The complex of non-crossing diagonals of a polygon

Given a convex n-gon P in R 2 with vertices in general position, it is well known that the simplicial complex θ ( 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, θ ( P ) is a ball of dimension n + 3 h − 4 . We also provide a new proof that θ ( P ) is a sphere when P is convex with vertices in general position.