The Complex of Non-Crossing Diagonals of a Polygon

Given a convex n-gon P in the Euclidean plane, it is well known that the simplicial complex \theta(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n-3. We prove that for any non-convex polygonal region P with n vertices and h+1 boundary components, \theta(P) is a ball of dimension n+3h-4. We also provide a new proof that \theta(P) is a sphere when P is convex.