On three dimensional stellar manifolds

Abstract

It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a stellar ball a*S. The study of S/~, two dimensional stellar sphere S with 2-simplexes identified in pairs leads us to the following conclusion: either a three dimensional manifold is homeomorphic to a sphere or to a stellar ball a*S with its boundary 2-simplexes identified in pairs so that S/~ is a finite number of internally flat complexes attached to a finite graph that contains at least one closed circuit. Each of those internally flat complexes is obtained from a polygon where each side may be identified with one or more different other sides. Moreover, Euler characteristic of S/~ is equal to one and the fundamental group of S/~ is not trivial.