Deformation spaces of 3-dimensional flat manifolds

Abstract

The deformation spaces of the six orientable 3-dimen-sional flatRiemannian manifolds are studies. It is proved that theTeichm"uller spaces are homeomorphic to the Euclidean spaces. Tostate more precisely, let Phi denote the holonomy group of themanifold. Then the Teichm"uller space is homeomorphic to (1)mathbb{R}^6 if Phi is trivial, (2) mathbb{R}^4 if Phi is cyclic with order two, (3) mathbb{R}^2 if Phi iscyclic of order 3, 4 or 6, and (4) mathbb{R}^3 if Phi congmathbb{Z}_2 times mathbb{Z}_2.