  In a seminal 1994 paper, Lusztig extended the theory of total positivity by
introducing the totally non-negative part (G/P)_{\geq 0} of an arbitrary
(generalized, partial) flag variety G/P. He referred to this space as a
"remarkable polyhedral subspace", and conjectured a decomposition into cells,
which was subsequently proven by the first author. Subsequently the second
author made the concrete conjecture that this cell decomposed space is the next
best thing to a polyhedron, by conjecturing it to be a regular CW complex that
is homeomorphic to a closed ball. In this article we use discrete Morse theory
to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that
the boundaries of the cells are homotopic to spheres, and the closures of cells
are contractible. The latter part generalizes a result of Lusztig's that
(G/P)_{\geq 0} -- the closure of the top-dimensional cell -- is contractible.
Concerning our result on the boundaries of cells, even the special case that
the boundary of the top-dimensional cell (G/P)_{> 0} is homotopic to a sphere,
is new for all G/P other than projective space.
