Flat Riemannian manifolds are boundaries

Abstract

In this paper we prove that each compact flat Riemannian manifold is the boundary of a compact manifold. Our method of proof is to construct a smooth action of (ℤ2) k on the flat manifold. We are independently preceded in this approach by Marc W. Gordon who proved the flat Riemannian manifolds, whose holonomy groups are of a certain class of groups, bound. By analyzing the fixed point data of this group action we get the complete result. As corollaries to the main theorem it follows that those compact flat Riemannian manifolds which are oriented bound oriented manifolds; and, if we have an involution on a “homotopy flat” manifold, then the manifold together with the involution bounds. We also give an example of a nonbounding manifold which is finitely covered byS 3 ×S 3 ×S 3.