Abstract:
In this paper we explore the eight geometries of Thurston's geometrization conjecture. We begin by discussing group actions, covering space topology, fiber bundles, and Seifert fiber spaces. We make precise the notion of a geometric structure. We then discuss the two dimensional geometries including a brief proof of the uniformization theorem. We list each of the eight geometries of Thurston's geometrization conjecture and then explore in depth the geometries of the two sphere cross the real line and of the three sphere.
