On the Classification of Closed Flat Four-Manifolds

Abstract

In this thesis, we discuss the classification of closed flat four-manifolds. First, we give a brief treatment of the history of this problem, with a discussion of the advantages, disadvantages, and inconsistencies among the previous classifications. Then, we discuss the process by which we reconcile the different classifications. The procedure is to exhibit presentations as isometry groups of the 74 groups corresponding to the manifolds. We determine fiber-bundle or twisted $I$-bundle decompositions of the manifolds, along with their homology and holonomy groups. Also, we determine the orientable double-covers for the nonorientable manifolds. In addition, we match our list of manifolds to those of the previous primary sources.