The University of Michigan Combinatorics Seminar
We give a new criterion for determining whether a finite CW complex is regular. This involves both combinatorial conditions on the closure poset and also topological conditions on the codimension one cell incidences. As an application, we prove a conjecture of Fomin and Shapiro on regularity of certain stratified totally positive spaces. This completes the solution of a problem, posed by Bjorner in 1984, of constructing a naturally arising regular CW complex whose closure poset is the Bruhat order of a finite Weyl group. The proof involves showing that parametrizations of totally positive spaces due to Lusztig yield the characteristic maps.