Eta invariants for flat manifolds

TL;DR

This paper computes eta invariants for the signature operator on most 7-dimensional flat manifolds with cyclic holonomy, revealing they are always integers, based on Donnelly's results.

Contribution

It extends the calculation of eta invariants to a broad class of 7-dimensional flat manifolds, showing they are integral, which was previously unestablished.

Findings

Eta invariants are integers for these manifolds

Calculations are based on Donnelly's theorem

Focus on manifolds with cyclic holonomy group

Abstract

Using H. Donnelly result from the article "Eta Invariants for G-Spaces" we calculate the eta invariants of the signature operator for almost all 7-dimensional flat manifolds with cyclic holonomy group. In all cases this eta invariants are an integer numbers. The article was motivated by D. D. Long and A. Reid article "On the geometric boundaries of hyperbolic 4-manifolds, Geom. Topology 4, 2000, 171-178