Thurston’s eight homogeneous geometries formed the building blocks of 3-manifolds in the Geometrization Conjecture. Filipkiewicz classified the 4-dimensional geometries in 1983, finding 18 and one countably infinite family. I have recently classified the 5-dimensional geometries. I will review what a geometry in the sense of Thurston is, outline the classification in 5 dimensions, and point out new features that appear as the dimension increases. This includes some wacky examples such as an infinite family of inequivalent homogeneous spaces with the same diffeomorphism type. The classification touches a number of topics including foliations, fiber bundles, representations of compact Lie groups, Lie algebra cohomology, algebraic number fields, and conformal transformation groups. I hope to give some indication of how all of these come into play. |