The stratified homotopy type of the reductive Borel-Serre compactification

Abstract

We identify the exit path $\infty$-category of the reductive Borel-Serre compactification as the nerve of a $1$-category defined purely in terms of rational parabolic subgroups and their unipotent radicals. As an immediate consequence, we identify the fundamental group of the reductive Borel-Serre compactification, recovering a result of Ji-Murty-Saper-Scherk, and we obtain a combinatorial incarnation of constructible complexes of sheaves on the reductive Borel-Serre compactification as elements in a derived functor category.