An almost flat manifold with a cyclic or quaternionic holonomy group bounds

Abstract

A long-standing conjecture of Farrell and Zdravkovska and independently S.~T.~Yau states that every almost flat manifold is the boundary of a compact manifold. This paper gives a simple proof of this conjecture when the holonomy group is cyclic or quaternionic. The proof is based on the interaction between flat bundles and involutions.

8 pages, to appear in the Journal of Differential Geometry. New version of Lemma 2.5: A manifold bounds if there is an involution on TM whose fixed bundle is full rank