Title: Relative $\mathbb{A}^1$-Contractibility of Koras-Russell Prototypes and Exotic Motivic Spheres Show Chinese title

Title: 相对$\mathbb{A}^1$-可收缩性以及Koras-Russell原型和外星动机球面

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. Show Chinese abstract

Abstract: Koras-Russell三折是光滑、仿射可缩三折的一个特定族，在代数几何背景下表现出“奇特”的行为。 本文的目标是将它的$\mathbb{A}^1$-可缩性从一个域扩展到一般的基概形。 作为结果，我们还提供了一种一般策略，以在一般基概形上扩展高维Koras-Russell原型的$\mathbb{A}^1$-可缩性。 作为主要结果，我们在无限完美域上所有至少4维的情况下建立了“奇特”动机球面的存在性。