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The complex of non-crossing diagonals of a polygon. (English) Zbl 1196.52008
Summary: Given a convex $$n$$-gon $$P$$ in $$\mathbb R^2$$ with vertices in general position, 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 with vertices in general position.
##### MSC:
 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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