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An exposition of Poincaré’s polyhedron theorem. (English) Zbl 0808.52011

Poincaré’s Theorem is an important, widely used and well-known result. There is a number of expositions in the literature; however, there is no source which contains a completely satisfying proof that applies to all dimensions and all constant curvature geometries. Sometimes hypotheses are included that unnecessarily restrict the range of validity of the theorem. This paper contains a literature review of other expositions of Poincaré’s Theorem at the end.

The main part of the paper emphasizes on algorithmic aspects of Poincaré’s Theorem, extending the work of R. Riley [Math. Comput. 40, 607-632 (1983; Zbl 0528.51010)]. Procedures are given to decide whether a finite set of finite-sided convex polyhedra and face-pairings do or do not give rise to an orbifold, or, a tesselation of X n . One way in which the treatment in this paper differs from previous ones is that it is not assumed that an embedded fundamental domain is already given. Instead the fundamental domain is expressed as the union of convex cells, each of which can be embedded separately, without knowing that their union can be embedded.

MSC:
52B55Computational aspects related to geometric convexity