Homotopy invariance of tame homotopy groups of regular schemes

TL;DR

This paper proves that tame homotopy groups of regular schemes are invariant under homotopy, addressing limitations of étale homotopy groups in positive characteristic by using the tame topology.

Contribution

It establishes the homotopy invariance of tame homotopy groups for regular schemes, extending the understanding beyond characteristic zero.

Findings

Tame homotopy groups are homotopy invariant for regular schemes.

Addresses limitations of étale homotopy groups in positive characteristic.

Provides foundational results for algebraic topology in algebraic geometry.

Abstract

The \'etale homotopy groups of schemes as defined by Artin and Mazur have the disadvantage of being homotopy invariant only in characteristic zero. This and other related problems led to the definition of the tame topology which is coarser than the \'etale topology by disallowing wild ramification along the boundary of compactifications. The object of this paper is to show that the associated tame homotopy groups are indeed ($\mathbb{A}^1$-)homotopy invariant, at least for regular schemes.