| We prove a projective version of Poincare's polyhedron theorem for manifolds and orbifolds. Let P be a finite or countable collection of thick, convex polyhedra in Sn where n ≥ 2. A face-pairing (R, A) on P is a scheme for identifying the facets of P pairwise by projective transformations. The quotient space obtained by carrying out those identifications is the gluing Q = P /(R, A). We describe a universal cover Z of Q. If Z is locally embeddable into Sn around a codimension-2 face, then the complement of the codimension-3 skeleton of Z is a projective orbifold. The local embeddability condition takes the place of the usual angle sum condition. In dimension 3, if the universal cover Z is locally finite then the gluing Q admits a projective structure as an orbifold. These conditions are both necessary and sufficient. |
