The Classification of Five-Dimensional Geometries

We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The 5-dimensional geometries with irreducible isotropy are the irreducible Riemannian symmetric spaces, while those with trivial isotropy are simply-connected solvable Lie groups of the form R^3 \rtimes R^2 or N \rtimes R where N is nilpotent. The geometries with nontrivial reducible isotropy are mostly products, but a number of interesting examples arise. These include a countably infinite family L(a;1) ×_{S^1} L(b;1) of inequivalent geometries diffeomorphic to S^3 × S^2, an uncountable family in which only a countable subfamily admits compact quotients, and the non-maximal geometry SO(4)/SO(2) realized by two distinct maximal geometries.