russell Prototypes and Exotic Motivic Spheres. https://arxiv.org/abs/2510.21594

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