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

TL;DR

This paper proves that almost flat manifolds with cyclic or quaternionic holonomy groups are boundaries of compact manifolds, confirming a long-standing conjecture in specific cases using flat bundle and involution techniques.

Contribution

Provides a simple proof of Farrell and Zdravkovska's conjecture for manifolds with cyclic or quaternionic holonomy groups, expanding understanding of manifold boundaries.

Findings

Almost flat manifolds with cyclic holonomy are boundaries.

Almost flat manifolds with quaternionic holonomy are boundaries.

The proof utilizes flat bundles and involutions.

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.