Remarks on the homotopy type of intersections of two real Bruhat cells

TL;DR

This paper studies the homotopy types of intersections of real Bruhat cells, providing a stratification method that reveals all connected components are contractible, with detailed examples for the case n=4.

Contribution

It introduces a stratification approach to analyze intersections of Bruhat cells and determines their homotopy types, especially for the case n=4.

Findings

All connected components are contractible for the studied intersections.

The stratification effectively determines the homotopy type of the intersections.

Detailed computations for n=4 illustrate the method's application.

Abstract

In a companion manuscript, we introduce a stratification of intersections of a top dimensional real Bruhat cells with another arbitrary cell. This intersection is naturally identified with a subset of the lower triangular group: these subsets are labeled by elements of a finite group, the lift to the spin group of the intersection of a Coxeter group with the special orthogonal group. The stratification allows us to determine the homotopy type of the intersection. In this work we give more examples, particularly detailing the computation of the intersection of two open Bruhat cells in general position for $n=4$. It turns out that all connected components are contractible.