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

Krishna Kumar Madhavan Vijayalakshmi

TL;DR

The paper addresses the problem of characterizing $ abla$A^1$-contractibility for Koras-Russell varieties over general base schemes and constructs exotic motivic spheres in higher dimensions. It develops a base-change framework in motivic homotopy theory, employs Milnor-Witt $K$-theory and the motivic Brouwer degree, and leverages the Danielewski trick and purity to extend known field results to schemes like $ ext{Spec}( ablaZ)$ and Noetherian bases. The main contributions include establishing relative $ abla$A^1$-contractibility for Koras-Russell threefolds and their prototypes, proving the existence of exotic motivic spheres in all dimensions $ abla ext{≥}4$ over infinite perfect fields, and outlining gluing/relative strategies for global motivic phenomena. These results provide a robust framework for identifying exotic motivic objects and suggest new directions regarding Zariski cancellation and global motivic topology across base schemes.

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.

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

TL;DR

The paper addresses the problem of characterizing A^1K ext{Spec}( ablaZ) abla-contractibility for Koras-Russell threefolds and their prototypes, proving the existence of exotic motivic spheres in all dimensions over infinite perfect fields, and outlining gluing/relative strategies for global motivic phenomena. These results provide a robust framework for identifying exotic motivic objects and suggest new directions regarding Zariski cancellation and global motivic topology across base schemes.

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 -contractibility from a field to a general base scheme. As a consequence, we also give a general strategy to extend the -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.
Paper Structure (20 sections, 33 theorems, 88 equations, 1 figure)

This paper contains 20 sections, 33 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Historical Outset
  3. Motivic homotopy theory
  4. Preliminaries
  5. Functoriality in Motivic Homotopy Theory
  6. The Base Change
  7. Smooth Base Change
  8. Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes
  9. Milnor-Witt $K$-Theory
  10. Relative A1-contractibility of Koras-Russell threefolds
  11. $\mathbb{A}^1$-weak equivalence of $j$:
  12. $\mathbb{A}^1$-weak equivalence of $i$
  13. Relative $\mathbb{A}^1$-contractibility of $\mathcal{K}$
  14. Relative A1-contractibility of Koras-Russell Prototypes over a base scheme
  15. Exotic Motivic Spheres
  16. ...and 5 more sections

Key Result

Proposition 1

(A1-cont-over-perfect-fields) For any perfect field $k$, the canonical morphism $f:\mathcal{K}\to \operatorname{Spec} k$ is an $\mathbb{A}^1$-weak equivalence in $\operatorname{Spc}_k$.

Figures (1)

  • Figure 1: Fibers of the Koras-Russell threefolds projected to $\mathbb{A}^1_x$ with the generic point indicated in red.

Theorems & Definitions (80)

  • Proposition
  • Theorem
  • Corollary
  • Definition
  • Theorem
  • Corollary
  • Theorem
  • Definition 1
  • Definition 2
  • Definition 3
  • ...and 70 more