Stratified spaces formed by totally positive varieties

By a theorem of A.Bj\"orner, for every interval $[u,v]$ in the Bruhat order of a Coxeter group $W$, there exists a stratified space whose strata are labeled by the elements of $[u,v]$, adjacency is described by the Bruhat order, and each closed stratum (resp., the boundary of each stratum) has the homology of a ball (resp., of a sphere). Answering a question posed by Bj\"orner, we suggest a natural geometric realization of these stratified spaces for a Weyl group $W$ of a semisimple Lie group $G$, and prove its validity in the case of the symmetric group. Our stratified spaces arise as links in the Bruhat decomposition of the totally nonnegative part of the unipotent radical of $G$.