  This paper proves a conjecture of Fomin and Shapiro that their combinatorial
model for any Bruhat interval is a regular CW complex which is homeomorphic to
a ball. The model consists of a stratified space which may be regarded as the
link of an open cell intersected with a larger closed cell, all within the
totally nonnegative part of the unipotent radical of an algebraic group. A
parametrization due to Lusztig turns out to have all the requisite features to
provide the attaching maps. A key ingredient is a new, readily verifiable
criterion for which finite CW complexes are regular involving an interplay of
topology with combinatorics.
