A projective version of Poincare's polyhedron theorem
by Benoy, Benjamin Jackson, Ph.D., UNIVERSITY OF CALIFORNIA, SANTA BARBARA, 2009, 184 pages; 3390733

Abstract:

We prove a projective version of Poincaré's polyhedron theorem for manifolds and orbifolds. Let [special characters omitted] be a finite or countable collection of thick, convex polyhedra in Sn where n ≥ 2. A face-pairing (R, A) on [special characters omitted] is a scheme for identifying the facets of [special characters omitted] pairwise by projective transformations. The quotient space obtained by carrying out those identifications is the gluing Q = [special characters omitted]/(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.
 
AdviserDaryl Cooper
SchoolUNIVERSITY OF CALIFORNIA, SANTA BARBARA
SourceDAI/B 71-02, p. , Mar 2010
Source TypeDissertation
SubjectsMathematics
Publication Number3390733
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