  The existence of acyclic complete matchings on the face poset of a regular CW
complex implies that the underlying topological space of the CW complex is
contractible by discrete Morse theory.
  In this paper, we construct explicitly acyclic complete matchings on any
non-trivial Bruhat interval $[v,w]$ based on any reflection order on the
Coxeter group $W$. We then apply this combinatorial result to regular CW
complexes arising from the theory of total positivity.
  As an application, we show that the totally nonnegative Springer fibers are
contractible. This verifies a conjecture of Lusztig. As another application, we
show that the totally nonnegative fibers of the natural projection from full
flag varieties to partial flag varieties are contractible. This leads to a much
simplified proof of the regularity property on totally nonnegative partial flag
varieties compared to the proofs by Galashin-Karp-Lam and in our earlier work.
