Homotopy invariance of tame homotopy groups of regular schemes

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.