Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) 26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?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 Rietsch (1998) 33]. In Williams (2007) 40] 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 (1998) 28], that (G/P)_?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.