Homotopy Type of Intersections of Real Bruhat Cells in Dimension 6

We study the homotopy type of the intersection of two real Bruhat cells. This homotopy type coincides with that of an explicit submanifold of the group of real lower triangular matrices with diagonal entries equal to one. For (n+1) x (n+1) matrices with n up to five, these submanifolds are disjoint unions of contractible connected components. Our focus is on such intersections for 6 x 6 real matrices. For this, we study the connected components of Bruhat cells corresponding to permutations in the permutation group with dimension six with at most twelve inversions, using the structure of the associated dual CW complexes. New combinatorial and topological tools are developed to describe the structure of the spaces BL for certain permutations. As a consequence, we show that, among permutations with at most twelve inversions, all connected components are contractible except for the permutation [563412]. For this permutation, we identify a new non-contractible connected component with the homotopy type of the circle.